This article provides you with a glossary of math terms and definitions in order to simplify your search for a particular formula among the plethora of arithmetic vocabulary.

In the ocean of mathematics, there are uncountable drops of different terms, words, definitions, and glossary. When you start searching for a specific topic and its meaning, you seem to get lost in the amazing world of numbers. Mathematics is the queen of all sciences, and this is reflected in the usage of numbers in our every day life. There is hardly any field, be it biology, physics, chemistry, astronomy, or economics where numbers don’t come into play. Our lives would almost come to standstill without this subject. To help you look for the right expressions, this article has a glossary of math terms and definitions, which are alphabetically listed below.

**Basics**

Mathematical definitions are deduced from the extensive research and theories. Unless an explanation is not proved correct for an expression, it is always an area of research and debate. The terminology enlisted here have been picked up from a plethora of different branches, like Algebra, Trigonometry, Mensuration, Geometry, Calculus, etc.

**Branches**

This field has applications in almost all aspects of life and work. The operations of addition, subtraction, multiplication, and division form the platform for the higher order. Kinematics, Dynamics, Linear algebra, Ring theory, Calculus, and Integration are the most popular research areas. The magical world of Permutations and Combinations, not to mention Probability, has its own wonderful applications in the real world. Read the article below in order to enter this wonderful world.

**AA similarity**

According to the AA similarity, if two angles of a triangle are congruent to two angles of another triangle, then the triangles are said to be similar to each other.

**AAS Congruence**

AAS congruence is called the angle-angle-side congruence. If there are two pairs of corresponding angles and a pair of corresponding opposite sides that are equal in measure, then the triangle is said to be congruent.

**Abscissa**

The X-coordinate of a point on the coordinate system is called abscissa. For example, in the ordered pair P(2, 3, 5), 2 will be called the abscissa of the point P. In mathematical language, it will be called the length of the point (P) relative to the X-axis.

**Absolute Convergence**

A series that converges when all its expressions are replaced by their absolute values. To check if a series converges absolutely, it is only required to replace any subtraction in the series with addition. A series _{n=1}Σ^{n=∞}a_{n} is absolutely convergent if the series _{n=1}Σ^{n= ∞} |a_{n}| converges.

**Absolute Maximum**

The highest point of the function or relation over the entire domain is called absolute maximum. The first and second derivative tests are commonly used to find the absolute maximum of a function.

**Absolute Minimum**

The lowest point of the function or relation over the entire domain is called the absolute minimum. The first and second derivatives are the commonly used methods to find the absolute minimum. Global minimum is also called the absolute minimum.

**Absolute Value**

A general concept of absolute value is that it makes a negative number positive. Absolute value is also called a mod value. The absolute value of a number (say X) is denoted as |X|. Remember, the absolute value uses bars, so don’t use parenthesis or any other symbol, else the meaning changes. To put it simply, |-7| = 7 and |7| = 7. Positive numbers and zero are left unchanged in the absolute value. A better and more accurate way of understanding is that the absolute value of a number denotes the distance between the number and origin. So, |x-a| = b, where b>0, says the quantity x-a is b units from 0, x-a is b units right of 0(origin) x-a is b units left of 0(origin).

**Absolute Value of Complex number**

The absolute value of a complex number |a + bi| = √a^{2} + b^{2}. The absolute value of a complex number is the distance between the origin and the complex plane. For a complex number in the form of r(acosθ + bsinθ), the modulus is r, i.e., the value of radius of the circle carved by the trigonometrical equation.

**Acceleration**

The rate of change of velocity with time is called acceleration. Mathematically, the second derivative of the distance traveled by an object is called acceleration.

**Accuracy**

The measure of the closeness of a value to the actual value of a result is called accuracy.

**Acute Angle**

An angle whose measure is less than 90^{0} is called an acute angle.

**Acute-Angled Triangle**

A triangle in which all the interior angles are acute is known as an acute angled triangle.

**Addition Rule Of Probability**

Addition rule of probability is meant to find out the probability of occurrence of either or both the events.

If P(A) and P(B) are mutually exclusive events, then the probability P(A or B) = P(A) + P(B) else P(A or B) = P(A) + P(B) – P(A and B).

**Additive Inverse of a Matrix**

If the sign of every matrix element is changed, then the matrix is said to be an inverse of the original matrix. If A is the matrix, then -A will be the inverse of the matrix. If add a matrix and its inverse, then the sum would be zero, since the each element in the original matrix is negative of the other.

**Additive Property of Equality**

Simply stated, additive property states that same number can be added on both side of the equation. For example, x – 3 = 5 is same as x – 3 + 3 = 5 + 3.

**Adjacent Angles**

If the two angles share a common vertex and common plane and even have a same side, but if they don’t overlap or one of the angles is not contained in the other, then the angles are called adjacent angles.

**Adjoint Matrix**

When we take the transpose of the co-factor of the original matrix, then it is known as adjoint matrix.

**Affine Transformations**

Affine transformations refer to the combination of the process that can be performed on any co-ordinate system, like translation, rotation, horizontal and vertical stretches, and shrinks. It is to be kept in mind that concurrency and co-linearity are invariant under any type of transformation.

**Aleph Null**

The 1st letter of the Hebrew alphabet, Aleph (א)is used to denote the cardinal number of the infinite countable set. Basically, א_{0} with a subscript is generally used to denote the cardinality of the infinitely countable set.

**Algebra**

It is a branch of pure mathematics that uses alphabets and letters as variables. The variables are the unknown quantities whose values can be determined with the help of other equations. For example, 3X – 7 = 78, is an algebraic equation in one unknown variable (here it is X). Now, by applying algebra techniques we can solve the equation. More on algebra tips.

**Algebraic Numbers**

All rational numbers are algebraic numbers. Numbers that are roots of the polynomials with integer coefficients and are under the surd are also included as algebraic numbers. Any number that is not a root of polynomial with integer coefficients is not an algebraic number. These numbers are called transcendental numbers. e and Π are called the transcendental numbers.

**Algorithm**

Algorithm is a simple step by step to arrive at the solution of any problem.

**Alpha**

Alpha is the 1st letter of Greek alphabet. It is denoted by A (in an upper case) and by α (in lower case). It is frequently used in science as a variable to denote angles, etc.

**Alternate Angles**

If two or more parallel lines are cut by a transversal, then the angles formed in the alternate direction to each other are called alternate angles.

**Alternate Exterior Angles**

When two or more parallel lines are cut by a transversal, the alternate angles that are exterior to one another is called alternate exterior angle.

**Alternate Interior Angles**

When two or more lines are cut by a transversal then the alternate angles that lie interior to each other are called alternate interior angles.

**Alternate Series**

Alternating series is a series that consists of alternate positive and negative sides.

The alternating series is of the form:

1 – ½ + 1/3 – ¼ + 1/5…….up to infinity.

**Alternating Series Remainder**

The alternating series is as follows:_{n = 1} ∑^{n = ∞} = (-1)^{n+1}a_{n} = a_{1} – a_{2} + a_{3} + …

If the series converges to S, by applying the alternating series test, then, the remainder,

R_{n} = S – _{k=1}∑^{n}(-1)^{k+1}a_{k}, for all n ≥ N, is called the alternating series remainder.

Also, |R_{n}| ≤ a_{n + 1}.

**Altitude**

Altitude is the shortest distance between the base to the apex of a figure like cones, triangle etc.

**Altitude of a Cone**

The distance between the apex of the cone and its base is called the height or the altitude of the cone.

**Altitude of a Cylinder **

The distance between the circular bases of the cylinder or the length of the line segment between two of its bases is known as altitude of a cylinder.

**Altitude of a Parallelogram**

The distance between the opposite sides of a parallelogram is called altitude of a parallelogram.

**Altitude of a Prism**

The distance between the two bases of a prism is called the altitude of a prism.

**Altitude of a Pyramid**

The distance between the apex of the pyramid to the base is called altitude of the pyramid.

**Altitude of a Trapezoid**

The distance between the two bases of the trapezoid is called altitude of a trapezoid.

**Altitude of a Triangle**

The shortest distance between the vertex of the triangle and the opposite side is called altitude of the triangle.

**Amplitude**

It is the measure of half the distance between the maximum and minimum range. For example, if you consider a sine wave, then ½ of the distance between the positive and negative curves in called an amplitude. It is to be remembered that only periodic functions with bounded range have amplitude.

**Analytic Geometry**

Analytical geometry is a branch that deals with the study of geometric figures with the help of co-ordinate axes. The points are plotted and with the help of the points we can easily find out the required information.

**Analytic Methods **

If you are asked to analytically solve a problem, then it means that you are not supposed to use a calculator. Analytical methods are used to solve the problems by the help of algebraic and numeric methods.

**Angle**

Angle is defined as the figure formed by touching the ends of two rays. In other words, it means two rays sharing a common point.

**Angle Bisector**

The line that bisects an angle into two equal halves is called an angle bisector.

**Angle of Depression**

The angle below the horizontal line that a observer must see in order to site the object is called angle of depression. To understand it better, consider an observer at the top of the cliff, when he has to sight an object down at some distance from the base of the cliff, the angle subtended he will have to subtend in order to site the object is called angle of depression.

**Angle of Elevation**

Angle of elevation is geometrically congruent to the angle of depression. If a viewer is observing an object at some elevation, then he needs to raise his line of sight above a horizontal level, this is called the angle of elevation.

**Angle of Inclination of a Line**

The angle subtended by a line with the x-axis is called angle of inclination of the line. The angle of inclination is always measured in counter clockwise direction, that means positive direction of the x-axis. The angle of inclination is always between the range 0^{0} to 180^{0}.

**Annulus**

The area between two concentric circles of a ring (say) is called annulus.

**Anticlockwise**

The direction opposite to that of the movements of the watch. In this case, it is an assumption that the anticlockwise direction is always measured positive.

**Antiderivative of a Function**

If F(x) = 2x^{2} + 3, then, its derivative F'(x) = 4x. Here 4x is called the antiderivative of F(x).

**Antipodal Points**

In three dimensions, the points diametrically opposite on a sphere is called antipodal points.

**Apothem**

Apothem is the same as the inradius of an inscribed circle in a regular polygon. In other words, it would mean the distance from any of midpoint of the sides of the polygon to the center of the polygon.

**Approximation by Differentials**

By the rule of approximation of differentials, the value of a function is approximated and the principles of derivation are used in this method. The formula used in the approximation by differentials is, f(x + ∆x) = f(x) + ∆y = f(x) + f'(x)∆x, where f'(x) is the differential of the function.

**Arc Length of a Curve**

The length of the line of a curve is called the arc length. There are three formulas to determine the arc length of a curve. There is a rectangular form, polar form, and parametric form that can be used.

- Rectangular form- ds = [1 + (dy/dx)
^{2}dx]^{1/2} - Parametric form- ds = (dx/dt)
^{2}+ (dy/dt)^{2}dt]^{1/2} - In polar form- ds = [r
^{2}+ (dr/dƟ)^{2}]^{1/2}

**Area of a Circle**

The area of a circle is given by the formula Πr^{2}.

**Arccos**

The inverse function of a cosine function is called the arccos function. For example, cos^{-1}(1/2) (read as cos inverse half) or”the angle whose cosine is equal to ½. As we all know it nothing but 60^{0}.

**Arccosec**

The inverse of a cosec function is called arccosec function. For example, cosec^{-1}(2) means the angle whose cosecant is equal to 2. The answer is 30^{0}. It is to be noted that there can be many more angles with the cosecant equal to 30^{0}. What we want is the most basic angle that gives the cosecant equal to 30^{0}. For other angles, we need to consider the range of the function.

**Arccot**

Arccot is the inverse of the cotangent function. For example, cot^{-1}(1) means the angle whose cotangent is equal to 1. Cot^{-1}1 = 45^{0}.

**Arcsec**

The inverse of a secant function is called the arcsec function. For example, sec^{-1}2 means the angle whose secant is equal to 2. Sec^{-1}2 = 60^{0}.

**Arcsin**

The inverse of a sine function is called arcsin function. For example, sin^{-1}(1/2) = 30^{0}.

**Arctan**

The inverse of a tangent function is called arctan function. For example, tan^{-1}(1) = 45^{0}

**Area below a Curve**

The area occupied by any curve is called area that the curve forms together with the x and y axes. The area of a function y = f(x) is given by the definite integral _{a}ʃ^{b}, where a and b are the limits of the function.

Area = _{a}ʃ^{b} f(x) dx

**Area between Curves**

The area between two curves y = f(x) and y = g(x) is given by the formula,

Area = _{a}ʃ^{b} |f(x) – g(x)|dx, where f(x) and g(x) is the area bounded above and below the x and y axis whereas x= a and x=b, on the left and right.

**Area of a Convex Polygon**

If (x_{1}, y_{1}), (x_{2}, y_{2}), … , (x_{n}, y_{n}) are the co-ordinates of a convex polygon then the area of the polygon is found out by the determinant method. The expanded form of the determinant is as follows:

1/2[(x_{1}y_{2}) + x_{2}y_{3}+ x_{3}y_{1}+ … x_{n}y_{1})] – [y_{1}x_{2} + y_{2}x_{3} + … y_{n}x_{1}].

**Area of an Ellipse**

The area of an ellipse is given by the formula ∏ab, where a and b are the lengths of the major and minor axis of the ellipse. If the ellipse has its center at (h, k) then,

Area = [(x-h)^{2}/a^{2} + (y-k)^{2}/b^{2}]

**Area of an Equilateral Triangle**

The area of an equilateral triangle is given by:

a^{2}√3/4, where a = side of the equilateral triangle.

**Area of a Kite**

The area of a kite is given by:

½ (product of the diagonals) = ½ x d_{1}d_{2}.

**Area of a Parabolic Segment**

The area of a parabolic segment is given by 2/3 of the product of width and height.

**Area of a Parallelogram**

Area of parallelogram = height x base of the parallelogram.

**Area of a Rectangle**

Area of rectangle = length x breadth

**Area of a Regular Polygon**

Area of regular polygon = ½ x apothem x perimeter.

**Area of a Rhombus**

Diagonals of a rhombus are perpendicular to each other. Area = ½ x product of diagonals or Area= h x s, where h and s are the height and side of the rhombus.

**Area of a Segment of a Circle**

We all know the area of a circle, but what if the area of a segment is to be found out, well the formula for area of a segment of a circle is:

Area = 1/2r^{2}(θ – sinθ) (radians)

**Area of a Trapezoid**

Area of a trapezium = ½ x (sum of the non- parallel sides) x h = ½ x (b_{1} + b_{2}) x h

**Area of a Triangle**

There are various formulas to calculate the area of a triangle that are as follows.

- Area = A = ½ x base x height
- A = ½ x ab SinC = ½ x bc SinA = I/2 x ca SinB, where A, B and C are the angles of the triangle respectively.
- Given s= a+b+c/2 (semi perimeter), by Heron’s Formula, A= [s(s-a)(s-b)(s-c)]
^{1/2}. - If ‘r’ and ‘R’ are the inradius and circumradius of the incircle and outercirlce of a triangle, then the Area (A) = rs and A= abc/4R, a, b and c are the sides of the triangle.

**Area Using Polar Coordinates**

When the polar co-ordinates are involved in computation of the area then the area is given by:

The area between the graph r = r(θ) and the origin and also between the lines θ = α and θ = β is given by the formula:

Area = ½ _{α}ʃ^{β}r^{2}dθ

**Argand Plane**

The complex plane is called the argand plane. Basically, argand plane is used to denote the complex numbers graphically. The x-axis is called the real axis and the y-axis is called the imaginary axis.

**Argument of a Complex Number**

In order to describe the angle or inclination of a complex number on the argand plane, we use the term argument. Argument of a complex number is measured in radians. The polar form of a complex number is given by r(cosθ + isinθ) and the argument in this is given by θ.

**Argument of a Function**

The expression in which the function operates is called argument of the function. The argument of the function y= √x is x.

**Argument of a Vector**

The measure of an angle describing a vector or a line in the complex number analysis is called the argument of the vector.

**Arithmetic Mean**

The most simple average technique that we use in day-to-day life.

For example, if there are 4 quantities then there arithmetic mean is given by,

Arithmetic mean = (a + b + c + c + d)/4

**Arithmetic Progression**

A series that has same common difference among its terms. For example, 1, 3, 5, 7, 9 … up to infinity. The nth expression of an arithmetic progression is given by, T_{n} = a + (n-1)d, where a = 1st term, n = number of terms, and d= common difference. It is also called arithmetic sequence. The sum of an arithmetic progression is given by: S = n/2[2a + (n-1)d] or S = n(a_{1} + a_{n})/2, here n= number of terms.

**Arm of an Angle**

One of the rays/line forming an angle with the other is called the arm of an angle.

**Arm of a Right Triangle**

Any of the sides of the right angled triangle is called the arm of a right angled triangle.

**Associative**

The operation a + (b+c) = (a + b) + c is called associative operation. Addition and multiplication are associative while division and subtraction are not. For example, (4+5)+ 7 = 4 + (5+7)

**Asymptote**

An asymptote is a curve or line that approaches the curve very closely. There are horizontal and oblique asymptotes but not vertical asymptotes.

**Augmented Matrix**

The matrix representation of a set of linear equations is called the augmented matrix.

For example, 3x – 2y = 1 and 4x + 6y = 4, then in a matrix form 3, -2 and 1 (from 1st equation) and 4, 6 and 4 (from 2nd equation) form the elements of 3×3 matrix respectively.

**Average**

Average is same as the arithmetic mean.

**Average Rate of Change**

The change in the slope of a line is called the average rate of change of the line. Also, the change in value of a quantity divided by time is average rate of change.

**Average Value of a Function**

For a function y =f(x), in the domain [a,b] the average value is given by the formula (1/b-a)_{a}ʃ^{b}f(x)dx

**Axes**

The x, y, and z axes are known as the axes of a co-ordinate system.

**Axiom**

A statement that has been assumed to be true without any proof.

**Axis of a Cylinder**

The line that passes exactly through the center of the cylinder and also passes through the bases of the cylinder. Simply stated, the line that divides the cylinder into two equal halves vertically.

**Axis of Reflection**

A line across which the reflection takes place.

**Axis of Rotation**

An axis along which the rotation of the axis takes place.

**Axis of Symmetry**

A line along which the geometrical figure or the shape is symmetrical.

**Axis of Symmetry of a Parabola**

The axis of symmetry of a parabola is the line that passes through the focus and vertex of parabola.

**Back Substitution**

Back substitution is a technique that is used to solve a system of linear equations that has already been changed to row-echelon form and reduced row-echelon form. After changing the equation, the first equation is solved, then the second last, then the next before that and so forth.

**Base (Geometry)**

The bottom part of a geometrical figure, like a solid object or a triangle is called the base of the object.

**Base of an Exponential Expression**

Consider the expression a^{x}. Then ‘a’ can be called the base of the expression a^{x}.

**Base of an Isosceles Triangle**

The base of an isosceles triangle is the non-congruent side of the triangle. In other words, it is the side other than the legs of the triangle.

**Base of a Trapezoid**

The trapezoid has four sides with two sides parallel. Either of the two parallel sides can be considered as the base of the trapezoid.

**Base of a Triangle**

Base of a triangle is the side at which an altitude can be drawn. It is the side, which is perpendicular to the altitude.

**Bearing**

Bearing is a method used to represent the direction of a line. If there are two points A and B, then we can say that A has a bearing of θ degrees from point B, if the line connecting A and B makes an angle θ with a vertical line drawn through B. The angle is measured in clockwise direction.

**Bernoulli Trials**

In the field of statistics, Bernoulli trials are the experiments where the outcome can be either true or false. In Bernoulli trials, all events must be independent. The binomial probability formula is given by p (k successes in n trials) = ^{n}C_{r}p^{k}q^{n – k}, where,

n= number of trials,

k = number of successes,

n – k = number of failures,

p = probability of successes in trials,

q = 1 – p, probability of failure in one trial.

**Beta (Β β)**

A Greek letter used frequently as a symbol to denote variables.

**Biconditional**

It is the method of expressing a statement containing more than one condition, that means a condition and its converse. These statements are called biconditionals. They are represented by the symbol ⇔. For example, the following statements can be called biconditionals: “A given triangle is equilateral” is same as “All the angles of a triangle measure 60º.”

**Binomial**

A binomial can be simply defined as a polynomial, which has two terms, but they are not like terms. For example, 3x – 5z^{3}, 4x – 6y^{2}.

**Binomial Coefficients**

The coefficients of the various expressions in the binomial expansion of the binomial theorem are called binomial coefficients. Mathematically, a binomial coefficient equals the number of r items that can be selected from a set of n items. They are simply called binomial coefficients, because they are coefficients of the binomial expanded expressions. Generally, they are represented by ^{n}C_{r}.

**Binomial Coefficients in Pascal’s Triangle**

Pascal’s triangle is an arithmetic triangle that is used to calculate the binomial coefficients of the various numbers. The binomial coefficients (^{n}C_{r}) in the pascal’s triangle are called the binomial coefficients in pascal’s triangle. Pascal’s triangle finds major use in algebra and probability/binomial theorem.

**Binomial Probability Formula**

The probability of getting m successes in n trials is called binomial probability formula. The formula is given by:

Formula: P(m successes in n trials) = _{m}C^{n}p^{k}q^{n-k}, where,

n = number of trials

m = number of successes

n – m = number of failures

p = probability of success in one trial

q = probability of failure in one trial.

**Binomial Theorem**

A theorem used to expand the powers of polynomial and equations. It is given by:

(a + b)^{n} = ^{n}C_{0}a^{n} + ^{n}C_{1}a^{n-1}b + … +^{n}C_{n-1}ab^{n-1} + ^{n}C_{n}.

**Boolean Algebra**

Boolean algebra deals with the logical calculus. Boolean algebra takes only two values in the logical analysis, either 1 or zero. Read more on Boolean Origination.

**Boundary Value Problem**

Any differential equation that has constrained on the values of the function (not that on the derivatives) is called the boundary value problem.

**Bounded Function**

A function that has a bounded range. For example, in the set [2, 9], 9 the upper bounded number and 2 is the lower bounded number.

**Bounded Sequence**

A sequence that is bounded with upper and lower bounds. Like the harmonic series, 1, ½, 1/3, ¼, … up to infinity is a bounded function since the function lies between 0 and 1.

**Bounded Set of Geometric Points**

The bounded set of geometric points is called the figure or set of points that can be enclosed in a fixed space or co-ordinates.

**Bounded Set of Numbers**

A set of numbers with lower and upper bound. For example, [3, 7] is called the bounded set of numbers.

**Bounds of Integration**

For a definite integral, _{a}ʃ^{b} f(x)dx, a and b are called the bounds or limits of integration. The bounds of integration also indicate limits of integration.

**Box**

A rectangular parallelepiped is often referred to as a box. The volume of such a rectangular box is given by the product of length, breadth, and height.

**Box and Whisker Plot**

The box and whisker plot is a beginning lesson for starters, in order to let them understand the basics of handling data value. Box and whisker plot shows certain data, not the complete statistics of the recorded data. Five number summary is another name for visual representation of the box and whisker plot.

**Boxplot**

A data that displays the five number summary in a diagrammatic form represented as:

Smallest | 1st Quartile | Median | 3rd Quartile | Largest |

**Braces**

The symbolic representation {or} that is used to indicate sets etc.

**Brackets**

The symbol [ ] which signifies grouping. They work in a similar way parentheses do.

**Calculus**

The branch that deals with integration, differentiation, and various other forms of derivatives.

**Cardinal Numbers**

Cardinal numbers are used to indicate the number of elements in an infinite or finite sets.

**Cardinality**

It is same as cardinal numbers. It is to be noted that cardinality of every infinite set is same.

**Cartesian Coordinates**

The Cartesian coordinates are the axes that are used to represent the coordinates of a point. (x,y) and (x,y,z) are the Cartesian coordinates.

**Cartesian Plane**

The planes formed by horizontal and vertical axes like the x and y axis is called the Cartesian plane.

**Catenary**

The curve formed by a hanging a wire or a ring is called catenary. Generally, a catenary is confused with a parabola. However, though the looks are similar, it is not same as the parabola. The graph of a hyperbolic cosine function is called the catenary.

**Cavalieri’s Principle**

A method to find the volume of solids by using the formula V = bh, where b = area of cross section of the base (cylinder/prism) and h = height of the solid.

**Central Angle**

An angle in a circle with vertex at the circle’s center.

**Centroid**

The intersection point of the three medians of a triangle.

**Centroid Formula**

The centroid of the points (x_{1}, y_{1}, x_{2}, y_{2}, … x_{n}, y_{n}) is given by:

(x_{1} + x_{2} + x_{3}+ … x_{n})/n , (y_{1} + y_{2} + y_{3}+ … y_{n})/n

**Ceva’s Theorem**

Ceva’s theorem is a way that relates the ratio in which three concurrent cevians divide a triangle. If AB, BC, and CA are the three sides of a triangle, and AE, BF and CD are the three cevians of the triangle, then according to Ceva’s theorem,

(AD/DB)(BE/EC)(CF/FA) = 1.

**Cevian**

A line that extends from the vertex of a triangle to the opposite side like altitudes and medians.

**Chain Rule**

A method used in differential calculus to find the derivative of a composite function.

(d/dx)f(g(x)) = f^{‘}((g(x))g^{‘}(x) or (dy/dx) = (dy/du)(du/dx)

**Change of Base Formula**

A very useful formula in logarithm that is used to express a certain logarithmic function in a different base. That’s why it is called base change formula.

Base Change Formula: log_{a}x = (log_{b}x/log_{b}a)

**Check a Solution**

Checking a solution means putting the value of corresponding variables in the equation and verify if the equations satisfy the given equation or systems of equation.

**Chord**

A chord is a line segment that joins the two points on a curve. In a circle, the largest chord is the diameter that joins the two ends of the circle.

**Circle**

The locus of all points that is always at a fixed distance from a fixed point.

**Circular Cone**

A cone with a circular base.

The volume of circular cone is given by V = 1/3πr^{2}

**Circular Cylinder**

A cylinder with circle as the base.

**Circumcenter**

The center of a circumcircle is called circumcenter.

**Circumcircle**

A circle that passes through all the vertices of a regular polygon and triangles is called circumcircle.

**Circumference**

The perimeter of a circular figure.

**Circumscribable**

A plan figure that has a circumcircle.

**Circumscribed**

A figure circumscribed by a circle.

**Circumscribed Circle**

The circle that touches the vertices of a triangle or a regular polygon.

**Clockwise**

The direction of the moving hands of a clock.

**Closed Interval**

A closed interval is the one in which, both the first and last terms are included while considering the entire set. For example, [3,4].

**Coefficient**

The constant number that is multiplied with the variables and powers in an algebraic expression. For example, in 234x^{2}yz, 243 is the coefficient.

**Coefficient Matrix**

The matrix formed by the coefficients of a linear system of equations is called the coefficient matrix

**Cofactor**

When a determinant is obtained by deleting the rows and columns of a matrix in order to solve the equation, it is called the cofactors.

**Cofactor Matrix**

A matrix with the elements of the cofactors, term by term, in a square matrix is called the cofactor matrix.

**Cofunction Identities**

Cofunction identities are the identities that show the relation between the trigonometrical functions like the sine, cosine, cotangent.

**Coincident**

If two figures are superimposed on each other, then they are said to be coincident. In other words, a figure is coincident when all points are coincident.

**Collinear**

Two points are said to be collinear if they lie on the same line.

**Column of a Matrix**

The vertical set of numbers in a matrix is called the column of the matrix.

**Combination**

A selection of objects from a group of objects. The order is irrelevant in the selection of the object.

**Combination Formula**

A formula that is used to determine the number of possible combinations of r objects from a set of n objects. The formula involves the binomial coefficients and is given as:^{n}C_{r}. It is read as ‘n choose r’

**Combinatorics**

The branch that studies the permutations and combinations of objects and materials.

**Common Logarithm**

The logarithm to the base 10 is called common logarithm.

**Commutative**

An operation is said to be commutative if x ø y = y ø x, for all values of x and y. Addition and multiplication are commutative operations. For example, 4 + 5 = 5 + 4 or 6 X 5 = 5 X 6. Division and subtraction are not commutative.

**Compatible Matrices**

Two matrices are said to be compatible for multiplication if the number of columns of 1st matrix equals to the number of rows of the other.

**Complement of an Angle**

The complement of angle say 75º is 90º – 75º = 15º.

**Complement of an Event**

The set of all outcomes of an event that are not included in the event. The complement of set A is written as A^{c}. The formula is given as: P(A^{c}) = 1 – P(A) or P (not A) = 1- P(A).

**Complement of a Set**

The elements of a given set that are not contained in the given set.

**Complementary Angles**

If the sum of two angles is 90º, then they are said to be complementary angles. For example, 30º and 60º are complementary to each other as their sum equals 90º.

**Composite Number**

A positive integer whose factors are the numbers other than 1 and the number itself. For example, 4, 6, 9, 12 etc. 1 is not a composite number.

**Compound Fraction**

A compound fraction is a fraction that has at least one fraction term in the numerator and denominator.

**Compound Inequality**

When two or more than two inequalities are solved together it is known as compound inequality.

**Compound Interest**

While calculating compound interest, the amount that is earned as an interest for a certain sum/principal is added to the original principal, and from that, the interest is calculated on the new principal. Thus, the interest is not only calculated on the original balance, but the balance or principal obtained after adding the interest.

**Concave**

Concave is a shape of a figure or a solid that has a surface curving inwards or bulging inwards. It is also known as non – convex. Concave down or concave up are the other forms of concave shapes.

**Concentric**

The geometrical figures that are similar in shape and share a common center. Usually, the term concentric is used for concentric circles.

**Concurrent**

If two or more than two lines or curves intersect at the same point, then they are said to be concurrent at that point.

**Conditional Equation**

An equation that is true for some values of the variables and is false for other values of the variables. The equation has certain conditions imposed on it that are only satisfied by certain values of the variables.

**Cos ^{-1}**x

The inverse of cos function is read as ‘cos inverse x’. For example, cos^{-1}½ = 60º.

**Cot ^{-1}x**

By cot^{-1}x, we mean the angle whose cotangent is equal to x. For example, when we are asked to find the smallest angle whose cotangent is equal to 1? The answer is 45º. Thus, cot^{-1}1 = 45º.

**Cube**

A cube is a three dimensional figure bounded by six equal sides. The volume of cube is given by l^{3}, where l is the side of a cube.

**Cube Root**

A cube root is a number denoted as x^{⅓ }, such that b^{3} = x For example, (64)^{⅓} = 4.

**Cubic Polynomial**

A polynomial of degree 3 is known as the cubic polynomial. For example, x^{3} + 2x^{2} + x.

**Cuboid**

Cuboid is a three dimensional box that has length, width, and height. It is also called a rectangular parallelepiped.

**De Moivre’s Theorem**

De Moiver’s Theorem is a formula that is widely used in complex number system in order to calculate the powers and roots of complex numbers. It is given by:

[r(cosθ + isinθ)]^{n} = r^{n}(cosnθ + isin*n*θ).

**Decagon**

A 10 sided polygon is called decagon.

**Deciles**

In statistics, deciles are any of the nine values that divide the data into 10 equal parts. The first decile cuts off at the lowest 10% of the data that is called the 10th percentile. The 5th decile cuts off the lowest 50% of the data that is called 50th percentile or 2nd quartile or median. The 9th decile cuts off lowest 90% of the data that is the 90th percentile.

**Decreasing Function**

A function whose value decreases continuously as we move from left to right of its graph is called decreasing function. A line with negative slope is a perfect example of a decreasing function, where the value of the function decreases as we proceed on the x-axis. If the decreasing function is differentiable, then its derivative at all points (where the function is decreasing) will be negative.

**Definite Integral**

An integral that is evaluated over an interval. It is given by _{a}ʃ^{b}f(x)dx. Here the interval is [a, b].

**Degenerate Conic Sections**

If a double cone is cut with a plane passing through the apex of the plane, it is called the degenerate conic sections. It has the general equations of the form:

Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0

**Degree (angle measure)**

Degree is the measure of the slope or the angle that a line or a plane subtends. Degree is represented by the symbol ‘°’.

**Degree of a Polynomial**

The power of a highest term in an algebraic expression is called the degree of the polynomial. In the expression 2x^{5} + 3y^{4} + 5x^{3}, the degree of the polynomial is 5.

**Degree of a Term**

In 5y^{7}, degree of term is 7, in 5x^{2}4y^{3}, the degree of the term is the sum of the exponents of 5x and 4y, that means 5.

**Del Operator**

Del operator is denoted by symbol ∂(x, y, z)/∂x. The del operator ∇ = (∂/∂x, ∂/∂y) or ( ∂/∂x, ∂/∂y, ∂/∂z )

**Deleted Neighborhood**

Deleted neighborhood of a set is defined as a set {x: 0 < ;|x – a| < δ}. For example, one deleted neighborhood of 2 is the set {x: 0 < |x – 2| < 0.1}, which is the same as (1.9, 2) ∪ (2, 2.1).

**Delta (Δ δ)**

A Greek letter representing the basic discriminant of a quadratic equation.

**Denominator**

The lower part of a fraction is called denominator. In fraction (4/5), 5 is the denominator.

**Dependent Variable**

Consider an expression y = 2x + 3, here, x is the independent variable and y is the dependent variable. It is a general notion to plot the graph by taking independent variable on x axis and dependent variable on Y-axis.

**Derivative**

The slope of a line tangent to a function is called the derivative of the function. This is the graphical interpretation of the derivative. As a differentiation operation, consider f(x) = x^{2} then it’s derivative is f^{‘}(x) = 2x.

**Descartes’ Rule of Signs**

A method for determining the maximum number of positive zeros of a polynomial. According to this rule, the number of changes in the sign of the algebraic expression gives the number of roots of the expression.

**Determinant**

Determinants are the mathematical objects that are very useful in determining the solution of a set of system of linear equations.

**Diagonal Matrix**

A square matrix that has zeros everywhere except the main diagonal.

**Diagonal of a Polygon**

A line segment joining non-adjacent vertices of a diagonal. If a polygon is of n-sides, then the number of diagonals is given by the formula:

n(n-3)/2 diagonals.

**Diameter**

The longest chord of a circle is called diameter. It can be also defined as the line segment that passes through the center of the circle and touches both the ends of the circumference of the circle.

**Diametrically Opposed**

Two points directly opposite to each other on a circle.

**Difference**

The result of subtracting two numbers is called difference.

**Differentiable**

A curve that is continuous at all points of its domain is called a differentiable function. In other words, if a derivative exists for a curve at all points of the domains variable, it is said to be differentiable.

**Differential**

A tiny or infinitesimal change in the value of a variable.

**Differential Equation**

An equation involving the functions and derivatives. For example, (dy/dx)^{2} = y

**Differentiation**

Performing the process of finding a derivative.

**Digit**

Any of the numbers among the nine digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

**Dihedral Angle**

The angle formed by the intersection of two planes.

**Dilation**

Dilation refers to the enlargement of a geometrical figure by transformation method.

**Dilation of a Geometric Figure**

A transformation in which all distances are increased by some common factor. The points are stretched from a common fixed point P.

**Dilation of a Graph**

In graphical dilation, the x-coordinates and y-coordinates are enlarged by some common factor. The factor by which the transformation of the graph is done, must be greater than 1. If the factor is less than 1, it is called compression.

**Dimensions**

The sides of a geometrical figure are often referred to as dimensions.

**Dimensions of a Matrix**

The number of rows and columns of matrix is called the dimensions of the matrix. For example if a matrix has 2 rows and 3 columns, its dimensions will be 2X3 (read as two cross three).

**Direct Proportion**

When one of the variables is a constant multiple of the other, it is called direct variation. For example, y = kx (here y and x are the variables and k is a constant factor).

**Directrices of an Ellipse**

Two parallel lines on the exterior of an ellipse that are perpendicular to the major axis.

e is a transcendental number that has a value approximately equal to 2.718. It is frequently used while working with logarithms and exponential functions.

**Eccentricity**

A number that indicates the shape of a curve. It is represented by the small letter ‘e’ (This e is in no ways related to the exponential e = 2.718). In conic section, the eccentricity of the curves is a ratio between the distance from the center to focus, and either the horizontal or vertical distance from the center to the vertex.

**Echelon Form of a Matrix**

An echelon matrix is used to solve a system of linear equations.

**Edge of a Polyhedron**

One of the line segments that together make up the faces of the polyhedron.

**Element of a Matrix**

The numbers inside the matrix in the form of rows and columns is called the element of matrix.

**Element of a Set**

Any point, line, letter, number, etc. contained in a set is called the element of the set.

**Empty Set**

A set that doesn’t contains any element. The empty set is represented by {} or Ø.

**Equality Properties of Equation**

The equality properties of algebra that are used to solve the algebraic equations. The definitions of these equality properties are as follows:

x = y means, x is equal to y and y ≠ x means y is not equal to x. The operations of addition, subtraction, multiplication, and division all hold true for equality properties of equation.

Reflexive Property- x = x;

Symmetric Property- If x = y then y = x;

Transitive Property- If x = y and y = z then x = z

**Equilateral Triangle**

An equilateral triangle has all its three sides equal and the measure of each angle is 60º.

**Equivalence Relation**

Any equation that is reflexive, symmetric, and transitive.

**Equivalent Systems of Equation**

Two sets of simultaneous equations that have same solution.

**Essential Discontinuity**

It is a type of discontinuity in the graph that cannot be removed by simply adding a point. At the point of essential discontinuity, the limit of the function doesn’t exist.

**Euclidean Geometry**

The geometrical study of lines, points, angles, quadrilaterals, axioms, theorems, and other branches of geometry is called the Euclidean geometry. Euclidean geometry is named after Euclid, one of the greatest Greek mathematicians and known as the ‘Father of Geometry’. Read more on Famous mathematicians.

**Euler’s Formula**

Euler’s formula is given by e^{iπ} + 1= 1. It is a widely used formula in complex number analysis.

**Euler’s Formula in Polyhedron**

For any polyhedron, the following relation holds true:

[Number of faces(n)] – [number of vertices(v)] – [number of edges(e)] = 2.

This formula holds true for all convex and concave polyhedron.

**Even Function**

A function whose graph is symmetric about y-axis. Also, f(-x) = f(x).

**Even Number**

The set of all integers that are divisible by 2. E= {0, 2, 4, 6, 8……}

**Explicit Differentiation**

The derivative of an explicit function is called the explicit differentiation. For example, y = x^{3} + 2x^{2} – 3x. Differentiating it gives,

y’= 3x^{2} + 4x – 3.

**Explicit Function**

In an explicit function, the dependent variable can be totally expressed in terms of independent variable. For example, y= 5x^{2} – 6x.

**Exponent Rules**

The exponential rules are as follows.

Serial Number |
Exponential Formula |

1 | a^{n}a^{m} = a^{n+m} |

2 | (a.b)^{n} = a^{n}. b^{n} |

3 | a^{0} = 1 |

4 | (a^{m})^{n} = a^{mn} |

5 | a^{m/n} = n√a^{m} |

6 | a^{-m} = 1/a^{-m} |

7 | (a^{m}/a^{n} )= a(^{m-n}) |

**Extreme Value Theorem**

According to this theorem, there is always at least one absolute maximum and one absolute minimum for any continuous function over a closed interval.

**Extreme Value of a Polynomial**

The graph of a polynomial of degree n has at most n-1 extreme values (either maxima or minima)

**Face of Polyhedron**

Polygonal outer boundary of a solid object having no curved surfaces.

**Factor of an integer**

If the given integer is divided evenly by another integer then the resultant is called factor of an integer. For example: 2, 4, 8, 16, etc., are the factors of 32.

**Factor of polynomial**

If polynomial P(x) is completely divided into polynomial R(x) by Q(x), then Q(x) is called the factor of polynomial. For example: P(x)= x^{2}+6x+8, Q(x)=x+4 then P(x)/Q(x)= x+2. Q(x)=x+4 is the factor.

**Factor theorem**

When x-a is factor of P(x), the value x in P(x) is replaced with a, then if the resultant value is 0, such a theorem is called Factor theorem. For example: P(x)= x^{2}+6x+24. Q(x)= x-(-4). If x is replaced with a, that is -4, then P(x)= 0.

**Factorial**

The product of an integer with all the consecutive smaller integers is called a factorial. It is represented as “n!”. For example: 5! = 5*4*3*2*1= 120.

**Factoring Rules**

These are the formulas that govern the factorization of a polynomial. For example,

- x
^{2}-(a+b)x +ab= (x-a)(x-b). - x
^{2}+2(a)x+a^{2}=(x+a)^{2} - x
^{2}-2(a)x +a^{2}=(x-a)^{2}

Read more on factor by grouping.

**Fibonacci series**

It is a series of numbers where the next number is found by adding the previous two numbers in the series. The first two numbers of the series are 0 and 1. The series is 0,1,2,3,5,8 …

**Finite**

The term is used to describe a set in which all the elements can be counted using natural numbers.

**First Derivative**

A function F(a), which governs the slope of the curve at any given point, or the slope of the line drawn tangent to the curve from that point in the plane is called the first derivative. It is represented as F’. For F(x)= 5x^{2}. F'(x)=10x will be the slope of the curve.

**First Derivative test**

A technique which is used to determine the capacity of inflection point.(minimum, maximum, or neither)

**First Order of the differential equation**

A differential equation P(a) who’s order is 1. For example: P(a)=3a, here the order of a is 1.

**Flip**

It is also known as axis of reflection. It is a line, which divides the plane or a geometric figure into two halves that are mirror images of each other.

**Floor Function (Greatest Integer Function)**

It is a function F(x), which is responsible for finding the greatest integer less than the actual value of P(x). For example: P(x)= 5.5, here the greatest integer less than 5.5 is 5. The function which gives F(x)=5 becomes floor function.

**Foci of the Ellipse**

They are the fixed two points inside the ellipse such that the vertical curve is governed according to the equation L1+L2= 2a and horizontal curve according to equation L1+L2=2b, where L is the distance between the focal point and the curve, a is the horizontal radius and b is the vertical radius.

**Foci of hyperbola**

They are fixed two points inside of the curve of hyperbola such that the determinant of the L1-L2 is always constant. L1 and L2 are the distances between point P (which is the curve) and respective focus of the curve.

**Focus**

The curves of the conic sections are governed according to distances from a special point called focus.

**Focus of Parabola**

In a parabola, the distance between a point P on the curve and an arbitrary point inside parabola, which is equal to the distance between the same point P and directrix of the curve. Such an arbitrary point is called focus of the parabola.

**FOIL method**

FOIL is an acronym for First Outer Inner Last. It is method by which binomials are multiplied. The Multiplication order is

- First terms of Binomials
- Outer terms of Binomial
- Inner terms of binomials
- Outer terms of Binomials.

For example: (a+b)(a-b)= a.a+a.(-b)+b.a +b.(-b)

**Formula**

The relationship between various variables (sometimes expressed in the form of an equation) depicted using symbols. For example: a+b=7

**Fractal**

When every part of the figure is similar to every other part of other figure, then the figure is called fractal.

**Fraction**

It is a ratio between two numbers. For example: 9/11.

**Fraction Rules**

The rules of algebra used for uniting various the fractions.

**Fractional Equation**

The expression in the form of A/B on both the sides of equal sign is called fractional equation. For example: x/6= 4/3.

**Function Operation**

Various operations, such as additions, subtractions, multiplications, divisions and compositions which have a combining effect on various functions. For example: F(a/b) = F(a)/F(b).

**Fundamental theorem of Algebra**

Every polynomial characterized by single variable having complex coefficients, will have a minimum of at least one root which is also complex in nature.

**Fundamental Theorem of Arithmetic**

The statement that the factors of a prime number are always distinct and unequal is the fundamental theorem of arithmetic.

**Fundamental Theorem of Calculus**

Differentiation and integration are two most basic operations of the calculus. The theorem that establishes a relationship between them is called Fundamental theorem of Calculus.

**Gauss-Jordan Elimination**

A method of solving a system of linear equations. In this process, the augmented form of the matrix system is reduced into row echelon form by means of row operations.

**Gaussian Elimination**

A method of solving a system of linear equations. In the Gauss elimination method, the augmented form of matrix is reduced to row echelon form and then the system is solved by back substitution.

**Gaussian Integer**

Gaussian integers are the integers in the complex numbers that are represented by a + bi. For example, 3 + 2i, 5i and 6i + 5 are called Gaussian integers.

**GCF**

The largest integer that divides a certain set of numbers. Its full form is called Greatest Common Factor. For example, the GCF of 20, 30, and 60 is 10.

**General Form for the Equation of a Line**

The general form of equation of a line is represented by the equation-

Ax + By + C = 0, where, A, B and C are integers.

**Geometric Figure**

A geometric figure is a set of points on the plane or space that leads to the formation of figure.

**Geometric Mean**

Geometric mean is a method of finding the average of certain set of numbers. For example, if there are numbers a_{1}, a_{2}, a_{3}, … , a_{n}, then multiply the numbers and take the nth root of the product.

Geometric Mean = (a_{1}, a_{2}, a_{3}, … , a_{n})^{½}

**Geometric Progression**

A geometric progression is a sequence whose terms are in a constant ratio with the previous terms. For example, 2, 4, 8, 16, 32, … , 28 are the terms of a geometric progression. Here, the common ratio is 2. (as 4/2 = 8/4 = 16/8 … )

**Geometric Series**

Geometric series is a series whose successive terms are in a constant ratio. An example of geometric series is 2, 4, 8, 16, 32, …

**Geometry**

The study of geometric figures in two and three dimensions is called geometry.

**Greatest lower bound**

The greatest of all lower bounds of a set of numbers is called the GLB or greatest lower bound. For example, in the set [2,7], the GLB is 2.

**Glide Reflection**

A transformation in which a figure has to go through a combination of steps of translation and reflection.

**Global Maximum**

The highest point on the graph of a function or a relation (in the domain of the function). The first and second derivative tests are used to find the maximum values of a function. It is also called global maximum, absolute maximum, and relative maximum.

**Global Minimum**

The lowest point on the graph of a function or a relation. The first and second derivative tests are used to find the minimum values of a function. It is also called the global minimum, absolute minimum, or global minimum.

**Golden Mean**

The ratio (1 + √5)/2 ≈ 1.61803 is called the golden mean. The unique property of golden mean is that the reciprocal of golden mean is about 0.61803. Hence, the golden mean is one plus its reciprocal.

**Golden Rectangle**

If the ratio of length and breadth of a rectangle is equal to the golden mean, then the rectangle is called the golden rectangle. It is believed that this rectangle is most pleasing to the eyes.

**Golden Spiral**

A spiral that can be drawn inside the golden rectangle.

**Googol**

The number 10^{100} is called googol.

**Googolplex**

Googolplex can be written as 10^{100}^{100}.

**Graph of an Equation or Inequality**

The graph obtained by plotting all the points on the coordinate system.

**Graphic Methods**

The use of graphical methods to solve the mathematical problems.

**Great Circle**

A circle that is drawn on the surface of the sphere and shares a common center with the circle.

**Greatest Integer Function**

The greatest integer function of any number (say x) is an integer ‘less than or equal to x’. The greatest integer function is represented as [x]. For example, [3.4] = 3 and [-2.5] = 3

**Half Angle identities**

The identities of trigonometry that are used to calculate the value of sine, cosine, tangent, etc., of half of a given angle.

The trigonometric identities are as follows:

sin^{2}x = (1 – cos2x)/2

cos^{2}x = (1 + cos2x)/2

**Half Closed Interval/Half Open Interval**

It is a set of all numbers containing only one end point.

**Harmonic Mean**

The inversion of the summation of the reciprocals of a set of numbers. For example: (1, 2, 3) are in a set then their harmonic mean is 1/(1+ ½+ ⅓ )

**Harmonic Progression**

It is a sequence in which every term is the reciprocal of the natural number. For example 1, ½, ⅓, ¼.

**Harmonic Series**

The summation of all the terms in harmonic progression. For example: 1+ ½+ ⅓+ ¼

**Height**

The least measurable distance between the base and the top of a geometric figure is called the height. The top can be the opposite vertex, an apex, or even another base of the figure.

**Height of the Cone**

The distance between the center of the circular base and the vertex of the cone can be called the height of the cone.

**Height of Cylinder**

The distance between the centers of the circular bases of the cylinder is the height of the cylinder.

**Height of a Parallelogram**

The perpendicular distance between the parallel sides of a parallelogram (i.e. the base to the opposite side).

**Height of a Prism**

The length of the shortest line segment between the bases of the prism.

**Height of a Pyramid**

The shortest distance between the vertex and extended base of the pyramid.

**Height of a Triangle**

The length of the shortest line segment between a vertex and the opposite side of the triangle.

**Helix**

It is a spiral shape curve in three dimensional space.

**Heptagon**

A heptagon can be called a polygon, which has seven sides. Its other name is septagon, but heptagon is widely used.

**Hero’s Formula**

Suppose all the three sides of the triangle are known. The formula used to calculate the area of the triangle in this scenario is called Hero’s formula. For example: √[s(s-a)(s-b)(s-c)]

**Hexagon**

It is a special geometric figure, which has six sides and angles.

**Hexahedron**

A solid, which has no curved surfaces and the number of surfaces are equal to six.

**Higher Derivative**

The derivative of first derivative or the derivative of second derivative that have degree more than 1 are called higher derivatives.

**Homogeneous Equations**

When two or more linear equations have their constant term as zero, then such a set of equations are called homogeneous equations.

**Horizontal Line Equation**

It is an equation, which is used to describe a line parallel to Y-axis.

**Horizontal Line Test**

If a horizontal line intersects a graph twice [graph is made by the function F(x)], then the function is said be not one on one. This test to check one to one of function is called horizontal line test.

**Horizontal Reflection**

A geometric figure in the first or fourth quadrant, the reflection of which is present in second or third quadrant along X-axis and vice or versa is called horizontal reflection.

**Hyperbola**

A hyperbola is a geometric figure, which is a locus of two points called foci, where the difference between the distances to each point is constant.

**Hyperbolic Geometry**

Given two entities, a point and a line, there can be infinitely many lines passing through the point and are parallel to first point. This is called Hyperbolic geometry.

**Hyperbolic Trigonometry**

The trigonometric functions sine cosine tangent, etc., whose values are calculated using ‘e’. The explanations of hyperbolic trigonometry are as follows:

sinhx = (e^{x} – e^{-x})/2,

coshx = (e^{x} + e^{-x})/2

tanhx = (sinhx/coshx) = (e^{x} – e^{-x})/(e^{x} + e^{-x})/2

**Hypotenuse**

The hypotenuse is longest side of right-angled triangle.

**Hypotenuse-leg Congruence**

Two different right-angled triangles are said to be congruent when their hypotenuse and one of the corresponding legs are equal in length.

**Hypotenuse-leg Similarity**

In two right-angled triangles, when the ratio of the corresponding sides have equal ratios, then such triangles are having HL Similarity.

**i**

In complex number analysis, the letter i denotes iota. Iota is given by negative square root of 1, that means √-1. = i

**Icosahedron**

Icosahedron is a polyhedron with 20 faces. In the case of a regular icosahedron, the faces are all equilateral triangles.

**Identity (Equation)**

An equation that is true for any values of the variable. For example, the identity, sin^{2}θ + cos^{2}θ = 1 is true for all values of θ.

**Identity Function**

The function f(x) = x is called the identity function.

**Identity Matrix**

A square matrix that has 1 as its element in the principal diagonal and rest all elements are zero.

**Image of a Transformation**

The image obtained after performing the operations of dilation or rotation or translation.

**Imaginary Numbers**

A complex number like 7i, that is free of the real part is called the complex number.

**Imaginary Part**

Consider a complex number -7 + 8i, the coefficient of i called the imaginary part of the complex number.

**Implicit Function or Relation**

A function in which the dependent variable can’t be exactly expressed as a function of the independent variable.

**Implicit Differentiation**

Differentiating an implicit function. For example, consider 4x^{2} + 5y^{5} – 6x = 1. Here, y can’t be written explicitly as a function of x.

**Impossible Event**

An event that is impossible to happen or an event whose probability is zero.

**Improper Fraction**

A fraction that has denominator greater than its numerator.

**Improper Integral**

A integration in which the bounds of integration has discontinuities in the graph. They can also have limits between ∞ and -∞. The discontinuities between the bounds of integration makes the use of limits necessary in evaluating improper integrals.

**Improper Rational Expression**

If the degree of a numerator polynomial is more than or equal to the degree of a denominator polynomial than the rational expression is called the improper rational expression.

**Incenter**

The center of a circle inscribed in a triangle or a polygon. Geometrically, incenter is the point of intersection of the angle bisectors of a triangle.

**Incircle**

The largest possible circle that can be drawn inside a plane figure. All triangles and regular polygons have an incircle.

**Inconsistent System of Equations**

A system of equations that has no solutions.

**Increasing Function**

A function whose value increases continuously as we move from left to right of its graph is called an increasing function. A line with positive slope is a perfect example of increasing function where the value of the function increases as we proceed on the x-axis. If the increasing function is differentiable, then its derivative at all points (where the function is increasing) will be negative.

**Indefinite Integral**

I = ^{a}∫_{b}f(x) dx, is known as the improper integral

**Independent Events**

If the occurrence or non-occurrence of two events is independent of each other, it is called the independent event.

**Independent Variable**

The quantity in an equation whose values can be freely chosen in an equation without taking into consideration the values of the other variables.

**Indeterminate Expressions**

An undefined expression that cannot be assigned any value. There are various forms of indeterminate expressions:

- 0/0
- ±∞/±∞
- 0
^{0} - 1
^{∞} - ∞
^{0} - ∞ – ∞

**Induction**

A method of proving a problem by the help of a series of steps. Mathematical induction is used to prove complex problems.

**Independent Events**

Two or more events are said to be independent events if the occurrence or non-occurrence of any of these events doesn’t affect the occurrence or non-occurrence of others. By the principle of probability, if A and B are two independent events, then P(A|B) = P(A).

**Independent Variable**

Independent variables are those whose value can be chosen without any restriction. For example, in the equation Y = 2x^{2} + 3x, y is the dependent variable and x is the independent variable.

**Indirect Proof**

Proving a statement or a fact by the method of contradiction is known as indirect proof. This means that the conjecture is taken to be false, and then, it is proved that the statement contradicts the assumption made at the beginning of solving the problem.

**Jacobian**

This expression is used to denote the jacobian matrix in vector calculus. In vector calculus, the jacobian matrix is the matrix of the first order partial derivatives of a vector-valued function. Conceptually, the jacobian of a function represents the orientation or inclination of a tangent plane to the function at a given point.

**Joint variation**

When a quantity varies directly with the other quantity, then it is called the joint variation. For example, when we say x is directly proportional to the square of y, it means that x = ky^{2}, where k = proportionality constant.

**Kite**

A kite is nothing but a quadrilateral, with each pair of its adjacent sides congruent to each other and diagonals perpendicular to each other.

**L’Hospital’s Rule**

This is a technique that is used to find out the limit of the functions that evaluate to indeterminate forms, like 0/0 or infinity/infinity. The solution is found out by individually calculating the limits of the numerator and the denominator.

**Lateral Surface Area**

Lateral Surface Area is nothing but the surface area of the lateral surfaces of a solid. It does not include the area of the base(s) of the solid.

**Latus Rectum**

It is the line segment that passes through the focus of a conic section and is perpendicular to the major axis, with both its end points on the curve.

**Law of Cosines**

An equation that relates the cosine of an interior angle of a triangle to the length of its sides is called the law of cosines.

If a, b, and c are the three sides of a triangle, A is the angle between b and c, B the angle between a and c, and C the angle between a and b, then the law of cosines states that c^{2} = a^{2} + b^{2} – 2abcosC, b^{2} = a^{2} + b^{2} – 2accosB, and a^{2} = b^{2} + c^{2} – 2bccosA

**Law of Sine**

An equation that relates the sine of an interior angle of a triangle to the length of its sides is called the law of sines.

If a, b, and c are the three sides of a triangle, A is the angle between b and c, B the angle between a and c, and C the angle between a and b, then the law of cosines states that

sin A/a = sin B/b = sin C/c

**Least Common Multiple (LCM)**

The smallest common multiple to which two or more numbers can be divided evenly. For example, the LCM of 2, 3, and 6 is 12.

**Leading Coefficient**

The coefficient of a polynomials leading term or the term with the variable having the highest degree.

For example, the leading coefficient of 7x^{4} + 5x^{3} + 9^{2} + 2x +21 is 7.

**Leading Term**

It is the term of a polynomial, which contains the highest value of the variable.

For example, the leading coefficient of 7x^{4} + 5x^{3} + 9^{2} + 2x +21 is 7x^{4}.

**Least Common Denominator**

The least common denominator is the smallest whole number that can be used as a denominator for two or more fractions. It is nothing but the LCM of the denominators of the fractions.

For example, the least common denominator of 3/4 and 4/3 is 12. Since 3/4=6/8=9/12 and 4/3=8/6=12/9=16/12. Hence we see that the least common denominator is 12.

**Least Integer Function**

The least integer function of x is a step function of x, which is the least integer greater than or equal to x. This function is sometimes written with reversed boldface brackets **]**x**[** or reversed plain brackets ]x[.

**Least Squares Regression Line**

The Linear Squares Regression Line is the linear fit that matches the pattern of a set of paired data, as closely as possible. Out of all possible linear fits, the least-squares regression line is the one that has the smallest possible value for the sum of the squares of the residuals. It is also known as Least Squares Fit and Least Squares Line.

**Least-Squares Regression Equation**

An equation of any form (linear, quadratic, exponential, etc) that helps in fitting a set of paired data as closely as possible is called the least squares regression equation.

**Least Upper Bound of a Set**

The smallest of all upper bounds of a set of number is called the Least Upper Bound.

**Leg of an Isosceles Triangle**

Any of the two equal sides of an isosceles triangle can be referred to as the leg of the isosceles triangle.

**Leg of a Right Angle Triangle**

Either of the sides of a right angle triangle, between which the right angle is formed can be referred to as the leg of the right angle triangle.

**Leg of a Trapezoid**

Either of the two non parallel sides of a trapezoid that join its bases can be referred to as the leg of the trapezoid.

**Lemma**

More accurately called a helping theorem, a lemma helps in proving a theorem. But it is not important enough to be a theorem.

**Lemniscate**

A curve that takes form on the numerical number 8, in any orientation can be referred to as the lemniscate. Its equations are generally given in the polar coordinates. r^{2} = a^{2}cos2θ.

**Like Terms**

They are terms that have the same variables and with the same power. Their coefficients can be directly added and subtracted. For example, 5x^{3}y^{2} and 135x^{3}y^{2} are like terms, and hence, can be added directly to give the number 140x^{3}y^{2}.

**Limacon**

A limacon is a family of related curves usually expressed in polar coordinates.

**Limit**

The limit of a function is the value of the function as its variable tends to reach a particular value.

For example, for f(x)=lim_{x→5}1/x^{2}= 1/25. As x→5, the function f(x) tends to reach to 1/25.

**Limit Comparison Test**

The limit comparison test is performed to determine if a series is as good as a good series or as bad as a bad series. The test is used specially in cases when the terms of a series are rational functions.

**Limit from Above**

The limit from the above is usually taken in cases when the values of the variable is assumed to be greater than that to which the limit approaches. For example, lim_{x→0+}1/x=infinity, is taken such that the value of x>0. Limit from above is often referred to as limit from the right. This is a one-sided limit.

**Limit from Below**

The limit from the below is usually taken in cases when the values of the variable is assumed less than that to which the limit approaches. For example, lim_{x→0–}1/x=-infinity, is taken such that the value of x>0. Limit from below is often referred to as limit from the left. This is a one-sided limit.

**Limit Involving Infinity**

A limit involving infinity or an infinite limit is one whose result approaches infinity or the value of the variable approaches infinity.

**Limit Test for Divergence**

A limit test for divergence is a convergence test which is based upon the fact that the terms of a convergent series must have a limit of zero.

**Line**

A line is a geometric figure that connects two points and extends beyond both of them in both directions.

**Line Segment**

A line segment is nothing, but the set of points between any two points including those two points.

**Linear**

The world linear means like a line. It is nothing but a graph or data that can be molded by a linear polynomial.

**Linear Combination**

A linear combination is the sum of multiples of the variables in a set. For example, for the set {x, y, z}, one possible linear combination is 7x + 3y – 4z.

**Linear Equation**

An equation that can be written in the form “linear polynomial” = “linear polynomial” or “linear polynomial”=constant is known as a linear equation.

For example, 3x + 26y = 34 is a linear polynomial.

**Linear Factorization**

If a polynomial can be factorized such that the factors formed after the factorization are linear polynomials, then this factorization is known as a linear factorization. For example, x^{2}-9 can be factorized as (x+3) and (x-3).

**Linear Fit Regression Line**

Any line that can be used as a fit in the process to model the pattern in a set of paired data.

**Linear Inequality**

An inequality that can be written such that the value of a polynomial is greater than, less than, greater than equal to or less than equal to a particular number is called linear inequality. For example, 3x + 7y>9.

**Linear Pair of Angles**

When two lines intersect each other, then the adjacent angles formed due to intersection of the two lines are called linear pair angles. The linear pair angles formed are supplementary.

**Linear Polynomial**

A linear polynomial is a polynomial with degree 1. The highest power of the variables involved in the polynomial should be one. For example, 9x + 7 is a linear polynomial.

**Linear Programming**

The linear programming is an algorithm that is used for solving problems. The method of using linear programming is by asking the largest or smallest possible value of a linear polynomial. If there are any restrictions, then the system of inequalities is used to present any restriction to the equations.

**Linear Regression**

The process of finding a linear fit is referred to as the linear regression.

**Linear System of Equations**

If there are more than one equations such that each equation is a linear equation, then the system of equations will be known as linear system of equations.

For example, 2x + 3y – 5z

9x + 7y + 12x = 19

15x – 6y + 11z = 9 is a linear system of equations, that can be used to determine the values of x, y and z.

**Local Behavior**

The behavior of a function in the immediate neighborhood of any point is called the local behavior. The local behavior of geometric figures can also be studied with respect to a particular point.

For example, for the graph of the equation y=2x + 3, if studied closely can be said to have the local behavior of a straight line parallel to the x-axis and at a distance of 3 units from the origin.

**Local Maximum**

The local maximum is the highest point in a particular section of the graph. It is also often referred as the local max or relative maximum or relative max.

**Local Maximum**

The local minimum is the lowest point in a particular section of the graph. It is also often referred as the local min or relative minimum or relative min.

**Locus**

A locus is nothing but the set of points that form a particular geometric figure. For example, a circle with radius 2 cm is the locus of all points, which are at a distance of 2 cm from a particular point.

**Logarithm**

The logarithm of x with respect to the base c is the power to which the base c must be raised in order to be equal to x. For example, log_{c}x=z then c^{z}=x.

**Logarithmic Rules**

The logarithmic rules are the algebra rules that need to be used when working with logarithms. Some of them can be listed as under:

- If log x = y, then 10
^{y}=x. It means that if the base of the logarithm is not mentioned, then consider the base as 10. - If ln x = y, then e
^{y}=x. It means that when log is replaced by ln, then take the logarithm as natural logarithm and has the base e. - log 1 = 0, since whatever be the base, if raised to the power 0 then the result is always 1.
- log ab = log a + log b
- lob (a/b) = log a – log b
- log b
^{3}= 3log b - log
_{a}x = log_{b}x/log_{b}a

**Logarithmic Differentiation**

It is the type of differentiation that is used in special circumstances. For example, the equation y = x^{tan x} can be differentiated, more easily if the logarithm of both the sides are taken.

On taking the logarithm of both the sides, the equation can be reduced to log y = tan x. log x (using logarithmic formula). Hence, the process of differentiation becomes simple.

**Logistic Growth**

A logistic growth is shown by using an equation. It is used to determine the demand of products in situations where the demand increases initially, then the demand goes down, and finally, it reaches a particular upper limit.

**Long Division of Polynomials**

The process of dividing polynomials is known as polynomial long division. The polynomial long division is used to divide improper rational numbers into proper rational numbers or sum of polynomials. The process of polynomial long division is same as that of long division of numbers.

**Lower Bound**

The lower bound of a set is any number that is less than or equal to all the numbers in a set. For example 1, 2, and 3 are all lower bounds of the interval [4, 5].

**Low Quartile**

The low quartile is the number for which 25% of the number is less than the number.

**Least Upper Bound of a Set**

The smallest of all the upper bounds of a set of numbers is called the least upper bound of the set. For example, the least upper bound of the interval [9, 10] is 10.

**Maclaurin Series**

The power series in x for a function f(x) is known as Maclaurin series.

**Magnitude**

The magnitude is the absolute value of a quantity. Magnitude is a value, and it can never be a negative number.

**Magnitude of a vector**

The magnitude of a vector is the length of the vector.

**Main Diagonal of a Matrix**

It is the numbers of a matrix taken diagonally starting from the number at the upper left corner and ending at the lower right corner.

**Major Arc**

The longer of the two arcs between the two arcs of a circle is called the major arc of the circle.

**Major Axis of an Ellipse**

The line passing through the two foci, the two vertices, and the center of the eclipse is called the major axis of the ellipse.

**Major Axis of a Hyperbola**

The line passing through the two foci, the two vertices, and the center of the hyperbola is called the major axis of the hyperbola.

**Major Diameter of an Ellipse**

The line segment joining the two vertex of ellipse and passing through its center and two foci is known as the major diameter of the ellipse.

**Mathematical Model**

It is nothing but a system of equations that is used for representing a graphs, some data, or even some real world phenomenon.

**Matrix**

A matrix is a rectangular or square array of numbers. All the rows of the matrix is equal lengths and all the columns are also of equal lengths.

**Matrix Addition**

Two matrices with the same dimensions can be added using the process of matrix addition. The process of matrix addition is such that the element in the position Row 1, Column 1 must be added to the element at the location Row 1, Column 1 of the other matrix.

**Matrix Element**

Any number in a matrix is known as the matrix element. The position of the number in the matrix is defined by the row number and column number.

**Matrix Inverse**

The matrix inverse of a matrix is the one, which on being multiplied with the matrix gives the identity matrix. If the matrix is denoted by A, then its inverse is denoted by A^{-1}.

**Matrix Multiplication**

Two matrices can be multiplied only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

**Maximum of a Function**

The highest point in the graph of the function is often referred to as the maximum of the function.

**Mean**

It is nothing but another word for average. When the word mean is used, it is generally referred to the arithmetic mean of a function.

For example, the arithmetic mean of the numbers 1, 4, 6, 7, 8 is (1+4+6+7+8)/5.

**Mean of a Random Variable**

This is often referred to in the case of probability where a number of trials are performed to see the most expected result. The average of all the outcomes of all these trials is considered the mean of a random variable.

**Mean Value Theorem**

This is a theorem used in Calculus. It states that for every secant for the graph of a ‘nice’ function, there is a tangent parallel to the secant.

**Mean Value Theorem for Integrals**

The mean value theorem for integrals states that for every function there is at least one point where the value of the function equals the average value of the function.

**Measure of an Angle**

The value of an angle in radians or degrees is referred to as the measure of an angle.

**Measurement**

The process of assigning a value for any physical quantity (e.g. Length, breadth, height, area, volume, etc.) is called measurement.

**Median of a Set of Numbers**

The median of a set of numbers is the number, which is greater than half the numbers in the set and smaller than the remaining half. In case of two medians, simply find out the arithmetic mean of the two numbers.

**Median of a Trapezoid**

The line joining the two non parallel lines of the trapezoid and parallel to the base of the trapezoids is called the median of the trapezoid.

**Median of a Triangle**

The line segment joining the vertex of a triangle to the mid point of the opposite side is called the median of the triangle. It is very clear that every triangle has three medians.

**Members of an Equation**

For any equation, the polynomials on the two sides of the equation are referred to as the members of the equation. For example, for the equation, 3x^{2}+5=26x, the members of the equation are 3x^{2}+5 and 26x.

**Menelaus’ Theorem**

The Menelaus’ theorem is an equation that shows how the two cevians of a triangle divide the two sides of the triangle and each other.

For example, if A, B, and C are the three vertex of the triangles and BF is the line segment from B to the side AC intersecting AC at F, CD is the line segment from C intersecting at B, and BF and CD intersect at the point P, then, (AD/DB)(BP/PF)(FC/CA)=1.

**Mensuration**

The process of finding out the measurement of the physical quantities in geometry is referred to as mensuration.

**Mesh of a Partition**

In any partition, the width of the largest sub interval is called the mesh of the partition.

**Midpoint**

The point at exactly half of the distance from the two points on the line segment joining the two points.

**Midpoint Formula**

The midpoint formula states that for any two points (x_{1}, y_{1}) and (x_{2}, y_{2}), the mid point is given by ((x_{1}+x_{2})/2 ,(y_{1}+y_{2})/2).

**Max/Min Theorem**

The max/min theorem states that for any continuous function f(x) in the interval [a,b] there exist two numbers in the interval (say c and d), such that, for f(c) and f(d) the function has its absolute maximum and minimum.

**Minimum**

The process of finding out the smallest possible value of the variable in a function is referred to as the minimum of the function.

**Minimum of a function**

The minimum value of the function within a limited region or entire region of the function is referred to as the minimum of the function.

**Minor arc**

If the circumference of the circle is divided into two arcs, then the smaller arc is referred to as the minor arc of the circle.

**Minor Axis of an Ellipse**

The minor axis of an ellipse is the line passing through the center of the ellipse and perpendicular to the major axis.

**Minor Axis of a Hyperbola**

The minor axis of a hyperbola is the line passing through the center of the hyperbola and perpendicular to the major axis.

**Minor Diameter of an Ellipse**

The minor diameter of an ellipse is the line passing through the center of the ellipse and perpendicular to the major diameter

**Minute**

A minute is a measurement equal to 1/60th of a degree. It is represented by the symbol ‘. Thus 12°36’ is called 12 degree and 36 minutes.

**Mixed Number**

It is also called a mixed fraction. This is a way of representing improper fraction as the sum of a number and a proper fraction. For example, 31/4 can be written as the mixed number 7 ¾, since 7+3/4 is 31/4.

**Mobius strip**

A mobius strip is a figure that can be represented as a strip of paper fixed at both the ends and with a half turn in the middle.

**Mode**

The number that occurs the maximum times in a list is referred as the mode of the number. For example, in the series 1, 3, 3, 3, 5, 6. 6 the mode is 3, since it occurs the maximum number of times.

**Modular Arithmetic**

When normal arithmetic operations are performed and the result is given in modular form, then the process is known as modular arithmetic.

For example, 15 – 3 = 12, but in mod(7) form the result is 15 – 3 = 5(mod 7).

**Modular equivalence**

Two or more integers are considered to be in modular equivalence if they leave the same integer on being divided by the same number. For example, 10 and 16 are both mod 3 equivalent numbers, because they leave the remainder 1 on being divided by 3.

**Modular Equivalence Rules**

The modular equivalence rules can be listed as under:

Suppose a and b are two mod n equivalent numbers,

- a+c and b+c are modular equivalent.
- Similarly, a-c and b-c are modular equivalent.
- a.c and b.c are modular equivalent. If ac and bc are modular equivalent numbers, then a and b are modular equivalent.

These were the modular equivalent rules for normal modular arithmetic operations.

**Modulo n**

Modulo n or mod n of a number is the remainder of the number when divided by n. For example, the number 7 when written in mod 3 form can be written as 7 ≡ 1 (mod 3).

**Modulus of a Complex Number**

The modulus of a complex number is the distance of the number from the origin on the complex plane. For example, for the number a+bi, the modulus of the number is given by (a^{2} + b^{2})^{½}. If the number is given in polar coordinates and the number is rcos θ + irsin θ, then the modulus is given by r.

**Modus Ponens**

Modus Ponens is a form of logical argument. For example, * if the pen is working, the pencil is working*. Now, if the argument is that the pen is working, then we can conclude that the pencil is working.

**Modus Tolens**

Modus Tolens is a form of logical argument that employs the proof of contradiction. For example, *if the pen is working, then the pencil is working. The pen is not working, hence the pencil is not working.*

**Monomial**

A polynomial with one term is called monomial.

**Multiplication Rule**

The multiplication rule is used in probability to find out if two events have occurred. For example, if there are two events A and B then, P(A and B) = P(A)P(B) or P(A and B)=P(A).P(B|A).

**Multiplicative Inverse of a Number**

The multiplicative inverse of a number is nothing but the reciprocal of the number. In other words, it is 1 divided by the number. For example, the multiplicative inverse of the number 3/5 is 1/(3/5)=5/3.

**Multiplicative Property of Equality**

The multiplicative property of equality states that if a and b are two numbers such that a = b, then a.c = b.c.

**Multiples**

Multiples are the numbers that can be evenly divided by the number whose multiple we are considering. For example, 16 is a multiple of 4 because 16 can be evenly divided by 4.

**Multiplicity**

The multiplicity of a polynomial is the number of times the number is zero for the given polynomial. For example, in the function f(x) = (x + 3)^{2}(x-2)^{4}(x – 7)^{3}, the number -3 has multiplicity 2, 2 has multiplicity 4 and 7 has multiplicity 3.

**Multivariable**

Any problem that involves more than one variable is called a multivariable problem.

**Multivariable calculus**

If the problems in calculus involve two or more independent or dependent variables, then the calculus is called multivariable calculus.

**Mutually Exclusive**

If the outcome of two events in probability have no common outcomes, then the events are called mutually exclusive.

**n-dimensions**

It is the property of space that indicates that n mutually perpendicular directions for movement of any particle in that space is possible.

**Natural Domain**

It is an alternative term for domain. It is the range within which the function exists or has definite values. The word natural domain is used to signify that the domain is not restricted.

**Natural Logarithm**

When the logarithm of a number is taken with respect to the base e, then the logarithm is said to be natural logarithm. The natural logarithm is represented as *ln a*, where a is the number.

**Natural Numbers**

All integers greater than 0 are called natural numbers.

**Negative Direction**

The negatively associated data is often described in the form of a scatterplot. This way of describing natural numbers is known as negative direction.

**Negative Exponent**

A negative exponent is used to describe the reciprocal of the number. For example, 5^{-2}=1/5^{2}

**Negative Number**

Any real number less than 0 is called a negative number.

**Negative Reciprocal**

The process of taking the reciprocal of a number, and then its negative, is called the negative reciprocal. For example, the negative reciprocal of ¼ is -4.

**Negatively-Associated Data**

If in a set of paired data, the value of one side increases with the decrease in the other, then the data is referred to as the negatively associated data.

**Neighborhood**

The neighborhood of any number a is the open interval containing the number. For example, the neighborhood of a can be written as (a + d, a – d).

**n – gon**

A polygon with n number of sides is called n – gon. For example, a hexagon can also be called 6-gon.

**Not Adjacent**

Two angles or lines are said to be not adjacent to each other, if they are not near to each other.

**Nonagon**

A polygon having nine sides is called a nonagon.

**Noncollinear**

The points that do not lie in a single line are said to be noncollinear points.

**Non-Euclidean Geometry**

To understand Non-Euclidean geometry, we need to understand the parallel postulate. This postulate states that for a given point, say P, and a line l not passing through P, there is exactly one line that passes through P, which is parallel to l. The Non-Euclidean Geometry, thus refers to that branch of geometry that does not obey the parallel postulate principle. The hyperobolic geometry and elliptic geometry fall in the class of Non-Euclidean Geometry.

**Nonnegative**

Any quantity that is not less than zero is referred as nonnegative.

**Nonnegative Numbers**

The set of integers starting from 0 to infinity in the positive direction of the X-axis is referred to as whole numbers.

**Non-overlapping sets**

Two sets of numbers which do not have a single element in common are called non-overlapping sets.

**Non real number**

Any complex number of the form a + bi, where b is not equal to 0 is called a non real number. In other words, any number with an imaginary part is called non real number.

**Nonsingular Matrix**

Nonsingular matrix is also called Invertible Matrix. Any square matrix, whose determinant is not 0 is called a nonsingular matrix.

**Nontrivial**

The solution of an equation is said to be nontrivial, if the solution does not include zeroes.

**Nonzero**

Any positive or negative number is a nonzero number.

**Normalizing a vector**

The process of finding out a unit vector parallel to the given vector and of unit magnitude is called normalization of the vector. The process is carried out by dividing the vector with its magnitude.

**n th derivative**

The process of taking the derivative of a function n times is called nth derivative. If the derivative of f(x) is taken n times, then its nth derivative will be represented as f^{n}(x).

**n th Partial Sum**

The sum of the first n terms in an infinite series is called the nth partial sum.

**n th Root**

The n^{th} root of a number is the number, which when multiplied with itself n times gives the number in question. The n^{th} root of 5 can be represented as 5^{1/n}.

**Null Set**

Any set with no elements in it is called a null set.

**Number Line**

A line representing all real numbers is called the number line.

**Numerator**

The top part of any fraction is called the numerator. In case of integers, the number itself is the numerator, as it is divided by 1.

**Oblate Spheroid**

If we revolve an ellipse about its minor axis, then the 3 dimensional sphere obtained will be of the shape called oblate spheroid. Earth is an example of oblate spheroid.

**Oblique**

A line or a plane that is neither horizontal nor vertical, but is tilted at some specific angle is called oblique.

**Oblique Cone**

An oblique cone is a cone in which the center of the base of the oblique cone is not aligned (not in line) with the center of the apex of the cone.

**Oblique Cylinder**

If the bases of the cylinder are not aligned just one above the other, it is called the oblique cylinder.

**Oblique Prism**

A prism whose bases are not aligned directly one above the other is called oblique prism.

**Obtuse Angle**

An angle whose measure is more than 90º but less than 180º.

**Obtuse Triangle**

If one of the angles of a triangle is an obtuse angle, then it is called the obtuse triangle.

**Octagon**

A polygon with 8 sides is called octagon. It may have equal or unequal sides.

**Octahedron**

Octahedron is a polyhedron with 8 faces. An octahedron appears like two square based pyramids placed on one another. All the faces of an octahedron are equilateral triangles.

**Octants**

The eight parts into which the three dimensional space is divided by the co-ordinate axis.

**Odd/Even Identities**

Trigonometric identities show whether each trigonometric function is an odd or even function.

For example:

sin(-x) = sinx

cos(-x) = cosx

tan(-x) = tanx

csc(-x) = -cscx

sec(-x) = secx

tan(-x) = tanx

cot(-x) = -cotx

**Odd Function**

If the graph of a function is symmetric about x axis, then the function is said to be an odd function. Alternately, an odd function satisfies the condition, f(-x) = -f(x).

**Odd Number**

The set of integers that are not a multiple of 2. For example, {1, 3, 5, 7, 9, …)

**One Dimension**

A dimension of the space where motion can take place in only two directions, either backward or forward.

**One-Sided Limit**

Taking the limit of a term either from the left hand side or right hand side is called the one-sided limit.

**One-to-One Function**

A one-one function is type of function in which every element of the range corresponds to at least one element of the domain. A one-to-one function passes both the tests, the horizontal and vertical test.

**Open Interval**

A set interval excluding the initial and final numbers of the domain. For example, in the interval of (2, 5), 2, and 5 are the excluded from the set of numbers while performing any operation.

**Operations on Functions**

The operations on functions are as follows:

Addition: (f +g)(x) = f(x) + g(x)

Subtraction: (f – g) = f(x) – g(x)

Multiplication: (fg)(x) = f(x). g(x)

Division: (f/g)(x) = f(x)/g(x)

**Order of a Differential Equation**

The power on the highest derivative of a differential equation is called the order of differential equation.

**Ordered Pair**

Two numbers written in the form (x,y) are called the ordered pairs.

**Ordinal Numbers**

The numerical words that indicate order. The ordinal numbers are first, second, third, etc.

**Ordinary Differential Equation**

A differential equation free of partial derivative terms.

**Ordinate**

The y coordinate of a point is usually called the ordinate. For example, if P is a point (5,8) then the ordinate is the 8.

**Origin**

The reference point of any graph indicated by (0,0) in 2-D and (0,0,0) in 3-D.

**Orthocenter**

The point of intersection of three altitudes of a triangle is called orthocenter.

**Orthogonal**

Orthogonal means making an angle of 90º

**Outcome**

The result of an experiment, like throwing a dice or taking out a pack of cards from a set of cards.

**Overdetermined System of Equations**

An equation in which there are more equations than the number of variables involved.

**Pi**

Pi is defined as the ratio of circumference of a circle to its diameter. It is represented by the Greek letter Π. Many great mathematicians have done pioneering work in researching on the number pi, like Archimedes, Euler, William Jones, etc., to name a few.

**Point-Slope Equation of a Line**

y – y_{1} = m (x – x_{1}) is known as the point slope equation of a line, where m is the slope of the line and (x_{1}, y_{1}) represents a point on the line.

For example, equation of a line passing through (3,4) and making an angle of 45 degrees with the positive direction of x-axis is, y – 4 = 1(x – 3), here, (x_{1}, y_{1}) = (3,4) and slope = m = tan 45° = 1.

**Polar Axis**

The x axis is known as the polar axis.

**Polar Conversion Formulas**

The rules that are required to change the rectangular coordinates into polar coordinates are known as the polar conversion formulas.

**Conversion Formulas**

Polar to rectangular- x = rcosθ , y = rsinθ

Rectangular to polar- r^{2}= x^{2} + y^{2}

Tanθ = y/x

**Polar Curves**

Spirals, lemniscates, and limacones are the curves that have equations in polar form. Such types of curves with equations in the polar form are known as the polar curves.

**Polar Integral Formula**

Polar integral formula gives the area between the graph of curve r = r(θ ) and origin and also between the rays θ= α and θ= β (where α ≤ β).

**Polygon**

A closed figure bounded by line segments. The name of the polygon describes the number of sides of a polygon. Triangle, pentagon, hexagon, etc., are the examples of polygon.

**Polygon Interior**

All the points enclosed by a polygon is called the polygon interior.

**Polynomial Facts**

An expression of the form, p(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + … + a_{2} + a_{1}x + a_{0} is called the standard polynomial equation. Examples of polynomial equations are 3x + 2y^{2} = 5 and 5x^{2}+ 3y = 3.

**Polynomial Long Division**

Polynomial long division is useful method to express a n improper rational expression as the sum of a polynomial and a proper rational expression.

**Positive Number**

A real number greater than zero is known as a positive number.

**Positive Series**

A series that consists of only positive terms.

**Postulate**

A postulate is just like an assumption that is accepted to be true without proof.

**Power**

The number or variable (called base) that is raised to the exponent is called power.

**Power Rule**

Power rule is a formula used to find the derivative of power of a variable.

**Power Series**

A series that represents a function as a polynomial and whose power goes on increasing with every term. In other words, it has no highest power of x.

Power series in x is given by:_{n=0}∑^{n=∞}a_{n}x^{n} + a_{1}x+ a_{2}x^{2} + a_{3}x^{3} + …

**Prime Numbers**

A number that has exactly two factors, 1 and the number itself is a prime number. For example, 2, 3, 5, 7, 11.

**Probability**

The likelihood of occurrence of an event is called probability. It is one of the most researched areas of mathematics. There are some basic rules of probability:

- For any event A, 0≤ P(A) ≤ 1
- P = 1 for a sure event.
- P = 0 for an impossible event
- P (not A) = 1- P(A) or P(A
^{c}) = 1 – P(A)

**Proper Fraction**

If the numerator of a fraction is less than the denominator, then the fraction is said to be proper.

**Proper Rational Expression**

A rational expression having degree of the numerator less than the degree of denominator.

**Pythagorean Theorem**

According to Pythagoras theorem, the sum of squares of the two arms or legs of a right angled triangle is equal to the sum of the square of the hypotenuse. If AB, BC, and AC are the three sides of a right-angled triangle taken in same order, then AC^{2} = AB^{2} + BC^{2} .

Read more on history of pythagorean theorem.

**Q1**

Q1 or the first quartile is the median of the data, which are less than the overall median. For example, consider a set of data, 3, 5, 7, 8, 9, 10. The median of this set of data is 7. 3 and 5 are the only numbers less than the median. The median of the numbers 3 and 5 is 4, so the 1st quartile is 4.

**Q3**

Q3 or the third quartile is the median of the data, which is more than the overall median. For example, if we consider a set of data, 2, 3, 5, 6, 8 the median is 5. Now, 6 and 8 are the numbers in this set that are greater than the overall median. These are called Q3 or third quartile.

**QED**

QED stands for quod erat demonstrandum, which means “That which has to be proven”.

**Quadrangle**

It is a polygon with four sides.

**Quadrants**

The four sections into which the x-y plane is divided by the x and y axis.

**Quadratic**

A two degree polynomial equation represented by the equation,

ax^{2} + bx + c = 0, where, a ≠ o.

**Quadratic Polynomial**

Any polynomial of degree 2.

**Quadrilateral**

A closed figure bounded by four lines.

**Quadruple**

Four times any number or a value is called quadruple.

**Quartic Polynomial**

A polynomial of degree four.

Example: ax^{4} + bx^{3} + cx^{2} + dx + e = 0

**Quintic Polynomial**

A polynomial of degree 5

a^{5} + b^{3} + c

**Quintiles**

From a set of data, the 20th and 80th percentiles are called the quintiles.

**Quintuple**

Multiplying any number by a factor of 5.

**Radian**

It is the unit of measuring angles. For example, 180º = Π radians, 45º = Π/4 radians etc,

**Radical**

The designated symbol for the square root of any mathematical entity is called radical.

**Radicand**

The mathematical quantity whose nth root is taken. It is the number under the radical symbol.

**Radius of a circle**

The distance or the measure of the line segment between center of circle and any point on the circle is called the radius of the circle.

**Range**

The limit within which set of values reside. For example, the range of the function y = x^{2} is [0, ∞] or {y|y ≥ o}

**Ratio**

The resultant quantity derived by dividing one number with the other.

**Rational Exponents**

The exponents, which are composed of rational numbers are called rational exponents.

**Rational Function**

Given two polynomials, one divided by another, the resultant is expressed as a function, then it is called rational equation.

**Rational numbers**

The set of all ratios, made up of real numbers, which do not have zero as denominator.

**Rational root theorem**

All possible roots of a polynomial are provided by the rational root theorem.

**Rationalizing Substitution**

It is a method of integration capable of transforming a fractional integrand into more than one kind of root.

**Rationalizing the Denominator**

The process of adjusting a fraction is such a way that denominator becomes a rational number.

**Ray**

A line having only one end point and extending infinitely in the other direction is called a ray.

**Real numbers**

It is a set of all numbers consisting of positive, negative, rational, square root, cube root, etc. Real numbers form the set of all the numbers on the number line.

**Reciprocal Numbers**

One divided by the given number is the reciprocal of the number.

**Rectangle**

A rectangle is a quadrilateral having all equal angles. They are equal to 90^{0}.

**Rectangle Parallelepiped**

Rectangle Parallelepiped is a polyhedron where every face is a rectangle.

**Recursive Formula**

In a series of numbers, the next expression in the series is calculated by a formula, which uses previous expressions in the same series. This is called recursive term, and the process is called recursive formula.

**Reducing a fraction**

When numerator and denominator, both have common factors, we cancel out all of them until no common factor remains.

**Regular Octahedron**

A polyhedron which has eight faces is called regular octahedron.

**Regular Polygon**

A regular polygon is one in which all angles and sides are congruent to each other.

**Regular Prism**

It is a prism in which all the face comprises regular polygons.

**Regular Pyramid**

The pyramid who’s base is made up of regular polygon is called regular pyramid.

**Regular Right Prism**

A regular right prism is one whose bases are made up of right polygons

**Right Pyramid**

Right Pyramid is a pyramid where base is a regular polygon and the apex is directly on top of the center of the base of polygon.

**Regular Tetrahedron**

Regular Tetrahedron is a pyramid where all the faces of the polygon are triangles.

**Related Rates**

The set of all the problems, where the changes in various rates are calculated by means of differentiation.

**Relation**

The ordered pair of entities, which have some distinct abstraction between them is called a relation.

**Relative Maximum**

Relative maximum is a point in the graph, which is at the highest point for that particular section.

**Relative Minimum**

Relative minimum is a point in the graph, which is at the lowest point for that particular section.

**Relative Prime**

Those numbers, which have the greatest common factors as prime numbers are called relative prime numbers.

**Remainder**

The number ,which is left over after the division as an undivided whole number is called remainder.

**Residual**

The measure of a line, which is parallel to Y axis and one end of which is touching the data point is called residual.

**Rhombus**

The parallelogram having all equal sides is called rhombus.

**Reimann Geometry**

Reimann geometry is a type of geometry where all the lines are considered non parallel, intersecting and happening on the surface of the sphere.

**Right Circular Cone**

A right circular cone is a cone whose base is a circle and any radius is making right angle to the line segment from apex of the cone to center of the circle.

**Right Circular Cylinder**

Right circular cylinder whose bases are circular.

**Regular Hexagon**

A hexagon with all sides equal to each other is called regular hexagon.

**Rose Curve**

The leaves of the curve which have complete symmetry over the center of the curve is called a rose curve.

**Rotation**

When figure is transformed according to a fixed point is called rotation (generally in same plane).

**Rounding a Number**

Without compromising the degree of accuracy to a large extent, the approximation of number to the nearest value is called rounding of the number.

**Scalene Triangle**

Scalene Triangle is a triangle, wherein, all the sides of the triangle are unequal or of different lengths.

**Scalar**

A scalar is the one with magnitude, but with no definite direction. Examples of scalars are length, temperature and mass. Mathematically, a scalar is said to be any real number or any quantity that can be measured by using a single real number.

**Solid Geometry**

Solid geometry is a term used for the surfaces and solids in space. It includes the study of spheres, cones, pyramids, cylinders, prism, polyhedra, etc. It also involves the study of related lines, shapes, points, and regions.

**Segment**

A segment constitutes all points between two given points, including those two points.

**Segment of a Circle**

Segment of a circle is any internal region of a circle, bounded by an arc or a chord.

**SAS Similarity**

SAS similarity is side-angle-side similarity. When two triangles have corresponding angles as congruent and corresponding sides with equal ratios, the triangles are similar to each other.

**SSS Congruence**

When two triangles have corresponding sides congruent, the triangles are said to be in SSS congruency.

**Semicircle**

Semicircle is a half circle, with a 180 degree arc.

**Spherical Trigonometry**

Spherical trigonometry is used for the study of triangles on the surface of any sphere. The sides of these triangles are arcs of great circles. This study is useful for navigation purposes.

**Solving Analytically**

A technique of solving a mathematics problem, by using numeric or algebraic methods. This technique does not involve the use of a graphic calculator.

**Solve Graphically**

A technique of solving a problem by using graphs and picture. Graphic calculators are used to solve a problem graphically.

**Spheroid**

Spheroid actually refers to an oblate spheroid. But, in some cases, it refers to an ellipsoid that looks more or less like a sphere.

**Takeout Angle**

The angle cut out from a piece of paper, so that the paper can be rolled into a right circular cone is called the takeout angle.

**Tan**

The trigonometric function known as the tangent function, gives the ratio of opposite and adjacent side of a triangle.

**Tan ^{-1}**

The angle that has tangent equal to 1, therefore, tan^{-1}1 = 45º. In radians tan^{-1}1 = Π/4

**Tangent Line**

A tangent line touches the curve instead of just crossing it. A tangent line can also be defined as a line that intersects the differential curve at a point.

**Tautochrone**

Tautochrone is a Greek word that means at the same time. It has a shape of cycloid hanging downwards. Its peculiar feature is that a bead sliding down the frictionless wire will always take the same time irrespective of the fact that how high or low is the release point.

**Taylor Polynomial**

The Taylor polynomial is a partial sum of Taylor series. Using the Taylor’s polynomial, a function can be approximated to a very close value, provided the function possess sufficient number of derivatives.

**Taylor Series**

Taylor series is given by: f(a) + f'(a)(x – a) + f”(a)/2(x – a)^{2} + f”'(a)/3(x – a)^{3}+ … + f^{n}(a)/n(x – a)^{n}.

**Term**

The parts of a mathematical sequence or operations separated by addition or subtraction.

**Tetrahedron**

Tetrahedron is a polyhedron with four triangular faces. It can be viewed as a pyramid with triangular base.

**Three Dimensional Coordinates**

The right-handed system of coordinates that is used to locate a point in the three-dimensional space.

**Torus**

If we revolve a circle (In 3-D) about a line that does not intersect the circle, then the surface of revolution creates a donut shaped figure called torus.

**Transpose of a Matrix**

The matrix, which is formed by turning all the rows of the matrix into columns or vice-versa.

**Transversal**

A line that cuts two or more parallel lines.

**Trapezium**

A quadrilateral with one pair of parallel sides is called trapezium.

**Triple (Scalar) Product**

Multiplication of vectors using dot product.

If a, b, and c are three vectors, then triple scalar product is a. (b x c)

**Trivial**

Trivial solutions are the simple and obvious solutions of an equation. For example, consider the equation x + 2y = 0, here x= 0, y =0 are the trivial solutions and x = 2, y = -1 are the non-trivial solutions.

**Truncated Cone or Pyramid**

A cone or pyramid whose apex is cut off by intersecting plane. If the cutting plane is parallel to the base, it is called the frustum.

**Truncated Cylinder or Prism**

A cylinder or prism that is cut by a parallel or oblique plane to the bases. The other base remains unaffected by the cutting of the base.

**Truncating a Number**

A method of approximation wherein the decimals are dropped after a certain point instead of rounding. For example, 3.45658 would be approximated to 3.4565.

**Twin Primes**

Prime numbers that have a difference of two between each other. For example, 3 and 5.

**Unbounded Set of Numbers**

Unbounded set of numbers can be defined as the set of numbers, which is not bound either by a lower bound or by an upper bound.

**Under determined System of Equations**

Under determined System of Equations is defined to be a linear system of equations, wherein the equations are comparatively less than the variables. The system might be consistent or inconsistent. This depends upon the equations in it.

**Uniform**

Uniform means same, constant, or in the same pattern.

**Undecagon**

A polygon having 11 sides is called undecagon.

**Unit Circle**

Unit circle is defined to be a circle with radius one and is centered at the origin on the x-y plane.

**Uncountable**

Uncountable is a set that has comparatively more elements than the set of integers. It is an infinite set, in which one cannot put its elements into a one-to-one correspondence with its set of integers.

**Upper Bound of a Set**

Upper bound of a set is defined to be a number, which is greater than or equal to all the elements present in a set. For instance, 4 is an upper bound of the interval [0,1], similarly 3,2 and 1 also are the upper bounds of this interval.

**u-Substitution**

u-Substitution is a method of integration, that necessarily involves the use of the chain rule in its reverse form.

**Union of Sets**

Union of sets is defined as the combination of the elements of two sets or more than that. The union is denoted by the U symbol.

**Unit Circle Trigonometry Definitions**

It is the set of all the six trigonometry functions, such as the sine, cosine, tangent, cosecant, secant, and cotangent.

**Variable**

The independent quantity in an algebraic expression is called variable.

**Varignon Parallelogram of a Quadrilateral**

The parallelogram formed by joining the midpoints of the adjacent sides of any quadrilateral.

**Vector**

A quantity drawn as an arrow that has both magnitude and direction.

**Vector Calculus**

The problems involving calculus principles (derivatives, integrals etc) of the three dimensional figures.

**Venn Diagrams**

Venn diagrams are the pictorial representation of the set operations.

**Verify a Solution**

We verify a solution by putting the obtained values of the variables and checking if those values satisfy the expression.

**Vertex**

For a triangle, the meeting end of two sides is called a vertex.

**Vertex of an Ellipse**

The points on the ellipse where the ellipse takes a sharp turn. Mathematically, vertices of an ellipse are the points that lie on the line through the foci (or the major axis)

**Vertex of a Hyperbola**

The points at which the hyperbola takes its sharpest turns. Vertices of a hyperbola are the points that lie on the line through the foci.

**Vertex of a Parabola**

The point at which the hyperbola takes a sharp turn. The vertex of a parabola lies midway between the focus and directrix.

**Vertical Angles**

Vertical angles are the opposite angles that are formed due to the intersection of two lines.

**Vertical Compression**

Vertical shrinking of a geometrical figure is called vertical compression.

**Vertical Dilation**

Enlargement of a geometrical figure vertically is called vertical dilation.

**Vertical Line Equation**

The equation x = a is called the vertical equation of line.

**Vertical Line Test**

It is used to test if a relation is a function. It is a fact that if a vertical line cuts the graph of a relation at more than one point, then the given relation is not a function.

**Vertical Reflection**

A reflection in which a plane figure is vertically flipped. For a vertical reflection, the axis of reflection is always horizontal.

**Vertical Shift**

Shifting a geometrical figure vertically is called vertical shift.

**Vertical Shrink**

Vertical shrink is the shrink in which the plane figure is distorted vertically.

**Vertical Stretch**

Stretching the dimensions of a figure by a constant factor K in the vertical direction is called vertical stretch.

**Vinculum**

The horizontal line that is used in a fraction or radical.

**Washer**

The region between two concentric circles is called washer. The radii of the two concentric circles different.

**Washer Method**

Washer method is used to determine the volume of solid of revolution.

**Weighted Average**

A type of arithmetic mean calculation in which one of the sets among the various sets of observation carries more importance than others (weight).

**Whole Numbers**

The numbers 0, 1, 2, 3, 4, 5, … , etc.

**x-intercept**

The point at which a graph intersects the x-axis.

**x-y Plane**

The plane formed by the x and y axis of the coordinate system.

**x-z Plane**

The plane formed by the x and z axis of the coordinate system.

**y-intercept**

y-intercept is defined as a point where the graph intersects the y-axis.

**y-z Plane**

y-z plane is simply defined as the plane formed by the y-axis and z-axis.

**z-intercept**

The point at which a graph intersects the z-axis.

**Zero**

Zero is a digit and plays a crucial role in mathematics. Zero is considered as a neutral number, as it is neither positive nor negative. It is also an additive identity.

**Zero Dimensions**

When we talk of zero dimensions, it means that no motion is possible without leaving that point.

**Zero Matrix**

A matrix whose every element is zero.

**Zero of a Function**

If f(x) = 0, then the value of x which gives f(x) = 0, is called zero of a function.

**Zero Slope**

Any horizontal line has a slope equal to zero. A horizontal line has same y-coordinate, so from the formula (y_{2} – y_{1})/(x_{2} – x_{1}), we get the slope equal to zero.

**Zero Vector**

A vector with no magnitude and direction is called a zero vector.

This article with just a glimpse of the massive world of mathematics. Hope this is helpful to you in your endeavor to find the definitions of math terms. You may also like to read science- glossary of science terms and scientific definitions. I would like to acknowledge and thank my colleagues Shah Newaz Alam, Vipul Lovekar and Ujwal Deshmukh for helping me in completing it. Thank you friends!