Significant figures are a method to express the certainty regarding existing data and its related calculations. This ScienceStruck article will help you in understanding the rules for identifying significant figures with examples.
All measurements have a limited precision, and therefore, have a limited number of significant figures. However, exact numbers, for which no measurements are made using instruments, have an infinite number of significant numbers.
The term precision refers to how closely two numbers, which have been measured individually, are in agreement. Significant figures (or digits) are those digits that explain the precision of a number. This term is crucial in each and every measurement. Zeros placed before and after do not get counted in the same manner. The number of significant figures is the number of digits assumed to be correctly measured.
This concept is very useful in science and mathematics, and identifying them requires that you learn the basic established rules. They will help you solve some significant figures practice problems. The paragraphs below will give you a detailed idea regarding the rules as well as some applications in chemistry.
List of Rules and Examples
The digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 are always significant. This is the kind of result that a device (like a ruler) provides, and for which, you have specified the significance. Also, remember that an exact number ending in zero also has a small ambiguity regarding the significance of the zero, i.e., the zero need not be necessarily significant, though they are considered so, in most of the cases. For instance, the number 52300 may have 5 or 3 significant figures.
- 123.56784 has 8 significant figures.
- 746 has 3 significant figures.
- 126.67 has 5 significant figures.
All zeros between 2 non-zero numbers are significant.
This happens when there are zeros between 2 non-zero digits. According to the 1st rule, the non-zero digits will be significant. However, if a zero is present between the two of them, it is calculated as significant too.
- 103.4509 has 7 significant figures.
- 2032.700807 has 10 significant figures.
All zeros to the right of the decimal point as well as at the end of the number (at the same time) are significant.
When it means after the point and after the number simultaneously, it means that the zeros trailing after the decimal point are significant, provided they are placed after a non-zero-number. You will understand it better after the examples below.
- 11.19000 has 7 significant figures.
- 1.100 has 4 significant figures.
- 0.001900 has 4 significant numbers.
Zeros before a non-zero digit (the 1st non-zero digit) of a number are not significant.
This is very obvious, you must have got the drift from the previous examples. Any number of zeros placed before a non-zero-number have no value (unless they are between 2 non-zero digits). They just play the role of placeholders.
- 0.00015: The zeros in bold are not significant.
- 0.0032: The zeros in bold are not significant. The zero before the decimal point is placed according to convention, even in the previous example.
- 0.00501: The zeros in bold are not significant, but according to rule 2, the zero between 5 and 1 is significant and the number has 3 significant figures.
All zeros to the left of a decimal point in a number greater than or equal to 10 are significant.
This means that if the number to the left of the decimal point is 10, 100, 102, 1004, etc., the digits are all significant.
- 10.245 has 5 significant figures.
- 100.2301 has 7 significant figures.
- 1002.24001 has 9 significant figures.
Zeros to the left of a decimal point without any non-zero digit are not significant.
In plain terms, this means that decimal numbers starting with a zero do not consider the starting zero to be significant. They are merely indicators of the decimal point.
- 0.12 has 2 significant figures.
- 0.012 has 2 significant figures.
- 0.00102 has 3 significant figures.
|Note: Always write a number in scientific notation (standard exponential), this clearly indicates the number of digits in the ‘digit’ term. This is a standard convention followed in most scientific theories.|
The Rounding Theory
- The concept of significant figures is often relevant to rounding. We have been using this since we learned numbers, and yet, we forget how and why the basic concept originated.
- Consider calculating the area of a circle. The formula is : (pi X r2), where pi = 3.142, r = radius of the circle.
- If r = 2, the area = (3.142 X 22), which gives us the value 12.568. The first obvious thing we do involuntarily is to round off this figure to 12.6. This is due to the concept of rounding.
- While calculating answers using mathematical formula, this concept is used often.
- While counting significant figures too, the final answer from which we need to calculate the number of significant figures is rounded off.
Rules for Rounding:
- If the digit to be rounded is more than 5, the last retained digit is increased by one. For example, 15.6 is rounded to 16.
- If the digit to be rounded is 5, and if any digit that it is followed by is not zero, the last remaining digit is increased by one. For example, 15.52 is rounded to 53.
- If the digit to be rounded is less than 5, the last remaining digit is left as it is. For example, 15.3 is rounded to 15 only.
- If the digit to be dropped is 5 and is followed only by zeroes, there are two conditions you need to consider. The last remaining digit is left as it is if the number is even, but is increased by one if it is odd. For example, 16.5 is rounded to 16, but 15.5 is rounded to 15.
Addition and Subtraction
- For both these operations, the result should be rounded off to the last common digit to the extreme right.
- The result should be such that it has the same number of digits as the measurement.
- The rule is thus: The number of places after the decimal point in the result should be less than or equal to the number of decimal places in every term of the result. For this, you first need to line up the numbers based on the point, then perform the operation accordingly.
- 12.561 + 1.01 will give you 12.571, which can be rounded off to 12.6 and can be considered as having 3 significant figures.
- 0.621 + 0.00231 will give you 0.62331. This will contain 5 significant digits.
- 1500 – 172.44 will give you 1327.56. Round it off to 1328, and you have 4 significant digits in this case.
Multiplication and Division
- For both these operations, the number of significant figures in the result is determined by the factor, which has the least number of significant figures.
- The rule is thus: The number of significant digits will be the same number as the number with the least significant figures.
- 12.4 X 3.14 equals 38.936, i.e., 39/39.0.
- 15.01 X 2.250 equals 33.7725, ideally 33.8.
Importance of Significant Figures in Chemistry
This concept is widely used in chemistry for measuring purposes in the laboratory. Consider three lab devices, and the relation between their measurement and significant figures.
The beaker in this figure has its smallest division as 10. This means that one can have a reading error of +/- 1. In the above image, the measurement is exactly 180 ml. If it is in between 2 figures, it could be read as 181 or 179, perhaps. This means that our final result has 3 significant figures. We know the 1st two numbers for sure, the third, could have a variation of +/- 1.
Here, the graduated cylinder has its smallest reading as 1. This means a reading error of +/- 0.1 can be considered. In the above image, the bottom of the meniscus (the curve) is between 23 and 24. It can be read as 23 or 24, or a value like 23.4 to 23.9. In this case as well, the number of significant figures is 3.
Note that a burette always has its values in the descending order from the top. In the diagram, the smallest division is 0.1. This indicates a reading error of +/- 0.01. The measurement above can be read as 20.7, but can also be read as 20.69 or 20.71. In any case, the result has 4 significant figures (without the round-off).
The number of significant figures is always related directly to the measurement. A lesser number signifies a rough estimate, while a greater number signifies an almost precise estimate. This concept holds tremendous value in the scientific and arithmetic fields, where accurate calculations are of utmost importance. A small rounding error or some other mistake can lead to a great change in the overall value. That is why you need to understand very well, the rules for identifying significant numbers and their importance.