A Brilliant Explanation of the Distributive Property of Multiplication

Omkar Phatak Jun 3, 2019
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Multiplication is the most fundamental of mathematical operations. We discuss its distributive property in the following lines.
Of the many mathematical operations that you will learn about, one of the most important ones is multiplication. It is essential that you master the art of multiplying numbers, as it's a skill that you will need in day-to-day math calculations.
Every operation comes with its share of properties which define it to be what it is. Distributive property makes it easier for you to multiply numbers. After you have understood how this property is applied in calculation of multiplicative products, you will realize how useful it is.

Definition

Let us first deal with the semantics. Here, 'distributive' deals with the multiplication operation which is applied to a number, that is expressed as a sum of smaller numbers. The context of the word is best understood through a statement of the property itself.
"When you multiply a number with another number, which is expressed as a sum of two or more numbers, the end product, is the sum of the products of first number with all the constituent parts of the second number."
Okay, this property is not as difficult to understand, as the earlier definition might suggest. An expression of the property in terms of an equation will simplify the matter. For the variables or numbers a, b, c, and d:

a x (b + c + d) = (a x b) + (a x c) + (a x d)
So in a way, here the multiplication operation is distributed over the addition operation. So, if the first multiplicand is a small number, and the second is a large number, breaking the latter down into parts, makes the calculation simpler. That's the prime application of this property, which will be demonstrated in the following lines.
Here is an equation that displays the distributive property in a more complex and more general case.

(a + b + c) x (k + l + m) = a x (k + l + m) + b (k + l + m) + c x (k + l + m) = (a x k) + (a x l) + (a x m) + (b x k) + (b x l) + (b x m) + (c x k) + (c x l) + (c x m)
Thus, one big multiplication has been divided into a sum of smaller multiplications. In the following section, we demonstrate this property, applied to actual numbers. There is no substitute to actually solving problems, when it comes to understanding mathematics. It is the only way you can learn it. Here are some solved examples.
  • 2 x (194) = 2 x (100 + 90 + 4) = (2 x 100) + (2 x 90) + (2 x 4) = 200 + 180 + 8 = 388
  • 5 x 859 = 5 x (800 + 50 + 9) = (5 x 800) + (5 x 50) + (5 x 9) = (4000 + 250 + 45) = 4295
  • 6 x 10300156 = (6 x 10000000) + (6 x 300000) + (6 x 100) + (6 x 50) + (6 x 6) = 60000000 + 1800000 + 600 + 300 + 36 = 61800936
By breaking down the multiplicands into sums of smaller numbers, multiplying them becomes considerably easier. We suggest that you work out some multiplication examples based on the given property, which will help you understand the concept even more deeply and appreciate its utility.