MSC # 22 – The Difference Of Two Squares.
Our Math Special for today is the other half of “Using the average”. This time we will start with the difference of two squares or DOTS: a2 – b2 = (a + b) (a – b). In his book, Vedic Mathematics, Sri Bharati Tirthaji wrote that given this form, a number can be readily expressed as a difference of two squares because it is very easy to express any given number as a product of two numbers. Even if a number is prime, it can be expressed as a product of itself and 1, as in 11 = 11 x 1.
Now if we start with
(a + b)2 – (a – b)2 = (a2 + 2ab + b2) – (a2 – 2ab + b2) = 4ab, we can now express ab as equal to [(a + b)2 / 4] – [(a – b)2/4] or [(a + b)/2]2 – [(a – b)/2]2
If we want to express a number as a difference of two squares, we can list down its factor pairs. For example, 24 can be factored into 24 x 1, 12 x 2, 8 x 3 and 6 x 4. If we want to use only the squares of integers, we will choose the pairs which are both even or both odd numbers: 12 x 2 and 6 x 4, in this case.
For 12 x 2, we have (12 + 2)/2 = 7 and (12 – 2)/2 = 5; 72 – 52 = 24
For 6 x 4, we have (6 + 4)/2 = 5 and (6 – 4)/2 = 1; 52 – 12 = 24
If we used 24 x 1, we would get (25/2)2 – (23/2)2 = 12.52 – 11.52 . We can simplify the computation by using By One More Than the One Before: 12(13) – 11(12) = 2(12) = 24. (Note that 12.52 = 156.25 and 11.52 = 132.25; the decimal parts will cancel each other out).
On the other hand, if we used 8 x 3, we would have (11/2)2 – (5/2)2 = 5.52 – 2.52 = 5(6) – 2(3) = 30 – 6 = 24.
And since we can also express 24 as 48 x 1/ 2, 72 x 1/ 3, 96 x 1/ 4, etc. we can have an infinite number of ways to express a number as a difference of two squares.
An interesting application of this is our example: given one side of a rectangle measuring 7, we are asked to find the measures of the two other sides.
Squaring 7 would give us 49 which we could factor as 49 x 1. The hypotenuse then is (49 +1)/2 = 25 and the other side is (49 -1)/2 = 24. Thus 7, 24 and 25 form a Pythagorean triple.
