# 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**: **a ^{2} – b^{2} = (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} = (a^{2} + 2ab + b^{2}) – (a^{2} – 2ab + b^{2}) = 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**; **7 ^{2} – 5^{2} = 24**

For **6 x 4**, we have **(6 + 4)/2 = 5** and **(6 – 4)/2 = 1; 5 ^{2} – 1^{2} = 24**

^{ }

If we used **24 x 1**, we would get **(25/2) ^{2} – (23/2)^{2} = 12.5^{2} – 11.5^{2}** . We can simplify the computation by using

**By One More Than the One Before**:

**12(13) – 11(12) = 2(12) = 24**. (Note that

**12.5**and

^{2}= 156.25**11.5**; the decimal parts will cancel each other out).

^{2}= 132.25On the other hand, if we used **8 x 3**, we would have **(11/2) ^{2} – (5/2)^{2} = 5.5^{2} – 2.5^{2} = 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.