MSC #22 – Using the Average: the Difference Of Two Squares(DOTS)
Our MATH-Inic Special for today is the other half of “Using the average”. This time we will start with the difference of two squares: a2 – b2 = (a + b) (a – c). According to Sri Bharati Tirthaji, 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:
ab = (a + b)2 / 4 – (a – b)2/4 = [(a + b)/2]2 – [(a – b)/2]2
In our example, we want to express 24 as a difference of two squares. 24 can be factored into 24 x 1, 12 x 2, 8 x 3 and 6 x 4. We used only the pairs, 12 x 2 and 6 x 4 because we want to use the squares of integers.
If we used 24 x 1, we would get, using “by one more than the one before”, (25/2)2 – (23/2)2 = 12.52 – 11.52 = 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 a number as as difference of two squares.
An interesting application of this is, given a number, we can form a Pythagorean triple. For example, if we are given a side 7, squaring it would give us 49 which we could factor as 49 x1. The hypotenuse then is (49 +1)/2 = 25 and the other side is (49 -1)/2 = 24. Thus 7,24,25 is a Pythagorean triple.
More interesting applications will be included in chapter 22 of our forthcoming book, 30 Master Strategies in Computing
