**Algebra: Multiplying a * b Trick (Using the Difference between a and b)**

Can we find a formula to find products where two values are
an equal-distant apart

**The Values of a and b Differ by 2**

Let a and b be real numbers which differ by 2, that is b – a
= 2. Here I am assuming that b >
a.

Let n be the midpoint between a and b. That is:

n = b – 1 and

n = a + 1

Therefore:

b = n + 1

a = n - 1

Then:

a * b

= (n - 1) * (n + 1)

= n^2 - n + n – 1

= n^2 - 1

Example: 51 * 49

Notice that:

51 – 49 = 2, and

51 - 1 = 50

49 + 1 = 50

Hence:

51 * 49 = 50^2 – 1 = 2499

Can we expand this included products of a * b, where the
difference is b – a = 2 * w

**The Values of a and b Differ by 2*w**

Let’s look at a more general case.

Let b – a = 2*w

Then:

b = n + w and a = n – w

Then:

a * b

= (n – w) * (n + w)

= n^2 – n*w + n*w – w^2

= n^2 – w^2

Example: 37 * 43.

43 – 37 = 6

w = 6/2 = 3

Then:

n = 43 – 3 = 37 + 3 = 40

Then:

37 * 43 = 40^2 – 3^2 = 1600 – 9 = 1591

Try another example:
57 * 49

57 – 49 = 8

8 / 2 = 4

57 – 4 = 53, 49 + 4 = 53

Then:

57 * 49 = 53^2 – 4^2 = 2809 – 16 = 2793

In summary for a * b with b > a.

Let w = (b – a)/2 and n = a + w or n = b – w

Then a * b = n^2 – w^2

Eddie

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