Monday, July 23, 2018

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


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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1 comment:

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