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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