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Multiplying Two Integers (The Story of Maths)



I was watching The Story of Maths on Netflix. One of the episodes discussed how ancient people discovered (example below):

4*6 = (5^2) - 1

a*b = (a+1)^2 - 1, when b = a+2

The pattern here is that 4, 5, and 6 are all sequential integers. They also proved this pattern was valid for all integers. I found this interesting and made an excel sheet to check it out. The natural question was could this be extended for numbers farther apart? I found that it could. I discovered patterns for a*b when b = a+3, all the way to b = a+8. Then I realized it could all be simplified and written into an equation that seems to handle any distance:

a*b = (a+c)^2 - (d + c^2)

c = int(((b+1)-a)/2)
d = mod(b-a,2)*a

I've attached the excel sheet I made below. In it you can enter values for a and b. The displayed results are calculated by the equation as well as the verification of simply doing a*b. I found the equation can handle positive and negative numbers, as well as 0.

The point I'm at right now is trying to figure out what I have done. This might have been done 1,000's of years ago. It is also possible that it is something well known today, but twisted in a different form so that it just looks new. The closest thing I found was the Quarter Square Multiplication. Quarter Square Multiplication is an amazing method. Mine is not so amazing, but for the life of me I just want to know if it something new or very old.

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