Name: Bruno-Pierre

Who is asking: Student Level: Secondary

Question: I noticed the other day that if you substract two consecutive squared positive numbers, you end up with the same result as if you add up the two numbers.

Ex. 5 and 6 (2 consecutive positive numbers)
52 = 25
62 = 36
36 - 25 = 11 (Substraction of the squared numbers)
5 + 6 = 11 (Sum of the numbers)

A more algebric view:
a2 - b2 = a + b where a and b are consecutive positive positive numbers. (b = a + 1)

I wondered if this rule had a name, and who discovered it. Tanx in advance.


This is a really nice observation. I expect that the person who first discovered it did so exactly as you did. By making an observation when doing some arithmetic.

Your algebraic expression is the key to seeing what is going on. Consider

(a + b)(a - b) If you expand this you get (a + b)(a - b) = a2 - ab + ab - b2 = a2 - b2 That is a2 - b2 = (a + b)(a - b) You have a and b with a - b = 1 and hence you get a2 - b2 = a + b.

The expression

a2 - b2 = (a + b)(a - b) is called the Difference of Squares. It is one of the exprressions you will see when studying factoring.

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