This is a small trick that I found during a slightly dull maths class one time this year. It isn’t my creation, but a small use for something that is taught routinely in maths class in school.
This is the trick: One of the things that comes up a lot in maths classes is Pythagoras’ Theorem. Using this often requires you to be adding and squaring and subtracting numbers; here is an easy shortcut for part of this process. This trick is for finding out what the difference between two squares is.
If the gap between the roots of the numbers to square is 1:
x2-y2 = x+y (assuming that x is the larger number)
152-142 = 15+14, which equals 29.
16.592-15.592 = 16.59+15.59, which equals 32.18.
If the gap between the roots is larger (call this gap ‘a’), then the formula is:
x2-y2 = a(x+y)
132-102 = 3(13+10), which equals 69.
Why does this work? because of the difference of two squares rule in algebra (I know it is fairly obvious).
x2-y2 = (x+y)(x-y)
When the gap between the numbers is 1, x-y becomes 1 and cancels out, leaving the answer to be x+y. If the gap is larger than one, you just multiply x+y by the gap, and you get the answer. You still have to perform 3 calculations, but it becomes much simpler, because instead of performing 2 large multiplications and one large addition, you only have to add two small numbers and multiply them by another small number (small relative to the first method, which is to just figure out the squares and subtract them).