Thread: Multiplication trick (x-y)(x+y)=x^2-y^2

Hopefully this trick will prove useful to a few of you. I discovered an equation which makes certain multiplications much faster.

I discovered it when I was 8 years old with a tiny bit of help from my father. I was messing around with a calculator and noticed some interesting patterns.

9*11 = 99 = 10^2-1
99*101 = 9999 = 100^2-1
999*1001 = 999999 = 1000^2-1
etc

I was the sort of kid to not merely be content with finding a pattern, I wanted to know WHY the patterns existed so I tried some other multiplications.

4*6 = 24 = 5^2-1
7*9= 63 = 8^2-1
-5*-3= 15 = -4^2-1

The rule (x+1)(x-1) = x^2-1 seemed to hold true no matter what the numbers. Then I tried (x+2)(x-2) and (x+3)(x+3):

8*12 = 96 = 10^2-4

7*13 = 91 = 10^2-9

It became obvious that the rule was (x+y)(x-y)=x^2-y^2 but I couldn't figure out why.

This is where my dad came in and showed me how to simplify equations using the FOIL method.

Firsts x^2
Outsides -xy
insides xy (outsides and insides cancel out)
Lasts -y^2

x^2-y^2

This trick has helped my mental arrhythmic tremendously over they years and hopefully it will help you too.

I notice that your equations are all odd by odd or even by even numbers. The number that you're then squaring is the halfway point between the two numbers to be multiplied.

What if you try to multiply an odd number by an even number?

Originally Posted by Daecon

I notice that your equations are all odd by odd or even by even numbers. The number that you're then squaring is the halfway point between the two numbers to be multiplied.

Yep. X = the number to square and y is the difference between the numbers and x.

Originally Posted by Daecon

What if you try to multiply an odd number by an even number?

It works with an odd and an even, it just doesn't make it easier. For example:

24 * 41 would give you x = 32.5 and y = 8.5

1056.25 - 72.25 = 984

If you wanted to use my method you'd be better off doing 24*40 then adding 24

x^2 = 32^2 = 1024
y^2 = 8^2= 64

x^2-y^2 = 960
960 + 24 = 984.

This is why algebra is so useful. You noticed the pattern, but until you show it is true in general you'd never be sure. (Your dad's proof is correct by the way.)

Actually, the reverse is often more helpful when dealing with algebraic matters (like trying to simplify equations). If you have the difference of any two squares, you can separate it because .

BTW, which can also be useful, but maybe not so often for mental arithmetic.

Edit: Also BTW, mental arithmetic tricks like this are often called number sense around where I'm from and there are a lot more of them. They can all be broken down into algebra to prove they work, but some proofs can be surprisingly complex. (Sorry, no examples off the top of my head.)

Originally Posted by MagiMaster

This is why algebra is so useful. You noticed the pattern, but until you show it is true in general you'd never be sure. (Your dad's proof is correct by the way.)

Actually, the reverse is often more helpful when dealing with algebraic matters (like trying to simplify equations). If you have the difference of any two squares, you can separate it because .

BTW, which can also be useful, but maybe not so often for mental arithmetic.

Edit: Also BTW, mental arithmetic tricks like this are often called number sense around where I'm from and there are a lot more of them. They can all be broken down into algebra to prove they work, but some proofs can be surprisingly complex. (Sorry, no examples off the top of my head.)

Great link, it actually contained my method but in somewhat simpler terms.

The products of small numbers may be calculated by using the squares of integers; for example, to calculate 13 × 17, you can remark 15 is the mean of the two factors, and think of it as (15 − 2) × (15 + 2),

i.e. 15² − 2². Knowing that 15² is 225 and 2² is 4, simple subtraction shows that 225 − 4 = 221, which is the desired product.