Multiplication using difference of squares

2015-01-17

Many years ago, while lost in my thoughts, I discovered a new way to multiply a pair of numbers together in my head. It works well with certain pairs of numbers and is useless with others. Which pairs will work depends on:

Here’s my multiplication technique in words:

Example where x = 13 and y = 17:

I don’t know about you but I haven’t memorised 13 x 17, but I do know that 15

While writing all this down just now I’ve realised that my technique can be explained using something I learnt in high school algebra called the ‘difference of squares’:

Relating this to my technique and assuming x < y:

My multiplication technique is just a way of taking two numbers and fitting them into the ‘difference of squares’ format. So I didn’t actually discover anything new, I just discovered something new to me that was probably first discovered hundreds of years ago.

2015-01-17

Many years ago, while lost in my thoughts, I discovered a new way to multiply a pair of numbers together in my head. It works well with certain pairs of numbers and is useless with others. Which pairs will work depends on:

- The numbers should be positive integers.
^{(1)} - The difference between the numbers should be even.
^{(2)} - You need to be able to square their average in your head.

^{2}than the entire multiplication tables up to 30 x 30.Here’s my multiplication technique in words:

xy = (their average)

^{2}- (half their difference)^{2}Example where x = 13 and y = 17:

their average = 15

(their average)

their difference = 4

half their difference = 2

(half their difference)

xy = (their average)

xy = 225 - 4

xy = 221

13 x 17 = 221

(their average)

^{2}= 225their difference = 4

half their difference = 2

(half their difference)

^{2}= 4xy = (their average)

^{2}- (half their difference)^{2}xy = 225 - 4

xy = 221

13 x 17 = 221

I don’t know about you but I haven’t memorised 13 x 17, but I do know that 15

^{2}= 225, and I can halve 4 to 2, square it back to 4, and subtract it from 225 pretty easily in my head. Again, if you know your squares this technique is very useful. If you need a calculator to work out the squares then it’s useless to you.While writing all this down just now I’ve realised that my technique can be explained using something I learnt in high school algebra called the ‘difference of squares’:

(a - b) (a + b) = a

^{2}- b^{2}Relating this to my technique and assuming x < y:

(a - b) is x

(a + b) is y

a is the average of x and y

b is half their difference

(a + b) is y

a is the average of x and y

b is half their difference

My multiplication technique is just a way of taking two numbers and fitting them into the ‘difference of squares’ format. So I didn’t actually discover anything new, I just discovered something new to me that was probably first discovered hundreds of years ago.