Trick No. 4 – Squaring a Number

I read this trick in a book called What Counts? by Brian Butterworth.

To square a number, say “N”, add a small number “a” to it to make it a round number.

Then find (N+a) x (N-a) +a².

Here are some examples:

77²
= (77 + 3) x (77 – 3) + 3²  – [here our "a" is 3, because it rounds 77 to 80]
= (80) x (74) + 3² – [think "8 x 74" and then add a zero]
= 5920 + 9 – [okay, 8 x 74 isn't exactly easy, but it's easier than 77²!]
= 5929

32²
= (32 + 8) x (32 – 8) + 8²  – [here our "a" is 8, because it rounds 32 to 40]
= (40) x (24) + 8² – [think "4 x 24" and then add a zero]
= 960 + 64
= 1024

We could have done 32² this way:

32²
= (32 – 2) x (32 +2) + 2²  – [here our "a" is 2, because it rounds 32 to 30]
= (30) x (34) + 2² – [think "3 x 34" and then add a zero]
= 1020 + 4
= 1024

One more:
59²
= (59 + 1) x (59 – 1) + 1²  – [here our "a" is 1, because it rounds 59 to 60]
= (60) x (58) + 1² – [think "6 x 58" and then add a zero]
= 3480 + 1
= 3481

There are other ways to square numbers, but this is my current favourite, since I’ve just “discovered” it. Do you know a different trick for squaring a number? Leave a comment!