Mental math tricks

So, you want to do math in your head, and be fast about it? It takes practice, practice, practice, and a decent amount of memorization, but it can be worth it.

Contents 1 Memorize 2 Finding Squares 2.1 Method 1 2.2 Method 2 2.3 Method 3 3 Multiplication 3.1 Method 1 3.2 Method 2

Memorize

Memorize your multiplication tables as high as you can manage--that will help. If you only memorize the table from 0 to 9, you can use these tricks.

If you want to know why these work, I may put that up later, but the proofs are trivial for the most part.

Finding Squares

Method 1

Works for x² when x between 11 and 99

Steps:

1. Take the original number, and add the second digit to it.
2. Multiply the result by the first digit (of the original).
3. Multiply the result by 10.
4. Square the second digit.
5. Add these numbers together.

Example 1: 14²

14 + 4 = 18


42 = 16
180 + 16 = 196

Example 2: 37²

37 + 7 = 44
 (because the original number began with 3, that's what's used instead of 4)

72 = 49
1320 + 49 = 1369

Method 2

Works for numbers ending in '5'

Steps:

1. Take the number preceding the 5 and multiply by itself plus one
2. \times e.g. if original number is 35, multiply
3. \times if original number is 465, multiply
4. Multiply result by 100, and add 25 (i.e. append '25' to the result)

Example 1: 35²

 or
'12'+'25'=1225

Example 2: 465²

 (use method 1 to help with method 2)

 or
'2162'+'25'=216225

Method 3

Works when number to be squared is from a known square

Steps:

1. Call the known number x²
2. Double x
3. If desired square is
   1. (x + 1)², add it to x²;
   2. (x − 1)², subtract it from x²
4. Add one to the result

Example 1:36²

We determined above that 352 = 1225

1225 + 70 = 1295
1295 + 1 = 1296

Example 2:34²

We determined above that 352 = 1225

1225 − 70 = 1155
1155 + 1 = 1156

Multiplication

Method 1

Works when x and y are between 11 and 99

Steps:

1. Multiply second digits of each number
2. "Cross multiply" numbers and add
   1. Multiply first digit of first number by second digit of second number
   2. Multiply second digit of first number by first digit of second number
   3. Add these two
3. Multiply first digits of each number
4. Combine

Example 1:

  14
× 32
----



 (the extra 1 is carried from the previous step)
'4'+'4'+'8'=448 (combine by concatenating the last digit of results from the bottom up)

Example 2:

  73
× 81
----




'59'+'1'+'3'=5913

Method 2

Works for two digit numbers when the first digits are the same and the second digits add up to 10

Steps:

1. Multiply first digit x by x + 1
2. Multiply second digits with each other
3. Multiply first result by 100, add second result

Example 1:

Example 2: