Math

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Squaring two-digit numbers in your head [1]

If the number ends in a 5.

(example: 35^2)

  1. Multiply the first digit, by the first digit plus one
    For 35, you get 3 * 4 = 12__
  2. Join that number with 25 at the end.
    Five squared (second digit) is 25.
    35^2 = 1225.

If the number doesn't end in a 5:

(example: 42^2)

  1. Find the closest number ending in a 5 or 0.
    For 42, you'd have 40.
  2. Keep in mind the distance taken. Add that distance back to the original number in the opposite direction.
    42 - 2 was 40. In the opposite direction, 42 + 2 -> 44
  3. Multiply the numbers from 1 and 2, which should be manageable to do in your head.
    In this case, 44 * 40 = 1600 + 160 = 1760
  4. Take the square of the original distance you added.
    In this case, 2^2 = 4.
  5. Add the numbers from 3 and 4.
    1760 + 4 = 1764.

Multiply two-digit numbers in your head

Multiplying by 11 [1]

(example: 23 * 11)

  1. For any two digit number, sum the two digits.
    For 23, you get 2 + 3 = 5
  2. Put that number in between the two digits
    That gives us 253. Which is the answer.
  3. If the sum is greater than 9, add the carried 1 to the hundredths position.
    For example, with 89, you have a sum of 17. 8[17]9 becomes 979

Cube root of a number in your head

For cubes between 1 -> 99 (1 -> 970,299) [2]

Prerequisite: memorize the cubes of 1-10.

Root Cube
1 1
2 8
3 27
4 64
5 125
6 216
7 343
8 512
9 729

(example: 79,507)

  1. Find the cube (1, 8, 27, 64, 125, etc.) which shares the last digit as your number. Take the cube root..
    For example, if your number is 79,507, that shares a last digit with the cube 27. The cube root of which is 3.
  2. Now, ignore the last three digits. Find the closest cube to the leftmost remaining digits, without running over. Take the cube root.
    For 79,507, we'll ignore the last 3 digits: 79,507. That leaves us with 79. Which is closest to the cube 64 without running over. The cube root of which is 4.
  3. The answer is the number from step 2, followed by the number from step 1.
    For 79,507 we got 507 => 3, and 79 => 4. The answer is 43.


For Step 1, you don't need to think too much. Most of the numbers are the same as the last digit of their cube (1, 4, 5, 6, 9). The exceptions are 2<->8 and 3<->7, which are switched with each other.

References

  1. 1.0 1.1 Arthur Benjamin - Secrets of Mental Math. https://www.youtube.com/watch?v=1JW9BA57aR8
  2. MindYourDecisions. 12 Dec 2014. How To Calculate Cube Roots In Your Head. https://www.youtube.com/watch?v=Ds8ijPsg26g