PostTwiceDaily2 03/31/2010 (p.m.)

• Square Large Numbers in Your Head Quickly - Shortcuts - Lifehacker

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  • Observe that 27² = 30 x 24 + 3². Since you probably know that 3² = 9 this means you have to calculate 30 x 24 + 9 which is relatively easy because the multiplication involves a multiple of ten which means it's really 3 x 24 and then add a zero.

    
    

    So the rule is that if you want to square number X you first round it to the nearest multiple of 10, called that X + r, and then calculate X - r (i.e. round the same amount in the opposite direction). You calculate (X + r) x (X - r) and add back the square of the amount you rounded by, r², which will be 1, 4, 9, 16 or 25.

    
    

    This works because ( X + r ) x ( X - r ) + r² = X² - rX + rX - r² + r² = X².

  • Here's my trick for squaring two digit numbers. Let's take 37 as an example.
    
    
    
    Square the last number (7) and write down the last number. 7^2 = 49, so write down the 9 (and carry the 4)
    
    
    
    Multiply all the digits together (37^2). 3*7*2 = 42. Add the 4 that you carried over from step 1 and you get 46. Write down the 6 (and carry the 4 again). So far, you have "69" written down.
    
    
    
    Now, square the first digit (3) and add the "carry over" from the last step. 3^2 +4 = 13. Write that down so you now have 1369.
    
    
    
    37^2 = 1369
  • When dealing with squares and numbers around 100, there is an even easier way to do this:
    
    
    
    95 x 95 = 9025
    
    
    
    : 95 is 5 away from 100
    
    : 95 - 5 = 90 (the first two digits of the answer)
    
    : 5 x 5 = 25 (the second two digits of the answer)
    
    : Answer = 9025
    
    
    
    Another example: 92 x 92
    
    
    
    : 92 is 8 away from 100
    
    : 92 - 8 = 84 (the first two digits of the answer)
    
    : 8 x 8 = 64 (the second two digits of the answer)
    
    : Answer = 8464
    
    
    
    Just subtract below 100 and add above 100:
    
    
    
    Another example: 102 x 102
    
    
    
    : 102 is 2 away from 100 (so add)
    
    : 102 + 2 = 104 (the first three digits of the answer)
    
    : 2 x 2 = 04 (the last two digits of the answer)
    
    : Answer = 10404
  • 
    
    23^2
    
    
    
    Round to nearest multiple of 10, in this case 20. To convert from 23 to 20 we need a -3 factor, so our formula:
    
    
    
    (X+factor) . (X - factor) + factor^2 is:
    
    
    
    23^2 = (23+(-3)) . (23-(-3)) + (-3)^2
    
    23^2 = 20 . 26 + 9
    
    23^2 = 2 . 10 . 26 + 9 (20 is equal to 2 . 10)
    
    23 ^2 = 2 . 26 . 10 + 9
    
    23^2 = 52 . 10 + 9
    
    23^2 = 529

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