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How to Square Any Two-Digit Number In Your Head

Land surveyors are constantly crunching numbers in their head.  But if you are like me, math is "work" part of surveying.

Square Any Two-Digit Number In Your Head

To square any two-digit number X, decide what number N it will take to raise or lower X to P, the nearest multiple of 10. Add the opposite of N to X to get Q, multiply P times Q, and add N^2 to the result. For instance:

67 * 67 = (67 + 3) * (67 - 3) + (3 * 3) =  70      *  64      +  9      = 4489

The hard part is 70 * 64, but if you teach yourself to ignore the zero at the end of the 70 and multiply from left to right, it sounds like this:

"Seven sixes make forty-two, times ten makes four hundred and twenty. Seven fours make twenty-eight, plus four hundred and twenty makes four hundred and forty-eight, times ten makes forty-four hundred and eighty, plus three squared--that's nine--makes forty-four hundred and eighty-nine."

 

Presto, you've figured out the answer in less time than it takes to say it. Note: don't be discouraged if you forget what number you needed to add at the end, or what number you were originally squaring; they're going to drop out of your short-term memory storage until you practice enough.

 

Here's an easier one, 52 * 52:

52 * 52 = (52 + 2) * (52 - 2) + (2 * 2) =  54      *  50      +  4      = 2704

Numbers ending in 4 and 6 are the hardest, because you've got to add 16.

86 * 86 = (86 + 4) * (86 - 4) + (4 * 4) =  90      *  82      +  16     = 7396

Numbers ending in 1 are easy:

71 * 71 = (71 + 1) * (71 - 1) + (1 * 1) =  72      *  70      +  1      = 5041

... and numbers ending in 5 are trivial:

45 * 45 = (45 + 5) * (45 - 5) + (5 * 5) =  50      *  40      +  25     = 2025

This is math that the average fourth-grader can handle ... can you?  

 

Do you have any heady-math tricks you'd like to share with the community?

 

 

This is a tip from an insightful blogger that I read pretty regularly, Kent Brewster

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Replies

  • You can also use the perfect square technique that we learned in high school that goes "square of the first term + twice the product of the first and second terms + square of the last term". If we take 67^2 for example, we need to strategize the discompositiion of the 67 say 60+7. Applying the technique (60+7)^2=60^2+2(60)(7)+7^2=3600+840+49. Redistributing to make it easier, 3600+800+89=4489.

  • GEO Ambassador

    wow... nice Jobber!  this could be helpful..

    any more math tricks anyone?

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