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How to Square 3 Digit Numbers (In Your Head)

By Edited Nov 13, 2013 0 0
Credit: http://www.uber-facts.com/wp-content/uploads/2012/11/MP900439432.jpeg

Ah, math.  Everyone loves math, right?  Well, maybe not.  I think most people lost interest in math the moment they started adding letters and lines and triangles.  They teach you many things in school, some you will use, and some they won't.  What they failed to teach, however, are some mental math skills.  What kind you ask?  Well, how about squaring a 3 digit number?    You may find that hard to believe, but it is very possible to do such math in your head.  Not only is it possible, but it is actually very easy.  You do need to know a few things before you get started.


Squaring and Basic Two Digit Multiplication

If you want to square large numbers, you must obviously know what squaring is.  Squaring is the the action of multiplying a number by itself.  2 squared is 2 x 2, which is 4.  5 squared is 5 x 5, which is 25.  12 squared is 12 x 12, which is 144.  Most of you should already know this.  The other thing you need to know is how to multiply a single digit number by a double digit number.  As an example, lets use 24 x 3.  There are two very easy ways to do this.  One is to break it down into two seperate equations, and then add them together.  (20 x 3) + (4 x 3).  60 + 12 = 72.  The other way is to factor the larger number.  24 is equal to 6 x 4.  So instead of doing 24 x 3, you could do 6 x 4 x 3.  4 x 3 = 12.  12 x 6 equals 72.  Easy, right?

Squaring A Two Digit Number

Now that you know how to square a number and do some basic multiplication, you are ready to start squaring.  lets use 24 as an example.  There are three easy steps.

Step 1 - Change 24 into two easy to work with numbers.

Step 2 - Multiply the two numbers

Step 3 - Add the squared difference from step one to get your final answer.

Let's go back to 24.  You want to get 24 to the closest multiple of ten.  The closest multiple of 10 to 24 is 20.  So, you subtract 4 from 24 to get 20.  Whatever you add/subtract from one side, you must do the oppisite to the other.  If you subtracted 4 from 24 to get 20, you must also add 4.  24 + 4 = 28.  Your two numbers are 20 and 28.  That's step one.  Step two is to multiply them.  20 x 28 is really just 28 x 2 with an extra zero added to the end.  28 x 2 = 56.  Add the zero to get 560.  You're almost done, all that remains is step 3.  Think back to earlier in the equation.  What number did you add/subtract from 24 to get some easier number?  Four.  To finish the process, you must add the square of this number to your final answer.  4 squared is 16.  Your final answer would then be 560 + 16, which is 576.  24 squared is 576.  Simple, right?

Lets do a few more, just to make sure this makes sense.  62.  The closest multiple to ten is 60.  To get 62 to 60 you must subtract 2.  You must also add 2 to get 64.  64 x 60 is just 64 x 6, which is 384.  With the zero that would be 3840.  You changed 62 by 2, so you must add 2 squared, or 4 to your answer.  3840 = 4 is 3844.  62 squared is 3844.

Lets do a hard one.  97.  The closest multiple to 97 is 100.  to get there, you must add 3.  When you do the subtraction, that makes your equation 94 x 100.  This one is easy, just add the two zeros.  it becomes 9400.  Add 3 squared, or 9 to get 9409.  Couldn't be simpler.

Three Digit Numbers

Now what about a three digit number?  Believe it or not, it's the same three steps.  136.  Step one, find the nearest multiple of 100.  In this case it is 100.  136 to 100 is a difference of 36.  136 + 36 is 172.  Your numbers are 172 and 100.  Multiply these to get 17200.  now add the number you changed by squared, which is 36.  All you must do is go back to the 2 digit multiplication.  36 is 40 x 32, which is 1280.  Add 16 to get 1296.  The last thing you do is add.  17200 + 1296 is 18496.  It's surprising how easy this is.  The only thing that stops you from doing 4, 5 and 6 digit numbers is how long the answer will be, and how many digits you need to memorize.  I may do a tutorial on memorizing long numbers easily, but for now, practice with two and three digit numbers.  Try these for yourself, and check with a calculator to see if you're right.












Good luck with your new-found talent.



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