# How to Square Numbers Quickly

The Math Dude: Quick & Dirty Tips to Make Math Simpler

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Scientific American presents Math Dude by Quick & Dirty Tips. Scientific American and Quick & Dirty Tips are both Macmillan companies.

After all of the fun we’ve had recounting the tales of Knot Dude, Papa Knot, seafaring pyramid builders, and the loads and loads of algebraic goodness they all used in their adventures, it’s time to shift our attention elsewhere for a while and talk about some practical and easy techniques that you can use to do simple math in your head…and to do it fast! Leading things off this week, we’re going to learn how to quickly calculate the square of any number. Keep on reading to find out how to do it!

Why Mental Math?
Before we start turning you into a mental math genius, it’s worth taking a minute or two to talk about why learning mental math is useful…and actually really, really important. In short, learning to recognize the patterns behind simple mental math problems makes you mathematically savvier. Not only will this help you impress your friends, calculate tips, make sure the cashier gives you the correct change, and lots of handy things like that, being mathematically savvy will also help ensure your future financial security.

In fact, a recent study published in the Proceedings of the National Academy of Sciences found that people who lacked the ability to solve simple math problems were much more likely to default on their home mortgage. One of the easiest and most fun ways to get comfortable with numbers and basic math is to learn mental math tricks, which, according to this study, means that learning mental math skills is also one of the best ways to make sure that you invest wisely in the future.

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1. 1. newway 07:30 AM 9/4/13

there is a faster trick if the number ends with 5.
square the last digit, e.g. 5, you get 25
then multiply the rest of the number with a number that is 1 bigger, generally: if the number can be written as
"x5" (x=1,2,...∞)
then x5^2 = x*(x+1) and 25 put together:
e.g. 105^2 = 10*(10+1) join 25 = 11025

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2. 2. babby in reply to newway 10:38 AM 9/4/13

this is simple?

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3. 3. denisosu 04:02 AM 9/7/13

The people who find this helpful are probably the people who can already multiply in their heads anyhow. A much more helpful approach (IMHO!) in most cases is to show how you can get an approximate answer with VERY simple math.

For example a number xy squared is
100 x^2 if y = 0;
a bit more than 100 x^2 if y = 1,2 or 3;
about 100 x(x+1) if y = 4,5,6 or 7,
a bit less than (x+1)^2 if y = 8 or 9.

no need for pen and paper, no need for remembering lots of numbers in your head (which is the part people find difficult) and you get the answer in about a second.

A similar principle works for multiplying any two numbers.

xy times zw is approximately

100 x(y+1)

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4. 4. denisosu 04:08 AM 9/7/13

(sorry, hit enter by mistake when I meant to hit delete :( )

xy times zw is approximately

100 x(z+1) if w >> y
100 (x+1)z if w << y
a bit more than 100 xz if both w, y are small
a bit less than 100 (x+1)(z+1) if both w, y are large.

As you practice, you start to realise that calculating the "bit", or at least estimating it, is not hard. But as a starting point, this gives you an answer +/- about 5% - how often do we need to do mental math more accurately than that?

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5. 5. bucketofsquid 05:47 PM 9/12/13

I tried this and it took much longer to just figure out which numbers to break the larger number into than it would to just do the multiplication.

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6. 6. Quinn the Eskimo 11:05 PM 11/16/13

Enter the number and x^2 button. Put the calculator back in your pocket. Works with most ^ powers, also.

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