An easy way to square numbers without a calculator!

This article will show you a way to square numbers without a calculator, and with a little practice you should be able to do it in your head and amaze your friends and family!

But enough about you, lets talk about me.  In olden days of yore when I was but a mere lad there existed a world where most common people were required to manually do arithmetic in the most barbaric fashion, with a pencil and paper and their brains.

Then calculators were invented, and later “pocket” calculators.  The first pocket calculators were around $425 (over $2,500 in 2016 dollars!) and barely fit in your pocket.  You don’t even want to know how much the early non-pocket versions cost.  Luckily Moore’s Law worked for calculators as well as computers and after 6 years the price dropped to about $20 (or $76 in 2016 dollars). One short year later the cost dropped by one half to $10 (about $38 today).

We humans are a smart lot and when forced to do any type of manual labor we quickly look for shortcuts to lighten our load, especially when arithmetic is involved.  This is a very old example of a shortcut our ancestors used to square numbers the easy way long before the invention of the calculator:

The Easier Way:

Let’s say we want to find 93².  The first thing most people do today is whip out their smartphone and say “Siri, what is 93 squared?” and wait for her to tell them the answer.  For me that dredges up images from WallE of morbidly obese humans randomly drifting throughout the galaxy and letting computers take care of everything. But what if, horror of horrors, there was an EMP (Electromagnetic Pulse) blast and all electronics are now toast?  Or the singularity has happened and all electronic gadgets are now out to exterminate mankind? And you only have seconds to square 93 to find the numeric key that will disable a doomsday device?  That’s right, you better put pencil to paper, dust off that brain of yours, and figure it out yourself.  Here’s how you do it the easy way:

93² = (90 + 3)² = 90² + (90 x 3) + (90 x 3) + 3²

which can be simplified to

90² + 3² + 2(90 x 3)

I know that looks kind of complicated but all we’re doing is separating the numbers in the tens and one place:

93 = 90 + 3

and multiply them with the FOIL method (First, Outside, Inside, Last)

(90 x 3)(90 x 3)

It’s easy enough to square 90, just multiply 9 x 9 and add two zeros to get 8,100.  3 squared of course is 9.  Whenever you multiply by a number ending in zero you drop the zero and multiply the integers, in this case 9 x 3 = 27, and then add the zero back in to get 270.  Same thing when we go to multiply this by 2, just double 27 to get 54 and add back the zero to get 540.  So our answer is:

 8,100
         9
+  540
 8,649

Go ahead and grab your calculator and check my work, I know you’re dying to.  With a little bit of practice I think you will find you can do these in your head.

The Hard Way:

Lets try and do it the old-fashioned way:

      93
  x  93
    279
8,370
8,649

By my count that requires 4 separate multiplications (3 x 3, 3 x 9, 9 x 3, 9 x 9) and 4 additions to get your answer (and don’t forget to carry the two).  It may not look like it but the proposed easier way is easier because your working with numbers with lots of zeros, and the steps are simple enough you should be able to do most two digit numbers in your head.

Don’t take my word for it, go ahead and give it a spin for yourself and let us know what you think.

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