Page 1 (Two-Digits) |
Page 2 (Three-Digits) |
Page 3 (Four-Digits) |
Page 4 (Five-Digits) |
Page 5 (Six-Digits) |
Page 6 (Seven-Digits) |
Page 7 (Eight Digits) |

I stumbled upon this method when, by chance (as some would say), I just happened to noticed the relatively simple relationship between the digits of the numbers being squared. This said, it should be pointed out that this method is to be applied to numbers with two or more digits and has, as to date, been proven to work with numbers of up to eight digits.

First, let's review what a square is. Simply put; the square of a number is that number multiplied by itself. With single digit numbers; this is a fairly simple operation (as long as you know your tables).

Most of us should be familiar with these single-digit squares. We should have memorized them at some point during our scholarly career (back-in-the-day, for some of us); and tucked the answers away somewhere in the recesses of our mind.

They're there to pop up whenever we have some mathematical situation in our day-to-day that requires us, for whatever strange reason, to multiply a number by itself.

When it comes to multiplying double or triple digit numbers, some of us tend to stumble. I know that I have to think about it more than once if the double digit is above fifteen.

Everyone in my class, back in those days, knew that 15 X 15 = 225. No one had to pause to give it a second thought. I imagine that it's still the same. Teachers make sure that everyone has memorized what the square of any number from 15 down is. Then, for whatever reason, they don't worry about teaching us the rest.

It's no surprise, then, when we lazily turn to our calculators to find those higher-number squares. There's no need to think about it; after all, when the solution is only a few button-presses away.

With that said, if you're the type of person who finds your calculator to be the solution to your mathematical canundrums; then, you might prefer to invest your time with one of my Tween or Children's books; or one of the PEGUI Pups stories. However, if you're even just a little curious as to what this method is, then keep on reading.

I would like to invite anyone who has ever heard of this method to drop me a line. No one ever mentioned it to me during my scholarly career (and I did take plenty of math courses along the way!). So, if it's so easy (relatively speaking); then, why was it never mentioned? Is it because of those calculator gadgets that take away the fun and satisfaction of finding the solution on your own?

Who knows!

As with every good mathematical problem; it is always best to explain the solution with an example. However, since I did say that this methodology is proven for up to eight digit numbers (if you care to go higher than that; be my guest); then, I will provide examples for each (and, sometimes, two examples for numbers with an odd number of digit — you'll see why as we go along).

Say you have a two-digit number and you need to find its square. You could do a quick multiplication of the number times itself; or, you could have a little fun with it that might, just maybe, turn out to be another simple (or, simpler), solution to the problem.

Let's pick a number at random: 46.

This method is what we're all used to. It is, as the title implies, the multiplication of a number by itself.

In this particular example; as long as we know what 6 X 6 is (= 36), what 4 X 4 is (= 16), and what 4 X 6 is (= 24); we're good to go.

It does need us to be able to add, too.

Odds are, none of us encountered any problems when we verified that what is written to the left is actually the correct result of 46 X 46 (= 2,116).

When we use the new method, however; we're going to do something a little different.

We still need to know what 6 X 6 is; and, what 4 X 4 is.

Besides that, we also need to know what 6 X 8 is; what 6^{2}
is; and what 4^{2} is.

Basically, if you know your multiplication tables, you'll be fine.

We need to know them because the formula that we're going to use is:

Square of Any Two-Digit Number = 100 a^{2} +
20 ab + b^{2}

Where a is equal to the first digit (in this case 4); and b is equal to the second digit (in this case 6), as follows:

Amazingly enough, you'll see that the result is the same, 2,116!

You might have noticed that the way the formula is written out and the way
the problem is worked out are slightly different. That is because, in my mind,
I like to think of it as:

(a^{2} * 100) + (2ab * 10) + b^{2}

I think of it this way because, when I came up with this methodology, that's how I worked it out. Soon enough, I rearranged the variables to reflect the formula that is given; but, please note that it is in this way, the way in which I thought it up, in which all the problems will be worked out. Or, in other words, though the formula given will always be (relatively speaking):

100 a^{2} + 20 ab + b^{2}

The way in which it is worked out will always be (relatively speaking):

(a^{2} * 100) + (2ab * 10) + b^{2}

Now, you might be thinking: "No way that works with numbers of up to eight digits!" And, you'd be right. It doesn't; but then again, it does as long as you keep something in mind. As the numbers get bigger; the formula modifies slightly.

What modifies in the formula is what a^{2}
and 2ab are multiplied by; meaning that instead of
it being 100 and 10, respectively; it will be 10,000 and 100 up until a certain
point, and then increase to 1,000,000 and 1,000 up until another certain point.
It will go all the way up to 100,000,000 and 10,000; depending on how you choose
to solve the problem because, interestingly enough (and excluding the eight-digit numbers), you don't necessarily have
to use such large multipliers at all as these last two!

So, when do these multipliers change? And, what if there is an odd number of digits in the number (three digits, five digits, or seven digits)? Then, a and b are not so easily defined; or, are they?

Page 1 (Two-Digits) |
Page 2 (Three-Digits) |
Page 3 (Four-Digits) |
Page 4 (Five-Digits) |
Page 5 (Six-Digits) |
Page 6 (Seven-Digits) |
Page 7 (Eight Digits) |