Today is another triangular day! It is the fifteenth of December and 15 is the fifth triangular number.
Two weeks ago, I talked about how the triangular numbers are one of the many figurative families, which are groups of numbers whose elements form the corresponding equilateral polygon. The triangular numbers are the first of these families, because its elements form an equilateral triangle.
The next of these families were the square numbers, and they are the same square numbers you are thinking of. You can easily find a square number by doing n^2, where you square n to get the nth square number.
Today, I want to continue discussing square numbers, but in more of a fun way. But before we get to the fun part, I want to do a little algebra.
We may remember the binomial theorem from algebra. It said that:
(a + b)(a + b) = a^2 + 2ab + b^2
You also might remember the reverse of this.
(a  b)(a  b) = a^2  2ab + b^2
But what if we do one plus and one minus?
(a + b)(a  b) =
It isn't very obvious with the simplification of the other two examples. So, let's just do it out.
(a + b)(a  b)
a^2  ab + ab  b^2
a^2  b^2
(a + b)(a  b) = a^2  b^2
There is the simplification. It is commonly called the difference of two squares, and is an easy way to factor quadratic equations. For example, if you had:
f(x) = 4x^2  25
You could factor it easily into:
f(x) = (2x + 5)(2x  5)
This topic comes up in probably every Algebra II class, but nobody ever notices just a little alteration you can make to it.
Let me move to the subject I want to talk about today, which is squaring numbers. You may know that I perform a mental math stage show called Mathemagics, where I do many feats of mental math. One of the most popular of them is the one where I ask an audience member for a number and I square it in my head.
People always jump to the conclusion that it is either super easy (like a hidden calculator gimmick or an unexplainable gift) or super hard (like a complicated formula that no one would ever be able to learn without extensive training). The super easy explanations are completely inaccurate, and the super hard explanations are also fairly incorrect. They are right that I practiced a formula, but it really isn't that complicated. In fact, you are indirectly taught it in your Algebra II class.
Let's try an example, like 18^2. 18 isn't too bad to multiply by, but what number close to 18 is easier?
20. So, we will go up two to twenty and that means we have to go down two to sixteen.
20

18

16
So, the first thing we do is 20 x 16. That may sound tough, but remember that it is only 2 x 16 with a zero tacked onto the end.
20
 \
18 320
 /
16
We are almost done. All we have to do is add to that the square of whatever number we went up and down. 2^2 is four, so we do 320 + 4 to get 324. And there is the answer.
Let's try a tougher one, say 67^2. How about you try it in your head and see if you figure it out.
Okay, this is hard for a second example. 67 is close to 70, so we go up three to 70 and down three to 64.
70

67

64
Now, 70 x 64 doesn't seem any harder than 7 x 64, but you can pull it off. Rather than trying what you learned in school with right to left multiplication, you have to switch to left to right multiplication. This sounds weird, but it is the correct way to approach it mentally.
7 x 60 is 420. 7 x 4 is 28. 420 + 28, left to right, is 448. Tack on a zero to get 4480.
70
 \
67 4480
 /
64
Now what do we do? We add the square of what we went up and down, namely the square of three. 3^2 = 9, so we get:
70
 \
67 4480
 / + 9
64 4489
And there's the answer. For three digit numbers, you do the same general thing. Try 381^2. 381 is 19 away from 400, so we go up to 400 and down to 362.
400

381

362
400 x 362, left to right is 144800, and now we need to figure out 19^2.
400
 \
381 144800
 /
362
19 is one away from 20, so we go up to 20 and down to 18. 20 x 18 is 360 plus one squared is 361.
400
 \
381 144800
 / + 361
362 145161
And there is the answer. With just a couple weeks of practice, this will become second nature.
But why does this work? If you'll remember from our brief Algebra II review, it was right in our face.
a^2  b^2 = (a + b)(a  b)
We just have one tiny alteration to make. Let's add b^2 to both sides.
a^2 = (a + b)(a  b) + b^2
And there you go. The number we are squaring is a, and the number that we go up and down is b.
Lots of the stuff I put here is stuff that I say you can easily implement into school curricula, but this is probably the easiest thing to put in. All the teacher has to do is mention that if you add b^2 to both sides, you have yourself an easy squaring formula and maybe demonstrate it.
This was very abnormal compared to the other triangular days. But if you read the post of any triangular day, I would suggest reading this one.
If you'd like to see me actually squaring numbers, you can find video footage on my website ethanmath.com, or my greatest hits video which I have posted on the blog.
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