Squares, Cubes and More

And so it has been nearly two years since the last post. Talk about having a dry season!

No, it wasn’t that I had nothing to write about. (Hint: I am seriously backlogged.)

It was about the lack of motivation. The dearth of questions from readers isn’t encouraging. So, how about dropping me one of your kid’s tricky Maths problems?

But surprise, surprise. I got the inspiration for a great post early today, just past 7 am.

The Mind Wonders

My son is fascinated by squares — a number multiplied by itself is called a square. And so he’d gone on to memorize quite a few of them, mostly the single-digit numbers.

This morning, he surprised me by asking if there were such things as cubes. I told him, “Yes, it’s like squares, only that in a cube, the number in question now appears 3 times.”

This was a perfect opportunity to introduce him to the mathematical concept of “powers”, so I took it.

Early Confusion

However, the mention of “3 times” sort of confused him. For his P4 mind to wrap around the idea of squaring, I think that is commendable. Nope, I’m not trying to blow horns here, just stating my humble opinion.

I rattled on with some examples of simple cubes, like 33 and 103. Then I checked for his understanding. At times (pun intended), he would mistake cubing as a number being multiplied by 3!

Nothing to laugh about though — there’s a reason why such topics are only taught at the secondary school level.

And mind us, parents; some of us could have forgotten about such Maths lessons and may be struggling with our kid’s homework.

The (Simple) Powers That Be — Exponents in Their Own Right

Interesting, sonny boy then asked about “what’s more than cubes?” The need to talk about the fourth-power, fifth-power, etc, couldn’t be avoided. So, let’s look into it and perhaps refresh our knowledge…

As I’ve taught my boy, a square is a number multiplied by itself. To make it easier for him to visualize, I’d often said “the number appears 2 times” to him. So now he knows that 52 = 5 X 5 = 25, for example.

And that a2 = a x a.

Yeah, I know. That’s algebra. It can be confusing for the young mind.

Some time ago, there was “another” opportunity — ha, there always will be another — for me to bring up the elementary concept of using letters in place of (unknown) numbers and the idea of variables.

I remember saying this to him: “in a way, the learning of Models is an early view of how algebra works to solve problem sums.” Well, not in so many words probably — my memory isn’t as sharp as before, lol.

Back to our topic of powers.

So, 5 cubed would then be 5 X 5 X 5 = 125. “5 (raised) to the power of 3” or “5 to the third power” means exactly the same thing.

And “5 to the fourth power” is 5 X 5 X 5 X 5 = 625. It is also the square of 5 squared, as in (5 X 5) X (5 X 5) = (5 X 5)2 = 522 = 54.

It was tempting to go on, but I stopped right there before totally confusing the boy with the law of indices.

In general,

an = a x a x … x a x a
Horizontal Curly Bracket
n occurrences of a

It’s easy: the number a appears n times in the multiplication.

Note that n is a positive whole number here. Incidentally, it’s called an exponent in algebra. Or an index.

Special Cases

Now, there are exceptions. And the answers are not always intuitive.

When n = 1
This is probably not difficult:

a1 = a

In a sense, a appears once (because the exponent, n, is 1) but there is nothing to multiply.

When n = 0
What happens here? No need to scratch the head, just remember this law of indices…

a0 = 1

The condition is that a cannot be zero.

When n < 0
Is this possible? Sure it is. Negative numbers are another “curse” (in my personal view) of Maths to get at us mere mortals.

But this rule rules:

an = 1 / an

Here’s a fun explanation of negative exponents.

Never Stop Wondering

While we walked to school, I teased my son with the idea of raising a number to the power of zero. Of course, that completely stumped him — it’s to be expected.

You see, zero is a weirdo (hey, that rhymes!) in Maths that doesn’t seem to stop causing “trouble”.

But the Pandora’s box was opened. So, I told sonny that a0 = 1 when a isn’t zero. It’s funny how 30 and 70 are the same, when obviously 3 isn’t 7. That’s just the way it is with zero…

Just like zero divided by something is zero. “When you have nothing to give away, everyone (= something) has zero,” I said.

What if that something were also zero? And what about a number divided by zero? Infinity, we were told in school about the second case.

It is indeterminate for the former case — and the latter, too — as wikipedia says here.

Having Fun, As Always

With that, I quipped “having nothing to give to nobody can be troublesome” in relation to what 0 / 0 could mean. Yup, English may not be the perfect tool to explain Maths; but then, we had fun this morning!

And when you have fun learning, the knowledge sticks.

Parents, don’t you agree gaining knowledge this way is much better for your kids? Especially for a subject like Maths! Feel free to share your views below.

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