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Mathematical Journeys: The Unperformed Operation

Come with me on a journey of division.

I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of piles I'd made, which in this case is 4. You'd probably write that as:

32 ÷ 4 = 8

So there are 8 candies in each pile.

Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4 instead of piles of 8. But what if there were less candies than we needed piles – if there were less than 4 candies in the whole? What if, in fact, we had only one candy? There's still four of us, and we still need to share equally – I guess we're each getting less than one candy, right?

If we set up our equation the same way we did above, we'll get:

1 ÷ 4 = ?

Now hold on a minute. See that dividing sign there? That looks an awful lot like a fraction, doesn't it? If you just replaced the dots with numbers? That's because it is. The dividing sign they teach you first actually came after the invention of fractions. It's a way of indicating that the first number goes on top of the fraction and the second number goes on the bottom. But once you get into higher level math classes, that dividing sign disappears. Instead, we write division as a fraction – because that's what it is!

The reason we stop using the dividing sign is because writing division as a fraction allows you to deal with our one-M&M scenario from earlier. You simply write the division itself as a fraction, and that fraction becomes the result of the division.

So instead of writing that equation as:

1 ÷ 4 = ?

And being confused by your lack of an answer, you'd write:

1
4

And that fraction would become the answer.

What's important to remember here is that a fraction is not just a number. While it IS a number – there IS exactly one point on the number line that that fraction represents – it's also an indicator of an unperformed operation. By writing that number as a fraction, you are saying “I'm supposed to divide this number by this other number, but I don't want to do that calculation just yet.”

This might seem a bit strange, especially given that ¼ is an easy calculation to make, and that its decimal form, 0.25, is equally easy to work with. Okay, fine – I'm going to make one of your friends disappear!

Now there are only three of us fighting over that one candy. Your new fraction would be:

1
3

If you try to convert that into decimal form by dividing one by three, you'll get 0.33333333333... on into infinity. Now, I don't know about you, but I don't particularly like the idea of trying to work with a number that stretches on into infinity – my arms aren't that long! So I just won't let it out of the box – I'll keep it as a fraction as long as I possibly can, thus acknowledging the existence of another operation while refraining from performing it until I'm really ready.

You see this concept of the unperformed operation a lot once you get into higher level math concepts, particularly in the use of named constants. Take pi, for instance. Pi is a constant, described as the result of a specific calculation involving circles. No matter what dimensions you give a circle, when you perform this calculation you end up with the same number. So clearly it's important, and it makes sense that we should be able to work with it. Only one problem – it's an incredibly unwieldy number, a non-repeating, non-terminating decimal that stretches out into infinity. Working with such a number would be downright impossible unless we are willing to approximate and chop off most of the digits. So what do we do? We give it a name, assigning it to a letter of the greek alphabet and using this letter to represent the constant in full.

To make sure we are always working with the entire non-terminating number and not an approximation, we leave operations involving this number unperformed. We simply carry the symbol through the problem, attached to whatever other number it was supposed to be multiplied or divided by. Only at the very end do we ever actually perform the operation, and even then only if we need a numerical estimation. Much more frequently we simply express our answer “in terms of” this constant, leaving the symbol intact for the next mathematician to pick up and work with later.

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