Fact-orials: Fractions and Decimals

Thursday, November 15, 2018

Fractions and Decimals

Background

When I look at a number line, there's a lot of space between the numbers. Say like between 0 and 1. What goes in there?

Question

How many numbers are there between 0 and 1? What are some examples?

Answer

There is an infinite number of numbers between 0 and 1. One type of number that is in there are Fractions and Decimals, such as and 0.2

Analysis

As mathematical operations go, addition and multiplication are pretty straightforward - you take two (or more) numbers, you do the operation, and they increase in a nice orderly way.

2 + 3 = 5, 2 x 3 = 6

See?

But put in their inverses and things get weird. Replace addition with subtraction and suddenly we need a whole new class of numbers (the negative numbers, and by extension, the integers) to deal with the result:

2 - 3 = -1

So what happens if we replace multiplication with division?

Well... we end up needing a new class of numbers - Fractions, and by extension, the Rational Numbers - to deal with the result.

Let's develop the idea of what a fraction is exactly. To do that, let's start with a number line:

There's all that space between 0 and 1, right? Well... what if we were to take that space and divide it into two. Like put a dot in the space equally between 0 and 1:

See that red dot in between the black dots at 0 and 1? Good! Ok - so now let's talk about moving from the number 0 to the number 1. When we were doing this with whole numbers and addition, I would start at 0, then jump 1 space to the right, and end up at 1:

0 + 1 = 1

But this time I want to land on that red dot first before getting to 1. So I have to do two jumps. It'll look like this:

Ok - it takes 2 jumps to go from 0 to 1.

On the first jump, I've done 1 out of 2 jumps.
On the second jump, I've done 2 out of 2 jumps.

To express that I've done 1 out of 2 jumps, I can write that using a fraction - which simply means I'm going to write a division expression:

The technical terms for these numbers are:

So before I do the first jump, I can say that I haven't done any jumps yet, but need to do two to get from 0 to 1:

And we know, because we haven't moved yet, that 0 divided by 2 is 0:

Now let's do the first jump. We've done 1 jump out of the 2 we need to do:

We can say we're at point "1 out of 2". Another way to say it is to say we're "half-way".

And lastly, we do the 2nd jump:

And we know we're now at 1. We know that because we can see it on the number line, but also because anything divided by itself is 1:

So that's what a fraction is - it's an integer divided by a non-zero integer (remember from division that we can't divide by 0). We can any integer as a fraction (like we did with 0 and 1 above), and in addition we can make all sorts of other fractions, like we did with the one-half above).

Now let's talk about decimals.

Let's first look at putting 9 dots evenly between 0 and 1, so that it takes 10 jumps to get from 0 to 1:

Now let's say we do 3 jumps. If we express this is terms of a fraction, we've moved 3 out of 10 spots, so that is

.

However, we can handle this a different way. Remember how we handled, in a base 10 system, the placeholders as we moved up from ones to tens to hundreds and beyond:

There are 10 ones to make a 10.
There are 10 tens to make a 100.
And so on.

We can also go the other way - we can get 10 somethings to make a 1. And those things are called tenths.

To indicate a separation between tenths and ones, we put in a decimal point (which looks like a period) between them (I should note that this use of the decimal point is used in Western systems. There are other systems used in the world that uses a comma to separate the tenths from the ones):

We can also extend the decimals further. For instance, there are 10 hundredths to each tenth:

and so on.

And so we can say that

Vocabulary used:

Integer - the set of numbers that goes ..., -3, -2. -1, 0, 1, 2, 3,...
Inverse relation - two relations or operations that when done are opposites. For instance, addition and subtraction are inverse relations: 2 + 3 - 3 = 2

For more information check out these links (comment to add your favourite link):

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