Rational numbers/Introduction

Prerequisites
All four operations (+, -, *, /) for the integers

Meaning of Rational Numbers
A rational number is a number which is the ratio of two integers.

Difference between rational numbers and the integers
You may think that the rational numbers and the integers are the same. However, what is 5/3? 5 and 3 are integers, so 5/3 must be rational number. But, the key point is that 5/3 is not an integer; 1=3/3, too few and 2=6/3, too much. If you now think that they are complete opposites now, 2 is an integer, and 2=2/1, a ratio of integers. Repeating this same logic with other integers gives the conclusion that all integers are rational numbers, but not all rational numbers are integers.

Examples of rational numbers
Some examples of rational numbers include:

1/2

5/2

-1/12

-73/3

Different looks
Rational numbers have many looks. For example, 1/2=2/4 because 2 is twice the size of 1, meaning that 2 must be split into twice the amount that 1 needs to split. This means that for example, with 1/3, you can multiply both sides by any number, say 5, giving 5/15. The reason is that you have 5 times as many things to split, meaning that they have to be split into 5 times as many pieces to cancel out.

If that doesn't make sense, here's another way to think about this. With the 1/2 example, you can divide 1 pizza into 2 pieces. 2/4 could mean dividing 2 pizzas into 4 pieces, meaning that there are two pieces per pizza. But why can 1/2 mean dividing 1 pizza into two pieces? Well, for example in the integers, 6/2 means dividing 6 pizzas into 2 pieces, or 3 pizzas per piece. It is therefore natural to extend this logic to the rational numbers.

Addition
To add rational numbers, you make the bottom part look the same. Then you add the top parts together. So with 5/3+3/2, notice that we can multiply the first number to give 10/6, and the second to give 9/6. 10/6+9/6=19/6. Here is a nice visual to understand: Notice how there are 10 columns on the left but 9 columns on the right, in total giving 10+9=19 rows. Now, notice how a block is 1/6 of a column. Observing the first row, it's clear that there is 10/6 on one side (six copies of the partial row make 10) and 9/6 on the other (six copies of the partial row make 9). It's also clear that the full row has 19/6 (six copies of the full row make 19).

Subtraction
To subtract rational numbers, you make the bottom part look the same. Then you subtract the top parts. So with 2/3-1/2, notice that we can multiply the first number to give 4/6 and the second to give 3/6. 4/6-3/6=1/6. Here's another visual:

Multiplication
To multiply rational numbers, you multiply both the top and bottom parts. For example, 2/3*1/2=(2*1)/(3*2). To help explain the visual, we'll have to explain another fact: 2/3 is 2*(1/3), 10/7=10*(1/7) etc. Lets use an integer example to explain why it's natural for this to be the case. 6/3=2*(3/3) and 8/4=2*(4/4). It is an obvious thing to extend into the set of rational numbers. Notice how each row is one third of the block. The non-shaded area is two rows or 2*(1/3) of the whole block. As per the logic above, this means that 2/3 of the block is unshaded. Now, the dashed area is 1/2 of the 2/3, or 1/2*2/3. Notice how it's 2*1 mini-blocks out of 3*2 mini-blocks, or (2*1)*(1/(3*2). As per the logic above, this means that (2*1)/(3*2) of the block has dashes in its entries, exactly what we wanted to prove.

Division
To divide rational numbers, you turn it into a multiplication problem. To see what I mean, let's use the example of (2/3)/(4/5). We flip the top and bottom part of the second rational number, giving this: (2/3)*(4/5), which can then be solved normally. But why, one may ask. Well, let's use an example in the integers to show why this is a natural thing to do: (8/1)/(4/1)=(8/1)*(1/4). Using the rules we learnt in the multiplication section, it simplifies to (8*1)/(1*4) or 8/4.