# Rational numbers/Introduction

## Prerequisites

[edit | edit source]All four operations (+, -, *, /) for the integers

## Meaning of Rational Numbers

[edit | edit source]A * ratio*nal number is a number which is the ratio of two integers.

### Difference between rational numbers and the integers

[edit | edit source]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

[edit | edit source]Some examples of rational numbers include:

1/2

5/2

-1/12

-73/3

## Math of Rational Numbers

[edit | edit source]### Different looks

[edit | edit source]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

[edit | edit source]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:

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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

[edit | edit source]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:

3 columns | 4 columns | ||

3 columns divided into 6 pieces | 4 columns divided into 6 pieces | ||

1 remaining column divided into 6 pieces | |||

1 remaining column |

### Multiplication

[edit | edit source]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.

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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

[edit | edit source]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.