2.2 - Addition and Subtraction of Numbers

We will find it very useful to represent a number using an arrow on the number line. The tail of the arrow will be at the origin (i.e. at zero) and the head of the arrow will be at the location of the number that is being represented. Here is how we will represent the numbers −4, −1, 2 and 3:

Normally we would draw the arrows directly on the number line. But in this case we wouldn’t be able to distinguish between so many arrows, so we have drawn them slightly above each other.



Addition

To add two numbers on the number line you put their arrows end-to-end (because that’s what adding means) and look where the final arrow ends up.

For example in the picture below the arrows of length 2 and 3 represent the numbers 2 and 3. To add 2 + 3 we put the arrow for 3 at the end of the arrow for 2 and see that we end up at 5, so 2 + 3 = 5.


The next example shows that the order of the two numbers being added does not matter.
3 + 2 also equals 5. This is called the commutative property of addition.



The commutative law of addition. This law states that for any numbers a and b,
a + b = b + a
In other words the result is the same no matter which number is considered to be the “first number” and which number is considered to be the “added on” number.





The associative law of addition. If we need to add three or more numbers together then we do this by adding two at a time. The associative law of addition says that the result is the same no matter which two we choose to start with. It states that for any numbers a, b and c,
(a + b) + c = a + (b + c)



The next example shows how to add negative numbers, which are represented by arrows pointing to the left. The rule for adding arrows end-to-end remains the same. This example shows that (−1) + (−4) = −5.


The next example shows the addition of a positive number and a negative number. The rule for adding arrows end-to-end remains the same. This example shows that 2 + (−5) = −3.



Zero.   The number 0 is called the additive identity because adding it to any number a just gives back a :
a + 0 = a
For every number a there is a number −a, called the additive inverse. Added together they give zero:
a + (−a) = 0




Subtraction

There are at least two different ways of understanding subtraction.

One way is to replace the subtraction by the addition of an opposite. We will understand why this works when we look at multiplication in the next section. Here are several examples showing the method:
replace this subtraction by this addition of an opposite
2 5 = − 3 2 + (−5) = − 3
−5 3 = − 8 −5 + (−3) = − 8
−8 (−2) = − 6 −8 + 2 = − 6
Another way to view subtraction is to use the idea that we subtract to find out how different two numbers are. (This is why the result is called the difference!) Thus the difference between 5 and 2 is 3. The trouble with this method is that we have to be careful which number we subtract from which.



More advanced topics:
Addition and subtraction of expressions,
Addition and subtraction of algebraic fractions.