Introduction
Now that we are familiar with locating integers on the number line, we can think about how to use the number line to add and subtract with integers.
When we were adding and subtracting whole numbers, the result was always another whole number. In particular, the result of a subtraction was always greater than or equal to zero. If we imagine the expression 5 - 3 = 2 to represent taking 3 apples from a pile of 5 apples, then it doesn’t make much sense to ask how many apples are left after performing the subtraction 3-5.
It turns out that apple arithmetic is fine for whole numbers, but it is not good enough if we want to make full use of the integers, which include positive and negative numbers. To understand the addition and subtraction of integers we can instead use the arithmetic of arrows on the number line or integer chips (yellow for positive and red for negative). This will give us a way to move back and forth along the number line, and to make sense of expressions like 3 - 5.
Arrows on the number line - addition
To begin, we can imagine that for every integer on the number line there is a corresponding arrow going from 0 to that integer.
Exploration
Move the point on the number line below.
For a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.
The addition of integers can be represented by adding their arrows on the number line. When we combine the lengths and directions of two arrows, we get a third arrow whose length and direction corresponds to an integer.
The image below shows how 6 + 2 = 8 is represented using the addition of arrows on the number line. Can you see how the order of addition does not affect the result?
What if we want to add a negative integer? We use the same approach, the only difference being that the arrows are pointing in different directions. The image below shows that 4 + (-9) = -5, which is the same result that we get from (-9) + 4.
The examples above show how we can combine positive and negative integers using addition to produce any other integer we like.
Examples
Example 1
Find the value of -7 + 13.
Adding a positive integer means we move to the right.
Arrows on the number line - subtraction
Let's go back to our example of 3 - 5, this is actually the same as 3 + (-5) = -2, as shown below. In other words, subtracting 5 is the same as adding the opposite of 5.
Finally, we can use the idea that subtracting a number is the same as adding its opposite to make sense of the expression 7 - (-2). Taking away -2 is the same as adding the opposite of -2, which we can write as 7 + (-(-2)). Now, this number (-(-2) is “the opposite of the opposite of 2”, which we know is just 2. So we have 7 - (-2) = 7 + 2, which gives 9 from our now familiar addition of arrows.
Exploration
Use the applet below to explore how to add and subtract integers on the number line. What kind of arrows add together to give 0? What is the result of adding 0 to any other integer?
Same with addition, for a number line with the positive direction to the right, the positive integers have arrows that point to the right, and the negative integers have arrows that point to the left.
Examples
Example 2
Find the value of 3 - (-9).
Subtracting a negative integer means moving it to the right, and is the same as adding its opposite integer.
Adjacent signs
We have seen that any way we might want to move back or forth along the number line can be expressed as a sum of integers. Subtraction is just a particular kind of addition.
However, the concept of subtraction is still useful; in many cases it can make more sense to take away an integer rather than add its opposite. So it is common to see expressions involving the addition of a negative integer rewritten as the subtraction of a positive integer. This process is called combining adjacent signs.
For a number line with the positive direction to the right, we find the following results.
- Adding a positive integer means we move to the right:3 + (+5) \quad = \quad 8 \quad = \quad 3 + 5
- Adding a negative integer means we move to the left:3 + (-5) \quad = \quad -2 \quad = \quad 3 - 5
- Subtracting a positive integer means we move to the left:3 - (+5) \quad = \quad -2 \quad = \quad 3 - 5
- Subtracting a negative integer means we move to the right:3 - (-5) \quad = \quad 8 \quad = \quad 3 + 5
In an expression like 3 + (-5), the + and - are adjacent signs which are combined into one subtraction operation to give 3 - 5. This is a common way to simplify expressions. However, notice that these adjacent signs have different meanings. The first tells us the operation (either addition or subtraction), while the second tells us the sign of the integer (either positive or negative).
With this in mind, an expression like 3 - (-5) would best be read as "3 subtract negative 5", or "3 take away negative 5", or even "3 minus negative 5". We would like to avoid reading it as "3 minus minus 5", since "minus" is an operation, not the sign of the integer -5.
Examples
Example 3
Find the value of -7 + 12 + (-8).
For a number line with the positive direction to the right, we find the following results.
- Adding a positive integer means we move to the right.
- Adding a negative integer means we move to the left:.
- Subtracting a positive integer means we move to the left.
- Subtracting a negative integer means we move to the right.