1.01 Add and subtract integers on a number line
Introduction
We've previously learned about integers . Integers are made up of positive and negative whole numbers, as well as the number 0. We also know that integers can be represented visually using a number line.
If we know how to locate integers on a number line, we can then think about how to use the number line to add and subtract with integers.
Let's use the expression 5 - 3 = 2 to represent taking 3 apples from a pile of 5 apples. That expression makes sense.
What if we needed to take away 5 more apples? Does this 3-5 make sense?
It turns out that apple arithmetic is fine for whole numbers, but it isn't good enough for all integers, which include positive and negative numbers. To understand the addition and subtraction of integers we can instead model the arithmetic using arrows on a number line. 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 an arrow going from 0 to that integer.
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.
Exploration
Use the applet below to explore how to add 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?
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 + \left(-9\right) = -5, which is the same result that we get from \left(-9\right) + 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.
Integer addition can be represented by adding an arrow to the number line. Combining the length and direction of the two arrows yields a third arrow whose length and direction are integers.
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.
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.