Now that we are familiar with locating integers on the number line, we can think about how to use the number line to perform arithmetic with integers. In this lesson we will focus on addition and subtraction of positive and negative 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$5−3=2 to represent taking $3$3 apples from a pile of $5$5 apples, then it doesn’t make much sense to ask how many apples are left after performing the subtraction $3-5$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. This will give us a way to move back and forth along the number line, and to make sense of expressions like $3-5$3−5.
To begin, we can imagine that for every integer on the number line there is a corresponding arrow going from $0$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.
The length of the arrow represents the magnitude (or size) of the integer, and the direction of the arrow represents the sign (positive of negative) of the integer. There is also an arrow at $0$0, but it starts and ends at $0$0, so it points in neither direction.
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$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$4+(−9)=−5, which is the same result that we get from $\left(-9\right)+4$(−9)+4.
The examples above show how we can combine positive and negative integers using addition to produce any other integer we like. That is, addition is the only operation we need to move from one integer to another. This is in contrast to moving between whole numbers, for which we needed a special operation of subtraction because there were no negative whole numbers we could add to one whole number to get a smaller whole number.
The outcome of this is that what we can reimagine subtraction as just the addition of a negative integer. For example, $3-5$3−5 is the same as $3+\left(-5\right)=-2$3+(−5)=−2, as shown below. In other words, subtracting $5$5 is the same as adding the opposite of $5$5, which we know from our lesson on the number line is $-5$−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-\left(-2\right)$7−(−2). Taking away $-2$−2 is the same as adding the opposite of $-2$−2, which we can write as $7+\left(-\left(-2\right)\right)$7+(−(−2)). Now, this number $\left(-\left(-2\right)\right)$(−(−2)) is “the opposite of the opposite of $2$2”, which we know is just $2$2. So we have $7-\left(-2\right)=7+2$7−(−2)=7+2, which gives $9$9 from our now familiar addition of arrows.
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$0? What is the result of adding $0$0 to any other integer?
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.
In an expression like $3+\left(-5\right)$3+(−5), the $+$+ and $-$− are adjacent signs which are combined into one subtraction operation to give $3-5$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-\left(-5\right)$3−(−5) would best be read as "$3$3 subtract negative $5$5", or "$3$3 take away negative $5$5", or even "$3$3 minus negative $5$5". We would like to avoid reading it as "$3$3 minus minus $5$5", since "minus" is an operation, not the sign of the integer $-5$−5.
Find the value of $11+\left(-6\right)$11+(−6).
Think: Adding a negative integer to $11$11 means we will move to the left on the number line from $11$11.
Do: The integer $11$11 is eleven units to the right of $0$0. From here we want to move $6$6 units to the left.
The integer we end up at is $5$5. This means that $11+\left(-6\right)=5$11+(−6)=5.
Reflect: As an alternative approach, we could have first combined the adjacent signs to simplify the expression. This would make $11+\left(-6\right)=11-6$11+(−6)=11−6. Counting down $6$6 units from $11$11 gives $5$5, as expected.
Find the value of $-8+2-\left(-7\right)$−8+2−(−7).
Think: This expression has three integers, but our approach will be no different from dealing with two integers. First we can combine the adjacent signs, then use the number line to evaluate the resulting expression.
Do: Subtracting a number is the same as adding the opposite of that number. So we can combine the adjacent signs by writing
Adding the opposite of $-7$−7
The opposite of the opposite of an integer is that same integer
Now we have a much simpler addition. We can evaluate this expression by moving left to right, in the same direction that we read a sentence. So we can first evaluate $-8+2$−8+2, then add $7$7 to the result.
The number line below shows that $-8+2=-6$−8+2=−6.
Now we want to evaluate $-6+7$−6+7. Counting up $7$7 units from $-6$−6 gets us to $1$1. The whole evaluation can be performed on one number line, as shown below.
Reflect: The order of addition does not matter, so we could have started by summing the last two integers, then adding the result to the first integer. Here is the evaluation in steps:
Combining adjacent signs
Performing $2+7$2+7 first
Performing the final sum
Can you think of other ways we could rearrange the expression to make the evaluation easier?
Find the value of $-7+15$−7+15.
Find the value of $2-\left(-3\right)$2−(−3).
Find the value of $-12+16+\left(-18\right)$−12+16+(−18).
Apply an understanding of integers to describe location, direction, amount, and changes in any of these, in various contexts.