In this explainer, we will learn how to graph an integer on a number line, write its opposite, and derive the relationship between that integer and its opposite.
You are certainly familiar with a number line starting from zero and extending to the right (or upward) to larger numbers.
Now, let us imagine that zero is a mirror facing toward the arrow. The direction of the arrow in the mirror would be to the left. We will now draw this second number line, going from zero to the left, and we will add a negative sign in front of all numbers on the left-hand side of zero.
We know that when we move to the left on a number line, the numbers get smaller. It is still the case here; when we start from zero and move to the left, the numbers get smaller. Numbers smaller than zero are negative numbers. We will learn in another lesson what it can possibly mean for a number to be smaller than zero. Here, we just want to explore this new, extended number line with both a positive and a negative direction.
If we want to position, for instance, on the number line, we need to move from zero toward the left by 5. This number, , is the image in the mirror of number 5. They are both the same distance from zero. These pairs of numbers are called opposites.
Definition: Opposite of a Number
When two numbers are located on a number line at the same distance from zero but on either side of zero, we say that they are opposites. One number is the opposite of the other and vice versa.
For instance, the opposite of is 17, and is the opposite of 5. If we place these pairs of opposite numbers on a number line, we see that, for each pair,
- both numbers are on either side of zero, so their signs are different;
- the distances between zero and both numbers are equal, which means that zero is halfway between a number and its opposite.
Let us look at a first question to check our understanding.
Example 1: Finding the Opposite of a Negative Number
What number is the opposite of negative nine?
Answer
Negative nine is a negative number; therefore, it is located on the left-hand side of zero on a number line, at a distance of nine from zero. Its opposite is on the right-hand side of zero, located at the same distance, that is, nine. Hence, the opposite of is , written simply 9.
Example 2: Finding the Distance between Zero and the Opposite of a Number
Answer the questions about the points on the number line.
- What distance away from zero is the selected number?
- If you start at the selected number and move toward zero, do the numbers get smaller or larger?
- Smaller
- Larger
- If you start at the number that is opposite the selected number and move toward zero,
do the numbers get smaller or larger?
- Larger
- Smaller
Answer
We can count the “jumps” from zero to ; we find there are 6 of them, meaning that the distance between zero and is 6.
When we move from toward zero, we move to the right, so the numbers get larger.
The opposite of is the number located at a distance 6 from zero but on the other side of zero, so it is 6. When we start from 6 and move toward zero, we move to the left, so the numbers get smaller.
Example 3: Finding the Opposite of Zero
What is the opposite of zero?
- It does not have one.
- Zero
- Any number
Answer
To find the opposite of a number, we need to identify its distance from zero and on which side of zero it lies on the number line. Then, the opposite of this number is then the number at the same distance from zero on the other side of zero. Here, we want to find the opposite of zero, which is a very special case. Its distance from zero is zero because the number is zero.
Therefore, the number at the same distance from zero as zero and on the other side of zero is zero itself.
If we imagine that a mirror is located at zero, looking toward the positive number line, then the negative number line is the image of the positive number line in the mirror. As zero is on the mirror, its image is itself.
Hence, the opposite of zero is zero.
We have found here an important result.
Opposite of Zero
The opposite of zero is zero.
Let us now look at some other questions to deepen our understanding of a number and its opposite.
Example 4: Finding the Opposite of a Number by Writing a Negative Sign in front of It
The integer is the opposite of 7 because they are the same distance from 0 on a number line, but in opposite directions. Farida wrote that the opposite of is , what number is this?
Answer
Here, Farida uses the property that to change the sign of a number, we can write a negative sign in front of it. It means that to find the opposite of a number, we can simply write a negative sign in front of it. We see that the opposite of 7 is indeed . What is the opposite of written as ? The opposite of is a number located at a distance of 7 from 0 on the positive side, so it is 7. Therefore, we have
Example 5: The Opposite of the Opposite of a Number
A class were using number lines to find the opposite of a number by looking at the distance from zero. After doing some examples, a student said “the opposite of the opposite of a number will always be negative.” Is this true or false?
Answer
We can prove this is false by finding an example where it is false. Let us find the opposite of the opposite of 8. The opposite of 8 is , and the opposite of is 8. It is positive. So what the student said is false.
The opposite of a number is the number located at the same distance from zero but on the other side of zero. If we want to find the opposite of the opposite a number, we need to change side again, but still at the same distance from zero. It means that it is the original number. Therefore, it can be any number, not only negative numbers.
Key Points
- When two numbers are located on a number line at the same distance from zero but on either side of zero, we say that one number is the opposite of the other and vice versa.
- Zero is halfway between a number and its opposite.
- The opposite of zero is zero.
- The opposite of the opposite of a number is the original number.
