In this explainer, we will learn exactly how to graph an integer on a number line, compose its opposite, and derive the relationship in between that integer and also its opposite.
You are watching: Graph the integer and its opposite
You are absolutely familiar with a number line starting from zero and also extending to the appropriate (or upward) to bigger numbers.
Now, let us imagine that zero is a mirror dealing with toward the arrow. The direction the the arrow in the mirror would be to the left. We will certainly now attract this second number line, going native zero come the left, and we will include a an unfavorable sign in former of all numbers ~ above the left-hand side of zero.
We know that once we move to the left ~ above a number line, the numbers gain smaller. That is still the instance here; as soon as we begin from zero and also move come the left, the numbers obtain smaller. Numbers smaller sized than zero are an adverse numbers. We will find out in an additional lesson what it have the right to possibly median for a number to be smaller sized than zero. Here, we simply want to check out this new, extended number line with both a positive and a an adverse direction.
If we want to position, because that instance, −5 top top the number line, we need to move from zero towards the left by 5. This number, −5, is the image in the winter of number 5. They space both the exact same distance from zero. This pairs of numbers are called opposites.
Definition: opposite of a Number
When 2 numbers are situated on a number line at the very same distance native zero but on either next of zero, we say that they room opposites. One number is the opposite of the other and also vice versa.
For instance, the opposite of −17 is 17, and −5 is opposing of 5. If we location these pairs of opposite number on a number line, we watch that, for each pair,both numbers are on either next of zero, for this reason their signs are different;the distances between zero and also both numbers are equal, which way that zero is halfway in between a number and its opposite.
Example 1: detect the opposite of a an unfavorable Number
What number is the contrary of an adverse nine?
Negative ripe is a negative number; therefore, the is situated on the left-hand side of zero on a number line, in ~ a distance of ripe from zero. Its the opposite is on the right-hand next of zero, located at the very same distance, the is, nine. Hence, the contrary of −9 is +9, written simply 9.
Example 2: finding the Distance between Zero and the the contrary of a Number
Answer the questions about the points on the number line.
What distance away indigenous zero is the selected number?If you start at the selected number and also move toward zero, execute the numbers get smaller or larger?SmallerLargerIf you begin at the number that is the contrary the selected number and also move towards zero, execute the numbers get smaller or larger?LargerSmaller
We can count the “jumps” from zero come −6; we discover there room 6 of them, meaning that the distance between zero and also −6 is 6.
When we move from −6 towards zero, we relocate to the right, for this reason the numbers obtain larger.
The the opposite of −6 is the number located at a distance 6 native zero yet on the other side that zero, so that is 6. Once we start from 6 and move toward zero, we move to the left, for this reason the numbers gain smaller.
Example 3: finding the opposite of Zero
What is the opposite of zero?It walk not have actually one.ZeroAny number
To discover the the contrary of a number, we require to identify its distance from zero and on which side of zero that lies ~ above the number line. Then, the opposite of this number is then the number in ~ the same distance indigenous zero on the various other side the zero. Here, we want to discover the the opposite of zero, i beg your pardon is a very special case. Its street from zero is zero because the number is zero.
Therefore, the number in ~ the same distance native zero together zero and on the other side the zero is zero itself.
If us imagine that a mirror is located at zero, looking toward the hopeful number line, climate the an adverse number line is the picture of the hopeful number heat in the mirror. As zero is top top the mirror, its picture is itself.
Hence, the contrary of zero is zero.
We have uncovered here an important result.
Opposite of Zero
The the opposite of zero is zero.
Let us currently look at some other questions come deepen our expertise of a number and also its opposite.
Example 4: finding the the contrary of a Number by composing a an adverse Sign in former of It
The essence −7 is the contrary of 7 since they are the same distance native 0 on a number line, however in the contrary directions. Farida composed that the opposite of −7 is −(−7), what number is this?
Here, Farida supplies the building that to readjust the sign of a number, we can write a an unfavorable sign in former of it. It means that to uncover the the contrary of a number, we can simply create a an unfavorable sign in former of it. We watch that the opposite of 7 is indeed −7. What is the contrary of −7 composed as −(−7)? opposing of −7 is a number located at a street of 7 from 0 on the positive side, so it is 7. Therefore, we have actually −(−7)=7.
Example 5: the opposite of opposing of a Number
A course were making use of number currently to find the the contrary of a number by looking in ~ the street from zero. After doing part examples, a student claimed “the the opposite of the contrary of a number will constantly be negative.” Is this true or false?
We have the right to prove this is false through finding an example where the is false. Let us discover the the opposite of the opposite of 8. The opposite of 8 is −8, and the the contrary of −8 is 8. That is positive. Therefore what the student claimed is false.
The opposite of a number is the number located at the very same distance native zero but on the other side the zero. If we want to find the opposite of the opposite a number, we require to adjust side again, yet still in ~ the same distance from zero. It method that it is the initial number. Therefore, it can be any number, not only an adverse numbers.
See more: What Is The Inverse Of Natural Log Arithmic, And Trigonometric Functions