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The same distance from zero on the other side, and why the opposite of an opposite is itself.
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In this lesson you find opposites: the number the same distance from zero on the other side. You place negative numbers, fractions and decimals and their opposites on a number line, and you learn to read a minus sign as 'the opposite of', so that the opposite of an opposite is the number itself. That small fact matters a great deal once you start adding and subtracting negative numbers.
You have used a number line with 0 at the left end and numbers growing to the right. You can place whole numbers, fractions such as $\frac{3}{4}$ and decimals such as 2.5 on it. You may also have seen negative numbers on a thermometer, where $-5$ degrees means 5 degrees below zero. In this lesson the number line runs in both directions from zero, and every number gets a partner on the other side.
| Term | What it means |
|---|---|
| Positive number | A number greater than zero, to the right of 0 on a number line, such as 7 or 2.5. |
| Negative number | A number less than zero, to the left of 0, written with a minus sign, such as $-7$ or $-2.5$. |
| Integer | A whole number or the opposite of one: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ |
| Opposite | The number the same distance from zero on the other side. The opposite of 7 is $-7$. |
| Zero | The point that separates positive from negative. It is neither, and it is its own opposite. |
| Distance from zero | How many units a number is from 0, counted without a direction. Both 7 and $-7$ are 7 units from zero. |
Draw a number line and put zero in the middle. The positive numbers go to the right and the negative numbers go to the left. Each negative number is a copy of a positive number, flipped across zero: $-4$ is 4 units to the left, just as 4 is 4 units to the right.
Two numbers like 4 and $-4$ are called opposites. They are the same distance from zero, on opposite sides. The opposite of 4 is $-4$, and the opposite of $-4$ is 4. Fractions and decimals have opposites too: the opposite of 2.5 is $-2.5$, and the opposite of $-\frac{1}{3}$ is $\frac{1}{3}$. Zero is the only number that is its own opposite, because it is zero units from itself.
The minus sign can be read as the opposite of. So $-(-4)$ means the opposite of $-4$, which is 4. Taking the opposite twice brings you back to where you started, just as turning around twice leaves you facing the same way. Written with a letter, $-(-a) = a$ for every number $a$.
One more fact ties it together: a number and its opposite always add to zero. $4 + (-4) = 0$, because 4 steps right and 4 steps left bring you back to the start.
Another way: picture
Fold a paper number line at zero. The 3 lands exactly on the $-3$, the 1.5 on the $-1.5$, and the 6 on the $-6$. Every number lands on its opposite. The fold line, zero, stays where it is.
Another way: story
Stand on a sidewalk at a lamppost and call it zero. Walking 5 steps east puts you at 5; walking 5 steps west from the lamppost puts you at $-5$. Turning around and walking back the same number of steps is taking the opposite.
On a number line that runs both ways, the marks to the left of zero are labeled $-1, -2, -3$ and so on, getting smaller as you go left. That can feel strange at first: $-10$ looks like a big number, but it is further left than $-2$, so it is smaller. $-10 < -2$, just as 10 degrees below zero is colder than 2 degrees below zero.
Every number has two pieces of information. The sign says which side of zero it is on: no sign or a plus sign means right, a minus sign means left. The digits say how far from zero it is. The opposite of a number keeps the digits and changes the sign.
A minus sign in front of a number, such as $-6$, is part of the number's name: negative six. A minus sign in front of parentheses, such as $-(-6)$, is an instruction: take the opposite of what is inside. Both readings agree, because $-6$ is also the opposite of 6.
To work out a stack of minus signs, go from the inside out, one sign at a time. $-(-(-6))$: the inside $-6$ is negative six; $-(-6)$ flips it to 6; one more sign flips it back to $-6$. A quicker check is to count the minus signs. An even number of them lands on the positive side, and an odd number lands on the negative side.
The same rule works with a letter. If $a$ stands for 9, then $-a$ is $-9$. But if $a$ stands for $-9$, then $-a$ is 9. So $-a$ is not always negative: it is the opposite of whatever $a$ is.
To place $-2\frac{1}{4}$, first decide the side: it is negative, so it is to the left of zero. Then decide the size: it is more than 2 but less than 3, so it lies between $-2$ and $-3$, a quarter of the way from $-2$ toward $-3$. A common slip is to put it between $-1$ and $-2$, or on the wrong side of $-2$. Its opposite, $2\frac{1}{4}$, is the mirror point between 2 and 3.
When the marks on a line are halves or quarters, count marks instead of units. On a line marked in halves, 2.5 is 5 marks from zero, so its opposite is 5 marks on the other side.
On the right side of zero, 3 is less than 8. Now take the opposites of both: $-3$ and $-8$. On the left side, $-8$ is further from zero, so it is further left, and $-8 < -3$. Taking opposites turned the order around.
This always happens. The fold at zero is a mirror, and a mirror swaps left and right. If one positive number is to the left of another, their opposites are the other way around. So to order negative numbers, you can order their opposites first and then reverse the list. The numbers 2, 5 and 9 are in order from least to greatest, so $-9, -5, -2$ are also in order from least to greatest.
Zero sits between every number and its opposite. So a number and its opposite are never equal unless the number is zero, and the positive one of the pair is always the greater.
Many quantities come in two directions, and zero means a starting point or a level. Opposite numbers describe the two directions.
In each case the opposite of a change undoes it, which is the real-world meaning of $a + (-a) = 0$.
To find and place the opposite of any number:
Step 1: Read the side. A minus sign means left of zero; no sign means right. For a stack of minus signs, work from the inside out first.
Step 2: Read the distance. The digits give the number of units from zero, including any fraction or decimal part.
Step 3: Switch sides. Keep the distance and move to the other side of zero. Write the new number with the sign changed.
Step 4: Place it. Count the same number of units, or marks, on the other side. For a fraction or decimal, name the two whole numbers it lies between.
Step 5: Check.
If a check fails, the usual cause is a sign left unchanged or a point placed between the wrong pair of whole numbers.
Heights on maps are measured from sea level, which is zero. Places above sea level have positive elevations, and places below have negative ones.
Badwater Basin in Death Valley is the lowest point in North America, at an elevation of about $-282$ feet: 282 feet below sea level. The opposite elevation, 282 feet, is the height of a small hill. About 85 miles away is Mount Whitney, the highest peak in the lower 48 states, at about 14,505 feet.
How far would you climb from Badwater Basin to the top of Mount Whitney? Split the climb at sea level. From $-282$ up to 0 is 282 feet, the distance of $-282$ from zero. From 0 up to 14,505 is 14,505 feet. The whole climb is $282 + 14{,}505 = 14{,}787$ feet. Thinking of $-282$ as '282 feet on the other side of zero' is exactly what the idea of opposites gives you.
A bank records a deposit as a positive amount and a withdrawal as a negative one. Suppose Jordan has 120 dollars in a savings account. On Monday Jordan deposits 35 dollars, recorded as 35. On Friday Jordan takes out 35 dollars, recorded as $-35$.
These two amounts are opposites, so together they make zero: $35 + (-35) = 0$. The balance goes from 120 to 155 and back to 120. The withdrawal exactly undoes the deposit.
Opposites show up whenever one change undoes another: a 6-yard gain and a 6-yard loss in a football game, or a 5-degree rise and a 5-degree drop in temperature. In each case, the pair adds to zero and nothing has changed overall.
Thinking the opposite is always negative. The opposite of $-8$ is 8, which is positive. The opposite switches sides, whichever side you start on.
Reading $-a$ as 'a negative number'. $-a$ means the opposite of $a$. If $a$ is $-3$, then $-a$ is 3.
Leaving a double minus as negative. $-(-5)$ is 5, not $-5$. Each minus sign is one flip.
Mixing up opposite and reciprocal. The opposite of 4 is $-4$, not $\frac{1}{4}$. The reciprocal is a different idea that you will use when dividing fractions.
Placing negative fractions between the wrong whole numbers. $-3.5$ is between $-3$ and $-4$, not between $-2$ and $-3$. Its distance from zero is 3.5, so it is past $-3$.
Thinking a bigger digit means a bigger number on the left. $-9$ is less than $-2$, because it is further left.
Find the opposite of $-14$. Read its side.
$-14 < 0$
The minus sign puts $-14$ to the left of zero.
Read its distance from zero.
$-14 \text{ is } 14 \text{ units from } 0$
The digits give the distance; the sign only gives the direction.
Keep the distance and switch to the right side.
$14 \text{ units right of } 0 \text{ is } 14$
The opposite is the same distance on the other side.
Write the answer with the sign changed.
$\text{opposite of } -14 = 14, \text{ so } -(-14) = 14$
Reading the minus sign as 'the opposite of' writes the same fact in symbols.
Check by adding the pair.
$-14 + 14 = 0$
14 steps left and 14 steps right bring you back to zero.
Find the value of $-(-(-2.5))$. Start with the number inside.
$-2.5 \text{ is } 2.5 \text{ units left of } 0$
Working from the inside out handles one minus sign at a time.
Apply the next minus sign: take the opposite of $-2.5$.
$-(-2.5) = 2.5$
The opposite of a negative number is positive.
Apply the outside minus sign: take the opposite of 2.5.
$-(2.5) = -2.5$
The opposite of a positive number is negative.
Write the result.
$-(-(-2.5)) = -2.5$
Two of the flips cancel, and one is left over.
Check by counting the minus signs.
$3 \text{ signs: odd} \;\Rightarrow\; \text{negative}$
An odd number of flips leaves the number on the other side from where it began.
Place the answer on a number line.
$-3 < -2.5 < -2$
$-2.5$ is halfway between $-2$ and $-3$, to the left of zero.
A number line is marked in quarters. Place $-1\frac{3}{4}$. Read its side.
$-1\tfrac{3}{4} < 0$
The minus sign puts it to the left of zero.
Find the two whole numbers it lies between.
$-2 < -1\tfrac{3}{4} < -1$
It is more than 1 unit but less than 2 units from zero.
Change the distance into quarter marks.
$1\tfrac{3}{4} = \tfrac{7}{4} \;\Rightarrow\; 7 \text{ marks}$
Each mark is one quarter, so the distance in marks is the number of quarters.
Count 7 marks to the left of zero and mark the point.
$0 \to -\tfrac{1}{4} \to \cdots \to -\tfrac{7}{4}$
Counting marks avoids putting the point on the wrong side of $-1$ or $-2$.
Place the opposite: 7 marks to the right of zero.
$\text{opposite of } -1\tfrac{3}{4} = 1\tfrac{3}{4}$
Same distance, other side.
Find the distance between the two points in marks.
$7 + 7 = 14 \text{ marks}$
From one point to zero is 7 marks, and from zero to the other is 7 more.
Change the marks back into units and check.
$\tfrac{14}{4} = 3\tfrac{1}{2}, \qquad 1\tfrac{3}{4} + 1\tfrac{3}{4} = 3\tfrac{1}{2}$
The gap between a number and its opposite is twice its distance from zero.
Name the point.
$-4.5$
Left of zero means negative, and the distance is 4.5.
Name its opposite.
$4.5$
Keep the distance and switch to the right side of zero.
Find the distance between them.
Which pair of numbers are opposites?
Point P is 8 units to the left of zero. Point Q is the opposite of P. Point R is 5 units to the right of Q. Find each point, the opposite of R, and the distance from P to Q.
Write P as a number: it is left of zero, so it is negative.
$P =$ p
Numbers to the left of zero are negative, and the digits give the distance.
Q is the opposite of P, so jump to the other side of zero.
$Q =$ q
Same distance from zero, other side.
R is 5 units to the right of Q.
$R = Q + 5 =$ r
Moving right on a number line adds.
Take the opposite of R.
$\text{opposite of } R =$ s
R is right of zero, so its opposite is the same distance to the left.
Find the distance from P to Q through zero.
$\text{P to 0, then 0 to Q} =$ d
P and Q are each the same distance from zero, so the gap between them is two of those distances.
What is the opposite of -36, and how far is that opposite from zero?
The opposite of -36 is opp, which is dist units from zero.
A minus sign in front of a number or parentheses means 'the opposite of'. Find the value of each: −(−20), −(−(−20)) and −(−(−(−20))).
−(−20) = two, −(−(−20)) = three, −(−(−(−20))) = four
One winter morning in Duluth the temperature was -17 degrees Fahrenheit. By the afternoon it had risen to the opposite of the morning temperature. How many degrees did the temperature rise?
The afternoon temperature was high degrees Fahrenheit, so the temperature rose rise degrees.
Two numbers are opposites, and on the number line they are 10 units apart. What are the two numbers?
The positive number is pos, and its opposite is neg.
The marks on this line are half a unit apart. Move the marker to the opposite of 1.5.
-6 |——————————| 6
Mark the position with a cross, then write the value:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Two numbers are opposites, and on the number line they are 70 units apart. What are the two numbers?
The positive number is pos, and its opposite is neg.
You can find the opposite of a number and place both on a line. Without looking: what is the opposite of −7, and what is the opposite of that?
17. Your turn: a point is 4.5 units to the left of zero. Name it, name its opposite, and find the distance between them, step 3
$4.5 + 4.5 = 9 \text{ units}$
Each point is 4.5 units from zero, on opposite sides.