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Distance from zero, which is why it is never negative.
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In this lesson you meet absolute value: how far a number is from zero, whatever its direction. Because it is a distance it is never negative, which is the fact people trip over. You also learn to tell an absolute value question from an ordering question. In real situations absolute value is the size of something where the sign only says which way: a debt of 40 dollars and savings of 40 dollars are the same amount of money.
You can place negative numbers on a number line and find a number's opposite: the number the same distance from zero on the other side. You also know that further left means less, so $-9 < -4$. This lesson gives a name and a symbol to the distance part of a number, the part that stays the same when you take the opposite.
| Term | What it means |
|---|---|
| Absolute value | The distance of a number from zero on the number line. It is never negative. |
| Absolute value bars | The two straight lines in $\vert -6\vert $, read 'the absolute value of negative six'. |
| Magnitude | Another word for the size of a quantity, without its direction. Absolute value measures it. |
| Balance | The amount of money in an account. A negative balance means money is owed. |
| Debt | Money owed. A balance of $-40$ dollars means a debt of 40 dollars. |
| Elevation | Height above or below sea level. Below sea level is written as a negative number. |
Every number on the number line has a direction from zero and a distance from zero. The absolute value of a number is just the distance. We write it with two straight bars:
$$|-6| = 6 \qquad |6| = 6 \qquad |0| = 0$$
Both $-6$ and 6 are 6 units from zero, so they have the same absolute value. A number and its opposite always do.
Because absolute value is a distance, it is never negative. You cannot walk negative 6 steps; you can only walk 6 steps in some direction. So the bars do not 'make a number positive' by magic; they measure how far it is from zero, and a distance is always zero or more.
This matters most when we talk about size in a real situation. A temperature of $-15$ degrees is 15 degrees from zero. A balance of $-30$ dollars is a debt of 30 dollars. An elevation of $-80$ feet is 80 feet below sea level. In each case the sign tells you the direction and the absolute value tells you how big.
Be careful: absolute value is not the same as order. $-15 < -3$, because $-15$ is further left. But $|-15| > |-3|$, because $-15$ is further from zero. For negative numbers, the smaller number has the bigger absolute value.
Another way: picture
Draw a number line and put your pencil on zero. Draw an arrow to $-6$ and another to 6. The arrows point in opposite directions, but they are the same length, 6 units. Absolute value measures the length of the arrow and ignores which way it points.
Another way: story
Your school is at zero on a straight road. Maya lives 6 blocks west and Carlos lives 6 blocks east. They live in opposite directions, but each walks exactly 6 blocks to school. The walk is the absolute value.
For a positive number or zero, the absolute value is the number itself: $|17| = 17$ and $|0| = 0$. For a negative number, it is the number's opposite: $|-17| = 17$. That is all there is to the arithmetic. Fractions and decimals work the same way: $|-4.75| = 4.75$ and $\left|-\frac{2}{3}\right| = \frac{2}{3}$.
When a minus sign is outside the bars, work out the bars first and then use the minus sign. $-|8|$ means the opposite of $|8|$, which is $-8$. And $-|-8|$ is also $-8$: the bars give 8, and the minus sign outside makes it $-8$. So an expression with bars can be negative, but only because of a sign outside them. The absolute value itself never is.
Treat the bars like parentheses when you calculate. $|-9| + |3| = 9 + 3 = 12$, and $|-9| - |-3| = 9 - 3 = 6$.
Why does 'drop the minus sign' work for a negative number? The digits of a number already tell you how far it is from zero, and the minus sign only tells you which side it is on. Taking the absolute value keeps the first piece of information and throws away the second. It does not change the digits at all, so $|-305| = 305$, not 503 or 35. If you ever find yourself changing a digit, stop and count the distance on a number line instead.
Two different questions can be asked about any two numbers.
For two positive numbers, the answers agree: 9 is greater than 4, and 9 is further from zero. For two negative numbers, they disagree: $-9$ is less than $-4$, yet $|-9| = 9$ is greater than $|-4| = 4$.
Real questions tell you which one to use. 'Which is colder, $-9$ or $-4$ degrees?' is about order: $-9$ is colder. 'Which temperature is further from zero?' is about absolute value, and the answer is again $-9$, now because it is 9 degrees away.
Absolute value also helps you find the distance between two points on a number line or on a grid line.
If the two numbers are on opposite sides of zero, add their absolute values. From $-7$ to 5 is $|-7| + |5| = 7 + 5 = 12$ units, because the path passes through zero: 7 units on one side and 5 on the other.
If the two numbers are on the same side of zero, subtract the smaller absolute value from the larger. From $-7$ to $-2$ is $|-7| - |-2| = 7 - 2 = 5$ units, because both are measured from zero in the same direction and the shorter distance is part of the longer one.
The same idea works on the coordinate plane. The points $(4, -6)$ and $(4, 3)$ share their first coordinate, so they are on one vertical line. The y-values are on opposite sides of zero, so the distance is $|-6| + |3| = 9$ units. A quick sketch tells you whether to add or subtract, and the answer, being a distance, is never negative.
Many situations use a sign for direction and a number for size. Absolute value pulls out the size.
The phrase to listen for is 'how much', 'how far' or 'how big'. Those questions want an absolute value, and their answers are never negative.
Step 1: Read the question. Decide whether it asks about position (greater, less, higher, colder) or about size (how far, how much, how big). Size means absolute value.
Step 2: Find each absolute value. Keep a positive number or zero as it is; for a negative number, drop the minus sign, because the opposite of a negative number is its distance from zero.
Step 3: Deal with any sign outside the bars. Work the bars first, then apply the outside sign.
Step 4: Compare or calculate. Compare the distances, or add and subtract them.
Step 5: Check.
If a check fails, reread the question: most mistakes come from answering the order question when the size question was asked, or the other way around.
In golf, each course has a par, the number of strokes a good player is expected to need. Scores are often written relative to par: $-3$ means 3 strokes under par (good), and $+2$ means 2 strokes over (not as good).
At the 2000 U.S. Open at Pebble Beach, Tiger Woods finished at $-12$, 12 strokes under par. The players tied for second place finished at $+3$. How many strokes did Woods win by? Split the gap at par, which is zero: from $-12$ up to 0 is $|-12| = 12$ strokes, and from 0 up to 3 is 3 strokes. Woods won by $12 + 3 = 15$ strokes, a record for a major championship.
Absolute value also tells you how far a score is from par in either direction. A player at $-4$ and a player at $+4$ are both 4 strokes from par, but only one of them is having a good day. The sign carries the good or bad news; the absolute value carries the size.
Mount Everest rises about 29,032 feet above sea level. The Challenger Deep, the deepest known point in the ocean, in the Mariana Trench in the Pacific, is almost 36,000 feet below sea level, so its elevation is about $-36{,}000$ feet.
Which is higher? Everest, of course: $29{,}032 > -36{,}000$. But which is further from sea level? Now compare absolute values: $|-36{,}000| = 36{,}000$ and $|29{,}032| = 29{,}032$. The Challenger Deep is further, by roughly $36{,}000 - 29{,}032 = 6{,}968$ feet, or about 7,000 feet. If Mount Everest were set on the ocean floor at the Challenger Deep, its peak would still be about 7,000 feet under water.
Both questions are about the same two numbers, but they have different answers. 'Higher' is about order; 'further from sea level' is about absolute value.
Thinking absolute value can be negative. $|-7|$ is 7, never $-7$. A distance cannot be negative.
Thinking the bars just 'make it positive' everywhere. $-|7|$ is $-7$, because the minus sign is outside the bars. Work the bars first.
Mixing up order and size. $-20$ is less than $-2$, but its absolute value is greater. A more negative number is further from zero.
Forgetting a number has a partner. Two numbers have absolute value 5: 5 and $-5$. Only zero has just one number, itself.
Misreading 'less than' with debts. A balance less than $-50$ dollars means a debt more than 50 dollars. The balance goes down as the debt goes up.
Adding a sign to a distance. The answer to 'how many feet below sea level' is 60, not $-60$. The words 'below sea level' already carry the direction.
Find $|-23|$. Read the bars.
$|-23| = \text{distance of } -23 \text{ from } 0$
Absolute value means distance from zero.
Count the distance from $-23$ to zero.
$-23 \to 0: \; 23 \text{ units}$
$-23$ is 23 units to the left of zero.
Write the absolute value.
$|-23| = 23$
The distance is the answer; the direction is dropped.
Now find $|23|$ the same way.
$0 \to 23: \; 23 \text{ units}, \text{ so } |23| = 23$
23 is the same distance from zero on the right.
Check that the answers make sense.
$|-23| = |23| = 23 \geq 0$
Opposites share an absolute value, and an absolute value is never negative.
Compare $-12$ and $-5$ in two ways. First place them on a number line.
$-12 \text{ is left of } -5$
12 units left of zero is further left than 5 units left.
Write the order.
$-12 < -5$
The number further left is less.
Find the absolute value of each.
$|-12| = 12, \qquad |-5| = 5$
Each absolute value is that number's distance from zero.
Compare the absolute values.
$|-12| > |-5|$
12 units is a greater distance than 5 units.
Put the two facts side by side.
$-12 < -5 \quad \text{but} \quad |-12| > |-5|$
For negative numbers, the smaller number is further from zero.
Say what it means for temperatures in degrees Fahrenheit.
$-12^{\circ}\text{F is colder, and } 12 \text{ degrees below zero}$
Order answers 'which is colder'; absolute value answers 'how far below zero'.
Ava's balance is $-35.50$ dollars, Ben's is $20.00$ and Cal's is $-48.25$. Who owes more than 30 dollars? Find who has a debt at all.
$-35.50 < 0, \qquad 20.00 > 0, \qquad -48.25 < 0$
Only a negative balance is a debt; Ben has money in his account.
Find the size of Ava's debt.
$|-35.50| = 35.50$
The amount owed is the balance's distance from zero.
Find the size of Cal's debt.
$|-48.25| = 48.25$
Absolute value turns a balance into an amount of money owed.
Compare Ava's debt with 30 dollars.
$35.50 > 30$
Ava owes more than 30 dollars.
Compare Cal's debt with 30 dollars.
$48.25 > 30$
Cal also owes more than 30 dollars.
Check with the balances themselves.
$-35.50 < -30, \qquad -48.25 < -30$
A balance less than $-30$ is the same thing as a debt greater than 30.
Say who owes the most.
$48.25 > 35.50 \;\Rightarrow\; \text{Cal}$
The biggest debt is the balance with the biggest absolute value, which is also the lowest balance.
Find the first absolute value.
$|-9| = 9$
$-9$ is 9 units from zero.
Find the second absolute value.
$|-4| = 4$
$-4$ is 4 units from zero.
Subtract the second absolute value from the first.
Match each expression to its value.
| 11 | -11 | 15 | |
|---|---|---|---|
| |-11| | |||
| −|11| | |||
| |15| |
Sam's bank balance is -82 dollars and Lee's is -28 dollars. A negative balance means the person owes the bank money. Complete the solution to compare their debts. Then Sam deposits 19 dollars.
Find the size of Sam's debt.
$|\text{Sam's balance}| =$ sa
The debt is how far the balance is below zero, which is its absolute value.
Find the size of Lee's debt.
$|\text{Lee's balance}| =$ sb
Absolute value gives the amount owed without the sign.
Find how much more Sam owes than Lee.
$(\text{Sam's debt}) - (\text{Lee's debt}) =$ diff
Sam's balance is lower, so his debt is larger, and the difference of the sizes is how much larger.
Find Sam's balance after the deposit.
$(\text{Sam's balance}) + 19 =$ nb
A deposit moves the balance 19 to the right, but not far enough to reach zero.
Find the size of Sam's debt now.
$|\text{new balance}| =$ nd
The balance is still negative, so Sam still owes this much.
Which statement about -13 and -6 is true?
Find |-44| and |44|.
|-44| = neg and |44| = pos
Which two numbers have an absolute value of 16.9? Write the negative one first.
The numbers are neg and pos.
Elevations are measured from sea level, which is 0. A submarine is at -280 feet and a hot-air balloon is at 630 feet. How far is the submarine from sea level, and by how many feet is the further of the two further away?
The submarine is sub feet from sea level, and the further of the two is further by diff feet.
A machine cuts metal pins to a target length. Each pin's error is recorded in thousandths of an inch: positive if the pin is too long, negative if it is too short. Pin A reads 1, pin B reads -6 and pin C reads -4. The pin furthest from the target is rejected. How many thousandths of an inch from the target is the rejected pin?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Elevations are measured from sea level, which is 0. A submarine is at -860 feet and a hot-air balloon is at 460 feet. How far is the submarine from sea level, and by how many feet is the further of the two further away?
The submarine is sub feet from sea level, and the further of the two is further by diff feet.
You can find an absolute value and say what it means. Without looking: what is the absolute value of −9, and why can an absolute value never be negative?
16. Your turn: evaluate $|-9| - |-4|$, step 3
$9 - 4 = 5$
The bars work like parentheses: find each value first, then subtract.