Absolute value

Why this exists

A mountain hut stands at 1{,}850 meters above sea level, and the lowest point of a desert valley lies at -72 meters. An account that is overdrawn shows a balance of -45 dollars. Each number says two things at once: a side of zero (above or below sea level, money held or money owed) and a size (how far up, how far down, how much).

Very often only the size matters. Which place is closest to sea level? Who owes the most? Whose score is furthest from zero? “Greater” cannot answer these, because “greater” means further right on the number line, and -72 is less than -38 even though the valley floor is further from sea level than a spring at -38 meters. A separate measure is needed: the size of a number with its side set aside. That measure is the absolute value.

Definition

The absolute value of a number is its distance from 0 on the number line. It is written by putting the number between two upright bars, so \lvert -6 \rvert is read “the absolute value of negative six”.

A distance counts units and has no direction, so an absolute value is never negative. The definition works for every number on the line, fractions and decimals included: \left\lvert -\tfrac{2}{7}\right\rvert = \tfrac{2}{7} and \lvert -1.9 \rvert = 1.9.

A minus sign written outside the bars is not part of the absolute value. In -\lvert 6 \rvert the bars are worked first, giving 6, and then the minus sign takes the opposite, so -\lvert 6 \rvert = -6. The bars group what is inside them, just as brackets do.

In a situation. When a quantity is measured from a zero that has two sides, the sign of the number tells the side and the absolute value tells the size:

• an elevation of -72 meters is \lvert -72 \rvert = 72 meters below sea level;
• an account balance of -30 dollars is a debt of \lvert -30 \rvert = 30 dollars;
• a card score of -8 is 8 points below zero.

So “a debt greater than 30 dollars” means a balance further below zero than -30, which is a balance less than -30. Words about size (a bigger debt, a deeper point, further from zero) compare absolute values. Words about order (greater, less, higher, lower) compare the numbers themselves.

Intuition

Picture the number line folded at 0, as if a mirror stood there. Every number lands on its opposite, and the two are the same distance from the fold. Absolute value is what survives the fold: the distance, with the side forgotten.

Now take two numbers on different sides. 3 is greater than -7, because it lies to the right. But -7 is further from zero, 7 units against 3. Two different questions, two different answers.

For positive numbers the two questions always agree: further right is also further from zero. For negative numbers they always disagree: going further from zero means going further left, so the number with the larger absolute value is the smaller number.

Results

Worked examples

The lesson is taught in two parts. Examples 1 to 3 belong to the first: finding absolute values of integers, fractions and decimals, and reading the bars. Examples 4 to 6 belong to the second: absolute value in real situations, and comparing absolute values against comparing the numbers.

Boundaries

Distance from zero only. The bars measure how far a number is from 0. How far apart two other numbers are is a different question; on the coordinate plane it is taken up in horizontal and vertical distance.

The side is lost. Since \lvert -45 \rvert = \lvert 45 \rvert, the absolute value alone cannot say whether 45 dollars is owed or held. Use it only when the side does not matter, or keep the sign alongside it.

The size depends on where zero is. An elevation’s absolute value is its distance from sea level only because sea level was chosen as 0. Measured from the valley floor instead, every number, and so every absolute value, would change.

Absolute value is not the opposite. The opposite of 8 is -8, while \lvert 8 \rvert = 8. The two agree only for negative numbers and zero.

Where this goes next

Absolute value is the first time a number’s size is separated from its sign, and much of what follows leans on that split. Adding integers with different signs compares the absolute values of the two numbers to see which one is left over. Distance on a number line measures how far apart any two points are, and horizontal and vertical distance does the same along the grid lines of the coordinate plane. Mean absolute deviation averages distances from a mean to describe how spread out data are. Later, the absolute value graph and absolute value equations treat absolute value as a rule in its own right.

Exercises

Exercises 1 to 8 go with the first part of the lesson (Examples 1 to 3), Exercises 9 to 17 with the second (Examples 4 to 6), and the reasoning group draws on both.

Part 1: finding absolute values

1. Find \lvert -29 \rvert, \lvert 16 \rvert and \lvert -1 \rvert.
2. Find -\lvert -37 \rvert, \lvert -37 \rvert and -(-37).
3. Which two numbers are 15 units from 0? Write the absolute value of each with bars.
4. Find \left\lvert -\tfrac{7}{10}\right\rvert, \lvert 5.08 \rvert, \left\lvert -2\tfrac{2}{3}\right\rvert and \lvert -0.3 \rvert.
5. Name every number whose absolute value is \tfrac{3}{5}.
6. Which of these numbers have absolute value 8: 8, -8, \tfrac{1}{8}, -(-8), -\lvert 8 \rvert?
7. A point sits 13 units to the left of 0. What number is it, and what is its absolute value?
8. Write <, > or = in each box: (a) \lvert -17 \rvert \;\square\; 17 (b) -\lvert -17 \rvert \;\square\; \lvert -17 \rvert (c) -\lvert 6 \rvert \;\square\; \lvert -6 \rvert

Part 2: absolute value in situations

9. The floor of a desert valley is 54 meters below sea level. Write its elevation as a number, and say what its absolute value tells you.
10. In the card game, Emil’s score is -15 and Flora’s is 11. Who is ahead? Whose score is further from zero?
11. Hari’s account balance is -27.40 dollars. How much does he owe? Is his debt greater than 25 dollars?
12. Which of these balances, in dollars, show a debt greater than 50 dollars: -48, -52.75, -50, 10, -90?
13. A mountain hut is at 2{,}315 meters, a campsite at 18 meters, a salt pan at -26 meters and the lowest point of a desert valley at -61 meters. Which place is lowest? Which is closest to sea level? List the places by distance from sea level, nearest first.
14. A card game’s rule says a player whose score is more than 10 points from zero must skip a turn. The scores are Bryn -14, Dina 10, Hari -3 and Emil 12. Who skips? Who is last?

Part 2: comparing numbers and absolute values

15. Write < or >: (a) -11 \;\square\; -6 (b) \lvert -11 \rvert \;\square\; \lvert -6 \rvert
16. Which is further from 0, -1.2 or 0.9? Which is greater?
17. Put -3.4, \tfrac{1}{2}, -0.6, 2.25, -1.5 in order from least to greatest, and then in order of absolute value, smallest first.

Reasoning

18. Describe every account balance that means a debt of less than 12 dollars.
19. Explain why the absolute value of a number can never be negative.
20. True or false: “Of two numbers, the greater one always has the greater absolute value.” Give a reason.
21. Two different numbers have the same absolute value. What must be true of them? Explain.
22. Explain why, of two negative numbers, the one with the greater absolute value is the smaller number.
23. Which numbers are equal to their own absolute value, and which are not? Explain.