Real numbers: absolute value

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function.

Geometric definition of absolute value

The absolute value of a number a , denoted $\mid a\mid$ , is the distance from that number to the origin (zero) on the number line. Absolute value answers the question of “how far,” not “which way.” That distance is always given as a nonnegative number. In short:

• If a number is positive (or zero), the absolute value function does nothing to it: $\mid 3\mid =3$
• If a number is negative, the absolute value function makes it positive: $\mid -3\mid =3$

WARNING: If there is arithmetic to do inside the absolute value function, you must do it before taking the absolute value—the absolute value function acts on the result of whatever is inside it. For example, a common error is

$\mid 5+\left(-2\right)\mid =5+2=7$ ( Wrong! )

The mistake here is in assuming that the absolute value makes everything inside it positive. This is not true. It only makes the result positive. The correct result is

$\mid 5+\left(-2\right)\mid =\mid 3\mid =3$ (Correct)

Examples

Determine each value.

$\mid 4\mid =$ __

• The answer is 4:

  

$\mid -4\mid =$ __

• The answer is 4:

  

$\mid 0\mid =$ __

• The answer is 0.

$-\mid 5\mid =$ __

• The quantity on the left side of the equal sign is read as “negative the absolute value of 5.” The absolute value of 5 is 5. Hence, negative the absolute value of 5 is −5.

$-\mid -3\mid =$ __

• The quantity on the left side of the equal sign is read as “negative the absolute value of −3.” The absolute value of −3 is 3. Hence, negative the absolute value of −3 is −(3) = −3.

Geometric absolute value exercises

By reasoning geometrically, determine each absolute value.

Algebraic definition of absolute value

From the preceding exercises, we can suggest the following algebraic definition of absolute value. Note that the definition has two parts.

The absolute value of a number a is

The algebraic definition takes into account the fact that the number a could be positive or zero ( a ≥ 0) or negative ( a <0).

1. If the number a is positive or zero ( a ≥ 0), the first part of the definition applies. The first part of the definition tells us that, if the number enclosed in the absolute value bars is a nonnegative number, the absolute value of the number is the number itself.
2. The second part of the definition tells us that, if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.

Examples

Use the algebraic definition of absolute value to find the following values.

$\mid 8\mid$ = __

• The number enclosed within the absolute value bars is a nonnegative number, so the first part of the definition applies. This part says that the absolute value of 8 is 8 itself. $\mid 8\mid$ = 8

$\mid -3\mid$ = __

• The number enclosed within absolute value bars is a negative number, so the second part of the definition applies. This part says that the absolute value of −3 is the opposite of −3, which is −(−3). By the definition of absolute value and the double-negative property, $\mid -3\mid$ = −(−3) = 3

Algebraic absolute value exercises

Use the algebraic definition of absolute value to find the following values.

Module review exercises

For the following problems, determine each of the values.<<I figured out these answers; still have to QA>><<also to verify: have squares&cubes been covered in earlier modules? Subtracting negative numbers?>>