Absolute Value

When you want to know how far a number is from zero, without caring about the direction, you are looking for its absolute value. Whether you are 5 steps to the right of zero or 5 steps to the left, you are still 5 steps away. Absolute value captures that idea of pure distance, stripping away the positive or negative sign.

Absolute value comes up constantly in real-world contexts: the magnitude of a temperature drop, the size of a debt regardless of whether you owe or are owed, and the distance between two points.

Definition

The absolute value of a number is its distance from zero on the number line. Distance is always zero or positive, so absolute value is never negative.

We write the absolute value of a number $a$ using vertical bars:

$|a|$

Formally:

$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$

This definition says: if the number is already zero or positive, leave it alone. If it is negative, flip the sign to make it positive.

Evaluating Absolute Value

Finding the absolute value of a number is straightforward. Just ask: “How far is this number from zero?”

Example 1: Absolute Value of a Positive Number

$|7| = 7$

The number $7$ is 7 units from zero. It is already positive, so the absolute value is just $7$.

Example 2: Absolute Value of a Negative Number

$|-12| = 12$

The number $-12$ is 12 units from zero. The negative sign is removed, giving $12$.

Example 3: Absolute Value of Zero

$|0| = 0$

Zero is zero units from zero. This is the only number whose absolute value is $0$.

Example 4: Absolute Value in an Expression

Evaluate $|{-3}| + |{5}|$.

$|{-3}| + |{5}| = 3 + 5 = 8$

Evaluate each absolute value first, then perform the arithmetic.

Example 5: Negation Outside the Bars

Evaluate $-|{-9}|$.

Step 1: Find the absolute value: $|{-9}| = 9$.

Step 2: Apply the negative sign outside: $-9$.

$-|{-9}| = -9$

The negative sign outside the bars is not removed by the absolute value operation. The bars only affect what is inside them.

Common Misconceptions

Misconception: “Absolute value makes everything positive.” More precisely, absolute value gives you the distance from zero, which is always non-negative. The key distinction appears when there is a negative sign outside the bars, as in Example 5: $-|{-9}| = -9$, which is negative.

Misconception: “The absolute value of a number is always the number with its sign changed.” Not quite. If the number is already positive, its absolute value is the same number unchanged. For example, $|4| = 4$, not $-4$.

Comparing Numbers Using Absolute Value

Absolute value is useful when you want to compare the magnitude (size) of numbers without regard to sign.

Which is farther from zero, $-8$ or $5$?

$|{-8}| = 8, \quad |{5}| = 5$

Since $8 > 5$, the number $-8$ is farther from zero than $5$.

A checking account has a balance of -\230. A savings account has a balance of \180. Which account has the larger magnitude?

$|{-230}| = 230, \quad |{180}| = 180$

The checking account has the larger magnitude (230 vs. 180), even though its balance is negative.

Absolute Value and Distance Between Numbers

The distance between any two numbers on the number line can be found using absolute value:

$\text{Distance between } a \text{ and } b = |a - b|$

Example: Find the distance between $-4$ and $3$.

$|{-4} - 3| = |{-7}| = 7$

The distance is 7 units. Notice that $|3 - ({-4})| = |7| = 7$ gives the same result. Order does not matter.

Practice Problems

Problem 1: Evaluate $|{-15}|$.

Problem 2: Evaluate $|{4}| - |{-6}|$.

Problem 3: Evaluate $-|{-20}|$.

Problem 4: Which number is farther from zero: $-14$ or $11$?

Problem 5: Find the distance between $-7$ and $-2$ on the number line.

Key Takeaways

• Absolute value $|a|$ gives the distance of $a$ from zero on the number line. It is always zero or positive.
• If the number is positive or zero, the absolute value is the number itself. If negative, the absolute value is the number with its sign flipped.
• A negative sign outside the absolute value bars is not affected by the absolute value operation.
• You can use absolute value to compare the magnitudes of numbers, ignoring their signs.
• The distance between two numbers on the number line is $|a - b|$.

Return to Arithmetic for more foundational math topics.