CalculateY
Quick Answer
|-7| = 7. The absolute value of -7 is 7, because -7 is 7 units from zero on the number line.
Absolute value of a number
Absolute value of an expression
Common Examples
| Input | Result |
|---|---|
| |-5| | 5 |
| |3.14| | 3.14 |
| |0| | 0 |
| |-5 + 3| | |-2| = 2 |
| |2 * -4| | |-8| = 8 |
How It Works
The formula
The absolute value function is defined piecewise:
\[|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\]This means absolute value simply strips the negative sign from negative numbers and leaves positive numbers unchanged. Zero is its own absolute value.
Worked example
| Find | -23.5 | : |
| Since -23.5 is negative, apply the rule | x | = -x: |
The negative of a negative is a positive.
Geometric interpretation
| On a number line, absolute value measures the distance from a point to zero. Both -7 and 7 are exactly 7 units from zero, so | 7 | = | -7 | = 7. This interpretation extends to the distance between two numbers: the distance between a and b is | a - b | . For example, the distance between 3 and -4 is | 3 - (-4) | = | 7 | = 7. |
Properties of absolute value
| Property | Formula | Example |
|---|---|---|
| Non-negativity | |x| >= 0 | |-3| = 3 >= 0 |
| Identity of zero | |x| = 0 iff x = 0 | |0| = 0 |
| Symmetry | |-x| = |x| | |-5| = |5| = 5 |
| Multiplicative | |xy| = |x||y| | |(-2)(3)| = |-6| = 6 = 2 x 3 |
| Triangle inequality | |x + y| <= |x| + |y| | |3 + (-5)| = 2 <= 3 + 5 = 8 |
Related Calculators
Frequently Asked Questions
What is absolute value?
Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative number. The absolute value of -5 is 5, and the absolute value of 5 is also 5.
Can absolute value be negative?
No. By definition, absolute value is always zero or positive. |x| >= 0 for every real number x.
What is the absolute value of 0?
|0| = 0. Zero is the only number whose absolute value equals zero.
How do you solve absolute value equations?
To solve |x| = a where a > 0, split into two cases: x = a or x = -a. For example, |x| = 5 gives x = 5 or x = -5. If a = 0, then x = 0. If a < 0, there is no solution.
What does absolute value mean in real life?
Absolute value represents magnitude without direction. Common examples: temperature difference regardless of which is warmer, distance traveled regardless of direction, and error in measurements (always reported as a positive number).