Calculate the absolute value (magnitude) of any number, positive or negative.
The absolute value of a number is its distance from zero on the number line, regardless of direction. It represents the magnitude of a number without considering its sign. The absolute value is always non-negative, making it useful for measuring distances, differences, and magnitudes.
Denoted by vertical bars |x|, the absolute value converts negative numbers to positive while leaving positive numbers unchanged. For example, |-5| = 5 and |5| = 5. This concept is fundamental in mathematics, appearing in algebra, calculus, physics, and engineering applications where only the size of a quantity matters, not its direction.
The absolute value function is defined piecewise: if the number is positive or zero, the absolute value is the number itself. If the number is negative, the absolute value is the opposite (negation) of the number.
Example 1: Negative Number
Since -42 is negative, we remove the negative sign.
Example 2: Positive Number
Since 7.5 is already positive, the absolute value is the same.
Example 3: Zero
The absolute value of zero is zero (distance from itself is zero).
Example 4: Real-World Application
Temperature difference: If it's -15°F outside and 72°F inside, the temperature difference is:
Absolute value helps us measure the magnitude of change regardless of direction.
No, the absolute value is always non-negative (zero or positive). By definition, it represents distance from zero, and distance cannot be negative.
For real numbers, they mean the same thing. However, magnitude is also used for vectors (multi-dimensional), while absolute value typically refers to single numbers on the number line.
Absolute value is used to measure distances (GPS navigation), temperature differences, financial losses/gains, error calculations, signal processing, and anywhere you need to know 'how much' without caring about direction.
The absolute value of zero is zero: |0| = 0. Zero is neither positive nor negative, and it's exactly zero distance from itself on the number line.
Because absolute value measures distance from zero, which is the same in both directions. Whether you go 5 units left or 5 units right, you're still 5 units away from zero.
Absolute value inequalities like |x| < 5 describe a range of values (-5 < x < 5). They're essential for describing tolerance ranges, error bounds, and standard deviation ranges in statistics.
This calculator only accepts real numbers; for a complex number a + bi the modulus √(a² + b²) represents magnitude, so use a complex-math tool to compute that.
This is a fundamental property of absolute value: the absolute value of a product equals the product of absolute values. It's useful for simplifying expressions involving multiplication.