Absolute Value Calculator
Find the magnitude of any number - always returns a non-negative value
Tip: Works with integers, decimals, fractions, and negative numbers.
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Summary:
Understanding Absolute Value
What is Absolute Value?
The absolute value of a number is its distance from zero on the number line, without considering direction. It's always non-negative. Mathematically, |x| = x if x ≥ 0, and |x| = -x if x < 0.
Visual Representation
Absolute Value Properties
| Property | Mathematical Expression | Example |
|---|---|---|
| Non-negativity | |x| ≥ 0 | |-7| = 7 ≥ 0 |
| Positive-definiteness | |x| = 0 ⇔ x = 0 | |0| = 0 |
| Multiplicativity | |xy| = |x||y| | |2×-3| = 6 = |2|×|3| |
| Subadditivity | |x + y| ≤ |x| + |y| | |5 + (-3)| = 2 ≤ 5 + 3 = 8 |
Important: The absolute value function is continuous everywhere and differentiable everywhere except at x = 0.
Frequently Asked Questions
What's the absolute value of zero?
The absolute value of zero is zero (|0| = 0). Zero is neither positive nor negative, and its distance from itself on the number line is zero.
Can absolute value be applied to complex numbers?
Yes, for complex numbers (a + bi), the absolute value (or modulus) is calculated as √(a² + b²). This represents the number's distance from the origin in the complex plane.
How does absolute value work with fractions and decimals?
The absolute value works the same way for all real numbers. For example, |-¾| = ¾ and |2.5| = 2.5. The calculator handles all these formats automatically.
What's the difference between absolute value and magnitude?
For real numbers, they're the same. "Absolute value" is typically used for numbers, while "magnitude" is more general and used for vectors, complex numbers, and other mathematical objects.
Why is absolute value important in real-world applications?
Absolute value is crucial whenever direction doesn't matter but size does—like distances, errors, deviations, or differences. It's used in physics, engineering, statistics, and economics.
Practical Applications of Absolute Value
Distance Measurement
Used in GPS systems and mapping to calculate distances between points regardless of direction (|x₂ - x₁| gives distance along one axis).
Error Analysis
In scientific experiments, absolute value measures how far measurements deviate from expected values (|measured - expected|).
Financial Calculations
Used to calculate price changes, profit/loss margins, and other financial metrics where direction is less important than magnitude.
Signal Processing
In electronics, absolute value circuits (rectifiers) convert alternating current to direct current by taking the absolute value of the waveform.