Absolute Value Calculator

About Absolute Value Calculator

The Absolute Value Calculator computes the absolute value (modulus) of any real or complex number. Enter a value and get instant results with step-by-step solutions, interactive visualizations, and a detailed property breakdown.

How to Use the Absolute Value Calculator

1. Enter your number in the input field. You can type a real number like -7.5 or a complex number like 3+4i.
2. Click "Calculate |x|" or press Enter to compute the absolute value.
3. View the result displayed prominently with the mathematical formula.
4. Study the step-by-step solution to understand exactly how the calculation was performed.
5. Explore the visualization: a number line for real numbers showing distance from zero, or a complex plane diagram for complex numbers showing the modulus as the distance from the origin.

What Is Absolute Value?

The absolute value of a number is its distance from zero, regardless of direction. It is always non-negative. Mathematically:

• For a real number \(x\): \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\)
• For a complex number \(z = a + bi\): \(|z| = \sqrt{a^2 + b^2}\)

Absolute Value of Real Numbers

Think of it as "how far is this number from zero on the number line?" Both -5 and 5 are exactly 5 units away from zero, so \(|-5| = |5| = 5\).

• \(|7| = 7\) (already positive)
• \(|-3.14| = 3.14\) (flip the sign)
• \(|0| = 0\) (zero is zero distance from itself)

Absolute Value of Complex Numbers (Modulus)

For a complex number \(z = a + bi\), the absolute value — also called the modulus — represents its distance from the origin in the complex plane. It is computed using the Pythagorean theorem:

\(|a + bi| = \sqrt{a^2 + b^2}\)

For example, \(|3 + 4i| = \sqrt{9 + 16} = \sqrt{25} = 5\). This is a classic 3-4-5 right triangle.

Key Properties of Absolute Value

• Non-negativity: \(|x| \geq 0\) for all \(x\), and \(|x| = 0\) only if \(x = 0\)
• Symmetry: \(|-x| = |x|\)
• Multiplicativity: \(|xy| = |x| \cdot |y|\)
• Triangle inequality: \(|x + y| \leq |x| + |y|\)
• Square root relation: \(|x| = \sqrt{x^2}\)

Real-World Applications

• Distance calculations: The distance between two points \(a\) and \(b\) on a number line is \(|a - b|\).
• Error measurement: Absolute error = \(|\text{measured} - \text{actual}|\).
• Signal processing: The modulus of a complex signal gives its amplitude.
• Programming: Used in sorting algorithms, collision detection, and tolerance checks.

FAQ

What is absolute value?

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. For example, |-5| = 5 and |5| = 5.

How do you find the absolute value of a complex number?

For a complex number a + bi, the absolute value (modulus) is calculated as the square root of (a squared plus b squared). For example, |3 + 4i| = sqrt(9 + 16) = sqrt(25) = 5.

Is the absolute value always positive?

The absolute value is always non-negative. It equals zero only when the input is zero. For all other real or complex numbers, the absolute value is strictly positive.

What is the absolute value of zero?

The absolute value of zero is zero. |0| = 0. Zero is the only number whose absolute value equals zero.

What is the difference between absolute value and modulus?

For real numbers, absolute value and modulus mean the same thing: the distance from zero. For complex numbers, the term modulus is more common, referring to the distance from the origin in the complex plane, calculated as sqrt(a^2 + b^2).

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