Absolute Value Calculator

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absolute value calculator
Absolute Value of a Number |x|Absolute Difference |x – y|Absolute Value of a Sum |x + y|
Answer:
Result =
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Calculator Use

This absolute value calculator is a versatile tool designed to find the non-negative magnitude of any real number, the distance between two points on a number line, or the magnitude of a combined sum. Whether you are solving basic algebra problems or analyzing physical distances, this calculator provides instant results with step-by-step logic.

To use the calculator, simply select the type of operation you wish to perform and enter your numerical values. It handles positive numbers, negative numbers, and decimals with ease.

Absolute Value |x|
Calculates the distance of a single number from zero. For example, the absolute value of -10 is 10.
Absolute Difference |x – y|
Finds the distance between two distinct points on a number line. This is always a positive value regardless of which number is larger.
Absolute Value of a Sum |x + y|
Calculates the sum of two numbers first, then applies the absolute value to the total.

How It Works

In mathematics, the absolute value (or modulus) of a real number is its distance from zero on the number line, without regard to its direction. Since distance cannot be negative, the absolute value is always zero or positive.

If x ≥ 0, then |x| = x
If x < 0, then |x| = -x

The absolute value calculator follows these fundamental properties:

  • Non-negativity: |a| ≥ 0. The result is never negative.
  • Positive-definiteness: |a| = 0 if and only if a = 0.
  • Multiplicativity: |ab| = |a| × |b|.
  • Triangle Inequality: |a + b| ≤ |a| + |b|.

Calculation Examples

Example 1: Single Number
Find the absolute value of -15.75.

  1. Identify the number: x = -15.75
  2. Apply the rule: Since x is negative, multiply by -1.
  3. Calculation: -1 * (-15.75) = 15.75
  4. Result: 15.75

Example 2: Distance (Absolute Difference)
Find the distance between a mountain peak at 5,000 feet and a valley floor at -200 feet.

  1. Let x = 5000 and y = -200
  2. Formula: |x – y|
  3. Calculation: |5000 – (-200)| = |5000 + 200| = |5200|
  4. Result: 5,200 feet

Common Questions

Why is the absolute value always positive?

Absolute value measures distance. In the physical world, distance doesn't care about direction. Whether you walk 5 miles north or 5 miles south, the distance traveled is 5 miles. Mathematics uses absolute value to express this magnitude regardless of the sign.

Can the absolute value of a variable be negative?

No. By definition, the output of the absolute value function is always greater than or equal to zero. If you see an equation like |x| = -5, there is no real solution for x.

How is absolute value used in real life?

It is used frequently in error analysis (calculating the difference between a measured value and a true value), determining tolerances in engineering, calculating financial gains/losses relative to a baseline, and measuring deviations in statistical data.

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