Absolute Value Calculator
Calculate |x| (absolute value)
Example: -5, 3.14, -100, 0
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. The notation is |x|, read as "absolute value of x" or "modulus of x."
Definition
|x| = x if x ≥ 0
|x| = -x if x < 0
Key Properties
- |x| ≥ 0 (always non-negative)
- |x| = |-x| (symmetric)
- |x × y| = |x| × |y|
- |x + y| ≤ |x| + |y| (triangle inequality)
- |0| = 0
Examples
Example 1: |-5|
Distance from 0 to -5 is 5 units
= 5
Example 2: |7|
Distance from 0 to 7 is 7 units
= 7
Example 3: |0|
Distance from 0 to 0 is 0 units
= 0
Example 4: |-3.14|
Distance from 0 to -3.14 is 3.14 units
= 3.14
Applications
- Distance: Calculate distance between two points
- Error/Deviation: Measure magnitude of error regardless of direction
- Physics: Speed vs velocity (magnitude vs vector)
- Finance: Magnitude of profit or loss
- Temperature: Difference in temperature
- Engineering: Tolerance and precision measurements
How to Use This Calculator
Enter Any Number
Type any number (positive, negative, or zero) in the input field. Examples: -5, 3.14, 100, -7.5, 0.
Click Calculate
Press the "Calculate Absolute Value" button to find |x| (the absolute value).
View Result
See the absolute value displayed, along with a visual number line representation and distance explanation.
Formula
|x| = x if x ≥ 0
|x| = -x if x < 0
In simpler terms: |x| removes the negative sign (if any)
Example 1: |-5|
Since -5 < 0, use |x| = -x
|-5| = -(-5) = 5
Distance from 0 to -5 is 5 units
Example 2: |7|
Since 7 ≥ 0, use |x| = x
|7| = 7
Distance from 0 to 7 is 7 units
Example 3: |0|
Since 0 ≥ 0, use |x| = x
|0| = 0
Distance from 0 to 0 is 0 units
Example 4: |-3.14|
Since -3.14 < 0, use |x| = -x
|-3.14| = -(-3.14) = 3.14
Distance from 0 to -3.14 is 3.14 units
About Absolute Value Calculator
The Absolute Value Calculator finds the absolute value (magnitude) of any number. The absolute value of a number is its distance from zero on the number line, always non-negative. This concept is fundamental in mathematics, physics, engineering, and many other fields.
When to Use This Calculator
- Distance Calculations: Find distance between points on a number line
- Error Measurement: Calculate magnitude of error regardless of direction
- Physics Problems: Find speed, magnitude, or distance measurements
- Algebra Homework: Solve absolute value equations and inequalities
- Engineering: Measure tolerances and precision deviations
- Statistics: Calculate deviations from mean or median
Why Use Our Calculator?
- ✅ Visual Number Line: See the distance representation
- ✅ Works with All Numbers: Handles positive, negative, zero, and decimals
- ✅ Instant Results: Get absolute value immediately
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Educational: Helps understand the concept visually
- ✅ Completely Free: No registration required
Understanding Absolute Value
Absolute value represents magnitude without direction. Think of it as "how far" without caring about "which way." It's like asking for distance instead of displacement.
- Always returns zero or a positive number (never negative)
- Symmetric: |x| = |-x| for any x
- Distance from 0 to any number is its absolute value
- Used extensively in solving equations and inequalities
- Essential in vector mathematics and physics
Real-World Applications
Distance: To find distance between points A and B, calculate |A - B|. For example, distance between -3 and 5 is |-3 - 5| = |-8| = 8 units.
Temperature Change: If temperature drops from 20°C to 15°C, the change magnitude is |20 - 15| = 5°C, regardless of whether it increased or decreased.
Stock Price Change: A stock that drops $5 has the same absolute change as one that gains $5. Both have |$5| change magnitude.
Tips for Using This Calculator
- Absolute value always returns a non-negative number
- |x| = |-x| for any number x
- The absolute value of zero is zero
- For positive numbers, absolute value equals the number itself
- For negative numbers, absolute value removes the negative sign
- Use absolute value to find distance without direction
Frequently Asked Questions
Can absolute value be negative?
No! Absolute value is always zero or positive. It represents distance, which cannot be negative. |x| ≥ 0 for all x.
What is |0|?
The absolute value of zero is zero: |0| = 0. Zero is neither positive nor negative, so its absolute value remains zero.
Is |-5| the same as -|5|?
No! |-5| = 5 (removes the negative), but -|5| = -5 (keeps the negative outside). The negative sign placement matters!
How do I calculate distance between two numbers?
Use |a - b|. For example, distance between 3 and 7 is |3 - 7| = |-4| = 4 units. This works whether a > b or b > a.
What is the absolute value used for?
Measuring magnitude without direction—distances, errors, deviations, speeds, and more. Any situation where you need size without sign.
Can absolute value work with decimals and fractions?
Yes! Absolute value works with any real number: integers, decimals, fractions, irrational numbers. Examples: |3.14| = 3.14, |-1/2| = 1/2.
What about absolute value of complex numbers?
This calculator handles real numbers. For complex numbers, the absolute value (modulus) is calculated differently: |a + bi| = √(a² + b²).