Absolute Value Calculator – Find |x| of Any Number
Calculate the absolute value of any number or expression instantly with our free online absolute value calculator. Supports positive, negative, and decimal numbers.
Understanding Absolute Value
Absolute value measures distance from zero – nothing more, nothing less. It answers the question: "How far is this number from 0 on the number line?" Since distance can't be negative, absolute value is always zero or positive.
Think of it this way: if you're standing at position -5 on a number line, you're 5 steps away from zero. If you're at position +5, you're also 5 steps away. The direction doesn't matter for distance – that's what absolute value captures.
The Absolute Value Rule
|x| = x if x ≥ 0, and |x| = -x if x < 0
In plain English: keep positive numbers as-is, flip the sign of negative numbers
The notation uses vertical bars: |x|. You'll see this everywhere in math – from basic algebra to calculus to physics. When you see |−7|, read it as "the absolute value of negative 7" or "the distance of negative 7 from zero."
Worked Examples
Example 1: Absolute value of a negative integer
Find |−47|
−47 is negative, so we flip the sign
|−47| = −(−47) = 47
The absolute value of −47 is 47. It's 47 units away from zero.
Example 2: Absolute value of a positive decimal
Find |23.5|
23.5 is already positive, so it stays the same
|23.5| = 23.5
Positive numbers don't change. The absolute value of 23.5 is 23.5.
Example 3: Absolute value of a small negative number
Find |−0.001|
−0.001 is negative, so we flip the sign
|−0.001| = 0.001
Even tiny negative numbers become positive. The absolute value is 0.001.
Example 4: Absolute value with expressions
Find |5 − 12|
First simplify inside the bars
5 − 12 = −7
Then take the absolute value
|−7| = 7
Always simplify the expression first, then apply the absolute value.
Quick Fact
The vertical bar notation |x| was introduced by German mathematician Karl Weierstrass in 1841. Before that, mathematicians wrote out "abs(x)" or used other notations. The bars are elegant because they visually "trap" the negative sign – like cages that strip away the negativity, leaving only the magnitude.
Frequently Asked Questions
Can absolute value ever be negative?
No. Absolute value represents distance, and distance is never negative. The smallest possible absolute value is 0 (for the number 0 itself). Every other number – positive or negative – has a positive absolute value.
What's the absolute value of zero?
|0| = 0. Zero is already at the origin, so its distance from zero is... zero. It's the only number whose absolute value equals itself and is neither positive nor negative.
How do I handle absolute value in equations?
When you see |x| = 5, there are two solutions: x = 5 or x = −5. Both are 5 units from zero. For more complex equations like |2x − 3| = 7, split it into two cases: 2x − 3 = 7 and 2x − 3 = −7, then solve each separately.
Does absolute value work with imaginary numbers?
For complex numbers, we use "modulus" instead of absolute value, but the idea is similar – it's the distance from the origin in the complex plane. For a complex number a + bi, the modulus is √(a² + b²). For real numbers, this reduces to regular absolute value.
Why is absolute value useful in real life?
Absolute value shows up whenever you care about magnitude, not direction. Examples: calculating how far off a measurement is (error = |actual − expected|), finding the difference between two temperatures regardless of which is higher, or computing distances in physics and engineering.
What's the difference between absolute value and parentheses?
Parentheses ( ) are for grouping – they tell you what to calculate first. Absolute value bars | | are an operation – they transform the number inside. So (−5) is still −5, but |−5| becomes 5. Don't confuse them!
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