Absolute Value Calculator
Calculate the absolute value of any number or expression instantly.
Calculator
Absolute value calculator card
Absolute value removes the sign and gives distance from zero.
Supports integers, decimals, fractions, negatives, and simple expressions like 3-8.
Results
Live results dashboard
Absolute value of a number
Absolute value
7
|-7| = 7
Original input
-7
Sign
negative
Distance from zero
7
Decimal equivalent
7
Formula used
|x| = x if x >= 0, and |x| = -x if x < 0
|-7| = 7 because -7 is 7 units from zero.
Interpretation
Dynamic interpretation cards
What the result means
|-7| = 7 because -7 is 7 units from zero.
Distance is non-negative
Absolute value is never negative because it represents distance on the number line.
Signs and magnitude
Negative inputs become positive magnitudes; positive inputs usually stay the same.
Distance formula
|a - b| gives the distance between a and b, no matter which value is larger.
Formula
Formula block
Absolute Value Definition
|x| = x if x >= 0
Negative Input Rule
|x| = -x if x < 0
Distance From Zero
Distance from zero = |x|
Distance Between Two Numbers
Distance = |a - b|
Basic Absolute Value Equation
|x| = a means x = a or x = -a, when a >= 0
Notation
Variable and notation explanations
x
The input number or expression inside the absolute value bars.
|x|
The absolute value of x, or its distance from zero.
a and b
Two values being compared on a number line.
|a - b|
The distance between a and b.
Bars are not parentheses
Parentheses group expressions; absolute value bars measure distance or magnitude.
Non-negative output
Absolute value output is always zero or positive.
Number line
Number line explanation
-5 and 5 are on opposite sides of zero, but both are 5 units from zero. That is why |-5| = 5 and |5| = 5.
Examples
Worked examples
Find |-9|
Rule: Negative numbers become positive magnitude.
Substitution: |-9| = 9
Answer: 9
-9 is 9 units from zero.
Find |12|
Rule: Positive numbers keep the same value.
Substitution: |12| = 12
Answer: 12
12 is already 12 units from zero.
Find |0|
Rule: Zero is zero units from zero.
Substitution: |0| = 0
Answer: 0
Zero is the only number whose absolute value is zero.
Find |-3.75|
Rule: Remove the negative sign.
Substitution: |-3.75| = 3.75
Answer: 3.75
Distance does not depend on direction.
Find |-2/5|
Rule: A negative fraction becomes positive.
Substitution: |-2/5| = 2/5
Answer: 2/5
The magnitude of the fraction is 2/5.
Distance between -4 and 7
Rule: Distance = |a - b|
Substitution: |-4 - 7| = |-11|
Answer: 11
-4 and 7 are 11 units apart.
Solve |x| = 6
Rule: |x| = a gives two values when a is positive.
Substitution: x = 6 or x = -6
Answer: x = 6 or -6
Both values are 6 units from zero.
Explain why |x| = -3 has no solution
Rule: Absolute value cannot be negative.
Substitution: No distance can equal -3
Answer: No real solution
Distances are zero or positive.
Equations
Absolute value equations and inequalities
|x| = a
Two solutions when a is positive: x = a and x = -a.
|x| = 0
One solution: x = 0.
|x| = negative
No real solution because absolute value cannot be negative.
|x| < a
x is within a units of zero when a is positive.
|x| > a
x is more than a units from zero.
Screening idea
These are beginner rules, not a full algebra solver.
Mistakes
Common mistakes
Thinking absolute value always changes a number
Positive numbers and zero stay the same.
Writing |-5| = -5
The correct value is 5 because -5 is 5 units from zero.
Confusing bars with parentheses
Absolute value bars change the result to a distance or magnitude.
Forgetting distance meaning
Distance does not have a negative direction.
Solving |x| = -4
There is no real solution because absolute value cannot be negative.
Mishandling |3 - 8|
First simplify inside the bars: 3 - 8 = -5, then |-5| = 5.
FAQ
Frequently asked questions
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