Absolute Value Calculator

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Absolute value

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What is Absolute Value?

Absolute value measures how far a real number is from zero. A number five units to the left of zero and a number five units to the right are equally far away, so |−5| and |5| are both 5. The vertical bars are mathematical notation, not brackets asking you to multiply anything.

This calculator applies the absolute-value operation to the number you enter. Its current result panel formats answers to zero decimal places, so a decimal input can produce a rounded whole-number display. For work that needs exact decimal precision, keep the unrounded magnitude in your calculation.

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How to Use This Calculator

  1. Enter the signed number in the X field. Include a minus sign if the original value is negative.
  2. Choose Calculate. The operation keeps a nonnegative input as it is and changes a negative input to its positive counterpart.
  3. For an integer input, check that the displayed answer is the same distance from zero as the starting number.
  4. If you entered a decimal, account for the result panel’s whole-number rounding before comparing with handwritten work.

Use Reset to clear a previous example. A visible old answer is not a recalculation: after changing the input, choose Calculate again.

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Formula Used

|x| = x when x ≥ 0; |x| = −x when x < 0. The current display rounds this magnitude to a whole number.
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Variables Explained

QuantityExplanation
xThe original real number, including its sign. The value may be positive, negative or zero.
|x|The nonnegative magnitude of x. For example, the exact magnitude of −2.75 is 2.75, even though a whole-number display rounds it to 3.
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Worked Example

Start with x = −18. The number lies 18 units to the left of zero, so its absolute value is 18. Repeating the calculation with x = 18 gives the same answer. With x = 0, the answer is 0 because there is no distance to travel.

A decimal example and the display limitation

The exact value of |−2.75| is 2.75. This page currently formats that value with no decimal places, producing 3. Do not copy 3 into a calculation that requires the exact distance 2.75. This difference is caused by display rounding, not by the definition of absolute value.

Distance between two numbers

To find the distance from −3 to 8, first subtract: 8 − (−3) = 11. Then take |11| = 11. Reversing the subtraction gives −11, whose absolute value is also 11. The subtraction creates a displacement; absolute value gives its magnitude.

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Why Use This Calculator?

Absolute value helps separate size from direction. A signed difference can show whether a reading was above or below a reference, while the absolute difference shows how far apart the two readings were. If a recorded score is 47 and a reference is 50, the signed difference is −3 and the absolute difference is 3.

Keep the sign when it carries useful information

Removing a sign can also remove meaning. A change of −6 and a change of +6 have equal magnitudes, but one is a decrease and the other an increase. Keep both the original signed value and its absolute value if the direction matters to your explanation.

Absolute value is also useful in inequalities. The statement |x| < 4 means x lies strictly between −4 and 4. By contrast, |x| = 4 has two solutions, −4 and 4. This calculator evaluates one supplied number; it does not solve an equation or list every possible value of x.

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Common Mistakes

    • Do not put a minus sign in front of a final absolute value unless the expression explicitly includes an outside minus. The expression −|−5| is −5, but |−5| is 5.
    • Do the operation inside the bars first. For |3 − 8|, subtract to get −5, then take its magnitude.
    • Do not replace |a + b| with |a| + |b|. Opposite signs can cancel before the absolute value is taken.
    • Do not confuse absolute value with squaring. Both give nonnegative results for real inputs, but |−5| is 5 while (−5)² is 25.
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Frequently Asked Questions

Can an absolute value be negative?

No. Distance from zero is never negative. Zero is possible and occurs only when the original value is zero. An expression with a minus sign outside the absolute-value bars can still have a negative final value.

Why does |−2.75| display as 3 here?

The calculation takes the correct magnitude, but the current response formats it with zero decimal places. The mathematical absolute value is 2.75. Use that exact magnitude if your next step depends on decimal precision.

Is |a − b| the same as |b − a|?

Yes. Reversing the subtraction changes the sign of the difference, but not its magnitude. Both expressions measure the distance between the same two real numbers.

Does |a + b| always equal |a| + |b|?

No. Take a = 5 and b = −5. The left side is |0| = 0, while the right side is 5 + 5 = 10. Add the signed numbers before applying the absolute-value operation.

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Sources & References

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