How absolute value works (with examples)

Absolute value is one of the simplest (and most useful) ideas in all of math. It answers a single question: How far is a number from zero? That distance is always non‑negative, which is why absolute value is never negative. You’ll see absolute value in algebra, geometry, statistics, finance, physics, and coding— anywhere you care about magnitude more than direction.

This calculator takes any real number (positive, negative, or zero) and returns its absolute value. If you enable “show steps”, it also explains exactly which rule was used and why.

The absolute value of a number x is written |x|. Conceptually, |x| is the distance from x to 0 on the number line. Because distance can’t be negative, |x| is always ≥ 0.

Absolute value is defined using a “piecewise” rule:

• If x ≥ 0, then |x| = x (nothing changes).
• If x < 0, then |x| = −x (the sign flips).

That’s it. The entire calculator is powered by this definition. When x is negative, multiplying by −1 makes it positive, turning “−7” into “7”.

• Error size: In measurement and prediction, we often care about how wrong we are, not whether the error was above or below the true value.
• Distance: In 1D, distance between two numbers a and b is |a − b|.
• Volatility & change: In finance and analytics, “magnitude of change” is often modeled with absolute differences.
• Constraints: Many math problems use absolute value to say “stay within a range,” like |x| ≤ 3.

• |12| = 12 because 12 is already non‑negative.
• |−12| = 12 because we flip the sign of a negative number.
• |0| = 0 because zero is exactly zero units from zero.
• |−3.75| = 3.75 works the same way for decimals.

Try it with your own values: type a number, hit Calculate, then change the sign and compare. It’s a fast way to build intuition.

If you want a “picture in your head”, imagine a number line: negative numbers are left of zero, positive numbers are right of zero. Absolute value ignores left/right and keeps only the distance. So −9 and +9 are equally far from zero, which is why |−9| = |9| = 9.

That same idea explains distance between two numbers: the distance from 2 to 11 is |2 − 11| = |−9| = 9. Distance from 11 to 2 is |11 − 2| = |9| = 9. No matter the order, distance is the same.

Internally the calculator does the same thing you would do by hand:

1. Read your input value x.
2. Check whether x is negative.
3. If it’s negative, multiply by −1.
4. Display the result as |x|.

If you turn on “show steps”, you’ll also see which branch of the piecewise definition was used: either “x ≥ 0, so |x| = x” or “x < 0, so |x| = −x”.

• Absolute value is not “always make it positive” in every context: It makes the final result non‑negative, but you still have to apply algebra rules properly when it’s part of a bigger expression.
• Don’t drop parentheses: |−(3 + 2)| = |−5| = 5. If you incorrectly remove the negative too early, you can mess up signs.
• Square vs absolute: |x| and x² are different. Example: x = −3 → |x| = 3 but x² = 9.

• Is absolute value the same as turning a negative into a positive?

  For single numbers, yes: it removes the negative sign. But mathematically it’s better to think “distance from zero,” because that extends naturally to distance between two values (|a − b|) and to inequalities like |x| ≤ 3.

• Can absolute value ever be negative?

  No. The smallest possible absolute value is 0 (only when x = 0). Otherwise it’s positive.

• How do I solve equations with absolute values?

  Start by rewriting |x| using its piecewise definition. Example: |x| = 5 means x = 5 or x = −5. For expressions like |x − 2| = 7, you set x − 2 = 7 or x − 2 = −7, then solve both.

• How do I solve inequalities like |x| < 4?

  Translate it into a “between” statement: |x| < 4 means x is within 4 units of 0, so −4 < x < 4. Similarly, |x| ≥ 4 means x is at least 4 units away: x ≤ −4 or x ≥ 4.

• Does absolute value work with decimals and fractions?

  Yes. The same rules apply. Example: |−0.125| = 0.125 and |−3/7| = 3/7.

• What about complex numbers?

  This calculator is for real numbers. For complex numbers, “absolute value” usually means magnitude, like |a + bi| = √(a² + b²). If you need that, a Complex Magnitude calculator is a good next tool to add to your Math category.

• Why do some textbooks use double bars ‖x‖?

  Double bars usually indicate a norm (length) in higher‑dimensional math. In one dimension, the norm behaves exactly like absolute value.

If you’re studying for a test, a quick practice trick is to generate a random number (positive or negative), compute |x|, then verify by checking your answer is always ≥ 0.

Absolute value has a few rules that show up constantly in simplification and proofs:

• Non‑negativity: |x| ≥ 0 for all real x.
• Zero only at zero: |x| = 0 ⇔ x = 0.
• Symmetry: |−x| = |x| (flipping the sign doesn’t change distance).
• Multiplication: |ab| = |a||b| (magnitude of a product is the product of magnitudes).
• Division (b ≠ 0): |a/b| = |a|/|b|.
• Triangle inequality: |a + b| ≤ |a| + |b| (adding can’t create more distance than the sum of distances).

That last one, the triangle inequality, sounds fancy but it’s basically saying: if you walk 3 miles east and then 4 miles west, your final distance from the start can’t exceed 3 + 4 = 7 miles. Sometimes you end up closer.

If you graph y = |x|, you get a V‑shape with its point (vertex) at (0, 0). For x ≥ 0, the graph is just y = x (a line with slope 1). For x < 0, the graph is y = −x (a line with slope −1). The absolute value function “folds” the left side of the number line upward so everything becomes non‑negative.

This V‑shape is why absolute value shows up in optimization and “closest point” problems. Minimizing |x − a| means “pick x as close as possible to a.”

• Navigation: If a robot wants to stay close to a path, it measures how far off it is using absolute differences.
• Quality control: Factories track “deviation from target” as |measured − target|.
• Sports & rankings: “How far behind the leader” is often absolute difference in points or time.
• Budgets: The size of a budget miss is |actual − planned|, regardless of over/under.
• Machine learning: “Mean absolute error (MAE)” is an error metric based on absolute value.

Example 1: |−18|

1. Input: x = −18
2. Since x < 0, apply |x| = −x
3. |−18| = −(−18) = 18

Example 2: |4.2|

1. Input: x = 4.2
2. Since x ≥ 0, apply |x| = x
3. |4.2| = 4.2

Example 3: Distance between −3 and 10

1. Distance = |a − b|
2. Compute: |−3 − 10| = |−13|
3. Absolute value: |−13| = 13

If you want quick reps (great for students), try these:

• |−7| = 7
• |0| = 0
• |11| = 11
• |−2.5| = 2.5
• |3 − 9| = |−6| = 6
• |−8 + 3| = |−5| = 5

A nice challenge is to invent your own: pick any number, change its sign, and confirm the absolute value stays the same.

• Absolute value = distance from 0.
• Positive stays positive.
• Negative flips sign.
• Useful for distance: |a − b|.
• Useful for ranges: |x| ≤ k ⇔ −k ≤ x ≤ k.

A common homework pattern is something like |x − a| = b. This means: “x is b units away from a.” On a number line there are usually two points that are exactly b units away—one to the left and one to the right. So you solve it by creating two equations:

• x − a = b
• x − a = −b

Example: |x − 5| = 3

• x − 5 = 3 → x = 8
• x − 5 = −3 → x = 2

So the solution set is {2, 8}.

Inequalities with absolute value come in two main flavors:

• “Less than” (inside a band): |x − a| < b means x is within b of a, so a − b < x < a + b.
• “Greater than” (outside a band): |x − a| > b means x is more than b away from a, so x < a − b or x > a + b.

Example: |x + 1| ≤ 4

• Rewrite as −4 ≤ x + 1 ≤ 4
• Subtract 1: −5 ≤ x ≤ 3

Example: |2x − 6| > 10

• 2x − 6 > 10 → 2x > 16 → x > 8
• 2x − 6 < −10 → 2x < −4 → x < −2

In code, absolute value is so common that nearly every language includes it: “abs(x)” in Python, JavaScript, and many others. It’s used for things like clamping values, measuring error, sorting by closeness, and comparing floats safely. A practical trick: if you’re comparing two decimal numbers that should be “the same” within a tolerance, you check if |a − b| ≤ tolerance. That’s how software handles tiny rounding differences.

• Comparing two offers: If prices differ by |p1 − p2|, you see the gap instantly.
• Change in temperature: A change from −2°C to 5°C is |−2 − 5| = 7 degrees.
• Sports score swing: If your team was down by 12 and then up by 3, the swing is |−12 − 3| = 15 points.

If you like “viral math,” these quick “difference” examples are great for short-form posts: they feel like magic because absolute value makes the distance obvious.

Frequently Asked Questions

• What is the absolute value of a negative number?

  It becomes positive. Example: |−8| = 8 because the distance from −8 to 0 is 8.

• What is the absolute value of zero?

  |0| = 0, because zero is zero units from zero.

• Is |x| the same as x²?

  No. Example: x = −3 → |x| = 3, but x² = 9.

• Why do I see |a − b| in distance problems?

  Because distance is always non‑negative, and |a − b| measures how far apart a and b are on the number line.

• Does this work for decimals and big numbers?

  Yes. You can use decimals, very large values, and negatives. If you want rounding, choose a decimal setting.

Fast ways to sanity-check your answer

• The result is never negative.
• |x| = |−x| (changing the sign doesn’t change the absolute value).
• If you computed a distance |a − b|, swapping a and b won’t change it.
• If x is already ≥ 0, the output should equal the input.

Fans also use these

Turn this into a 2-minute practice drill

Use the Random Number Generator to produce a value, compute |x| here, then verify by checking your answer is never negative. Do 10 in a row and you’ll never miss an absolute value question again.

• If x is positive, keep it.
• If x is negative, drop the minus sign.
• Distance uses absolute value: |a − b|.
• Within a range: |x| ≤ k ⇔ −k ≤ x ≤ k.