What is the absolute value of a number?

The absolute value of a number, also called its modulus, is the distance of that number from zero on the number line. It is written with two vertical bars: |−4| = 4 and |4| = 4. Since a distance has no direction, an absolute value is never negative.

What is the absolute value of a negative number?

It is the same number without the minus sign, that is its opposite: |−7| = 7. Formally, for a < 0 we have |a| = −a, and the minus in front of a does not make the result negative — it flips the sign of a number that was already negative.

How do you compute the distance between two numbers on the line?

Subtract them and take the modulus: the distance between a and b is |a − b|. The order does not matter, because |a − b| = |b − a|. For instance, the distance between −3 and 5 is |−3 − 5| = |−8| = 8.

Is √(a²) equal to a?

Not always — the correct statement is √(a²) = |a|. For a = 5 both versions give 5, but for a = −5 the root gives 5 rather than −5, because a square root is non-negative by definition. This is the most common source of a lost sign in calculations with roots.

How do you solve an equation with absolute value?

The equation |x| = r with r > 0 has two solutions, x = r and x = −r, because two numbers lie at distance r from zero. For r = 0 there is one solution (x = 0), and for r < 0 there is none, since a modulus is never negative. The equation |x − a| = r reads as the numbers at distance r from a, that is x = a − r and x = a + r.

Intermediate level

Absolute value

Q: How do you compute the distance between two numbers on the line?

Subtract them and take the modulus: the distance between a and b is |a − b|. The order does not matter, because |a − b| = |b − a|. For instance, the distance between −3 and 5 is |−3 − 5| = |−8| = 8.

Q: Is √(a²) equal to a?

Not always — the correct statement is √(a²) = |a|. For a = 5 both versions give 5, but for a = −5 the root gives 5 rather than −5, because a square root is non-negative by definition. This is the most common source of a lost sign in calculations with roots.

Q: How do you solve an equation with absolute value?

The equation |x| = r with r > 0 has two solutions, x = r and x = −r, because two numbers lie at distance r from zero. For r = 0 there is one solution (x = 0), and for r < 0 there is none, since a modulus is never negative. The equation |x − a| = r reads as the numbers at distance r from a, that is x = a − r and x = a + r.

The absolute value of a number is its distance from zero — never negative, because a distance has no direction. Learn the case-by-case definition, the distance between two numbers written as |a − b|, the properties of the modulus and equations with absolute value.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Negative numbersNegative numbers are the numbers below zero — frost on a thermometer, an overdraft, a level underground. Learn the number line running both ways, opposite numbers, comparing negatives, and the sign rules for addition, subtraction, multiplication and division.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Shopping and complaints
A bag of flour should hold 1000 g; the shop scale reads 977 g. The deviation is |977 − 1000| = 23 g — and that is what decides the complaint, because against an allowance of 15 g it makes no difference whether the bag is lighter or heavier.
Weather forecasting
A forecast said 18 °C and the thermometer read 21 °C; the next day it said 15 °C and the reading was 12 °C. Both forecasts were equally wrong: |18 − 21| = 3 and |15 − 12| = 3. Without the modulus the first error would come out as −3 and cancel the second to zero on averaging.
Convert it here: Temperature
Quality control and metrology
A shaft must measure 25.00 mm within a tolerance of 0.05 mm. The caliper reads 24.97 mm, so |24.97 − 25.00| = 0.03 mm and the part passes. At a reading of 25.08 mm the modulus gives 0.08 mm and the part is scrapped.
Software development
Floating-point numbers are never tested for equality with an equals sign, because 0.1 + 0.2 comes out as 0.30000000000000004. The test used instead is |a − b| < 0.000001 — one number rather than two separate comparisons.

All formulas

Definition
$|a| = \begin{cases} a & \text{for } a \ge 0 \\ -a & \text{for } a < 0 \end{cases}$
a non-negative number is left alone, a negative one has its sign flipped
Non-negativity
$|a| \ge 0$
a distance is never negative
Symmetry
$|-a| = |a|$
opposite numbers sit equally far from zero
Distance between two numbers
$|a - b| = |b - a|$
the order of subtraction does not change the distance
Products and quotients
$|a \cdot b| = |a| \cdot |b| \qquad \left|\frac{a}{b}\right| = \frac{|a|}{|b|}$
the modulus distributes over multiplication and division
Root of a square
$\sqrt{a^2} = |a|$
not a, because a square root is always non-negative
Triangle inequality
$|a + b| \le |a| + |b|$
the modulus of a sum never exceeds the sum of the moduli

The lesson on negative numbers said that the distance between two numbers on the line is always positive — but we computed it in words back then: subtract the number further left from the one further right. Mathematics has one sign that replaces that instruction: the absolute value, also called the modulus.

Definition

The absolute value of a number $a$ is written with two vertical bars and defined case by case:

|a| = \begin{cases} a & \text{for } a \ge 0 \\ -a & \text{for } a < 0 \end{cases}

In words: a non-negative number is left alone, and a negative one has its sign flipped. The minus in front of $a$ on the second line does not make the result negative — it flips the sign of a number that was already negative, so $|-7| = -(-7) = 7$ .

Distance from zero

The same formula has a simpler description: $|a|$ is the distance of $a$ from zero on the number line.

From zero to −4 and from zero to 4 is the same number of steps, so |−4| = |4| = 4.

Two properties follow, and neither has to be memorised separately — the picture shows both:

|a| \ge 0 \qquad |-a| = |a|

A distance is never negative, and opposite numbers sit on either side of zero at the same distance from it. The only number whose modulus is zero is zero itself.

The distance between two numbers

If $|a|$ is the distance from zero, then the distance between any two numbers comes from subtracting them and taking the modulus:

|a - b| = |b - a|

The order of subtraction makes no difference precisely because the modulus removes the sign — and the sign is the only thing that would have changed.

|−3 − 5| = |−8| = 8 — the same as |5 − (−3)| = |8| = 8.

What is the distance between −12 and −4?

Properties of the modulus

The modulus distributes over multiplication and division:

|a \cdot b| = |a| \cdot |b| \qquad \left|\frac{a}{b}\right| = \frac{|a|}{|b|} \quad (b \neq 0)

Over addition it does not — and that is the point of the triangle inequality:

|a + b| \le |a| + |b|

For $a = 3$ and $b = -5$ the left side gives $|-2| = 2$ while the right gives $3 + 5 = 8$ . Equality holds only when both numbers share a sign (or one of them is zero): going once left and once right takes you less far from zero than going twice the same way.

The most-used property ties the modulus to the square root:

\sqrt{a^2} = |a|

Not $a$ — because a square root is non-negative by definition. For $a = -5$ we get $\sqrt{(-5)^2} = \sqrt{25} = 5$ , which is $|-5|$ and not $-5$ .

Equations with absolute value

The equation $|x| = r$ reads as a question: which numbers lie at distance $r$ from zero? The answer depends on $r$ :

$r > 0$ — two solutions: $x = r$ and $x = -r$ ,
$r = 0$ — one solution: $x = 0$ ,
$r < 0$ — no solutions, so the solution set is $\varnothing$ , since a modulus is never negative.

The equation $|x - a| = r$ reads the same way: the numbers at distance $r$ from $a$ , that is $x = a - r$ and $x = a + r$ .

Solve the equation |x − 3| = 4.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

|−7| =

Common mistakes

Taking $|a| = a$ as a general rule — it holds only for $a \ge 0$ . For a negative number the modulus flips the sign.
Reading $-a$ as a negative number — if $a = -7$ , then $-a = 7$ . A minus in front of a letter settles nothing about the sign.
Shortening $\sqrt{a^2}$ to $a$ — the correct result is $|a|$ ; for $a = -5$ the two differ by a sign.
Distributing the modulus over addition — $|a + b|$ generally does not equal $|a| + |b|$ ; only an inequality holds here.
Forgetting the second solution — the equation $|x| = 6$ has two solutions, $6$ and $-6$ , not one.

Formula card

Topic: Absolute value

Definition
$|a| = \begin{cases} a & \text{for } a \ge 0 \\ -a & \text{for } a < 0 \end{cases}$
a non-negative number is left alone, a negative one has its sign flipped
Non-negativity
$|a| \ge 0$
a distance is never negative
Symmetry
$|-a| = |a|$
opposite numbers sit equally far from zero
Distance between two numbers
$|a - b| = |b - a|$
the order of subtraction does not change the distance
Products and quotients
$|a \cdot b| = |a| \cdot |b| \qquad \left|\frac{a}{b}\right| = \frac{|a|}{|b|}$
the modulus distributes over multiplication and division
Root of a square
$\sqrt{a^2} = |a|$
not a, because a square root is always non-negative
Triangle inequality
$|a + b| \le |a| + |b|$
the modulus of a sum never exceeds the sum of the moduli

The numbers −4 and 4 lie on opposite sides of zero, both at a distance of 4 — which is why |−4| = |4| = 4.

From −3 to 5 is 8 steps — exactly what |−3 − 5| gives.

Frequently asked questions

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