In math, the **absolute value** or **modulus** of a number is its non-negative value or distance from zero. It is symbolized using vertical lines. Here is a look at the absolute value definition, examples, and ways to solve absolute value equations.

### Absolute Value Definition

Absolute value is the non-negative value of a number or expression. For real numbers, it is defined:

|*x*| = *x* if *x* is positive

|*x*| = −*x* if *x* is negative (because -(-*x*) is positive)

|0| = 0

Note that absolute value isn’t technically the “positive” value of a number, because zero has an absolute value, yet is not positive or negative.

### History

The absolute value concept goes back to 1806, when Jean-Robert Argand used the term *module* (meaning unit) to describe the complex absolute value. The English spelling was introduced in 1857 as *modulus*. Karl Weierstrass introduced the vertical bar notation in 1841. Sometimes the term *modulus* is still used, but *absolute value* and *magnitude* describe the same thing.

### Absolute Value Examples

Here are some absolute value examples:

- |9| = 9
- |-3| = 3
- |0| = 0
- |5.4| = 5.4
- |-22.3| = 22.3
- |0 – 1| =1
- |7 – 2| = 5
- |2 – 7| = 5
- |3 x -6| =18
- |-3 x 6| =18
- -|5 – 2| =-3
- -|2 – 5| =-3

### Teaching the Absolute Value Concept

The absolute value concept typically appears in the math curriculum around Grade 6. There are a few ways to introduce in ways that make sense to students and help them practice it.

- Have students identify equivalent absolute value expressions on a number line.
- Compare absolute value to distance. For example, say that two points may be in opposite directions, yet are the same distance from the student’s home, school, etc.
- Give students a number and ask them to come up with absolute value expressions that have the same value.
- Make a card game out of it. Write expressions on several index cards where some cards have the same values. For example, |
*x + 5*| = 20, |*x*| = 15, and |*-15*| all have the same value. Ask students to match equivalent expressions.

### Properties of the Absolute Value

The absolute value has four fundamental properties: non-negativity, positive-definiteness, multiplicativity, and subadditivity. While these properties may sound complicated, they are easy to understand from examples.

- |
*a*| ≥ 0:**Non-negativity**means the absolute value of a number is greater than or equal to zero. - |
*a*| = 0 ⇔*a*= 0:**Positive-definiteness**means the absolute value of a number is zero only if the number*is*zero. - |
*ab*| = |*a*| |*b*|:**Multiplicativity**means the absolute value of a product of two number equals the product of the absolute value of each number. For example, |(2)(-3)| = |2| |-3| =(2)(3) = 6 - |
*a + b*| ≤ |*a*| + |*b*|:**Subadditivity**says that the absolute value of the sum of two real numbers is less than or equal two the sum of the absolute values of the two numbers. For example, |*2 + -3*| ≤ |*2*| + |*-3*| because 1 ≤ 5.

Other important properties include idempotence, symmetry, the identity of indiscernibles, the triangle inequality, and preservation of division.

- ||
*a*|| = |*a*|:**Idempotence**says that absolute value of the absolute value is the absolute value. - |-
*a*| = |*a*|:**Symmetry**states that the absolute value of a negative number is the same as the absolute value of its positive value. - |
*a – b*| = 0 ⇔*a*=*b*:**Identity of indiscernibles**is an equivalent expression for positive-definiteness. The only time the absolute value of*a – b*is zero is when*a*and*b*have the same value. - |
*a – b*| ≤ |*a – c*| + |*c – b*|: The**triangle of inequality**is equivalent to subadditivity. - |
*a / b*| = |*a*| / |*b*| if*b*≠ 0:**Preservation of division**is equivalent to multiplicativeness.

### How to Solve Absolute Value Equations

It’s easy to solve absolute value equations. Just keep in mind a positive and negative number can have the same absolute value. Apply the properties of the absolute value to write valid expressions.

- Isolate the absolute value expression.
- Solve the expression inside the absolute value notation so it can equal both a positive (+) and negative (-) quantity.
- Solve for the unknown.
- Check your work, either graphically or by plugging the answers into the equation.

#### Example

Solve for x when |2x – 1| = 5

Here, the absolute value is already isolated (alone on one side of the equal sign). So, the next step is solving the equation inside the absolute value notation for both positive and negative solutions (2*x*-1=+5 and 2*x*-1=-5):

2*x*-1=+5

2x = 6

x = 3

2*x*-1=-5

2x = -4

x = -2

Now you know possible solutions are x = 3 and x = -2, but you need to verify whether or not both answers solve the equation.

For x = 3:

|2(3) – 1| = 5

|6 – 1| = 5

|-5| = 5

For x = -2:

|2(-2) – 1| = 5

|-4 – 1| = 5

|-5| = 5

So, yes, x = 3 and x = -2 are solutions to the equation.

### Absolute Value for Complex Numbers

The modulus concept originally applied to complex numbers, but students initially learn about absolute value as it applies to real numbers. For complex number, the absolute value of a complex number is defined by its distance from the origin on a complex plane using the Pythagorean theorem.

For any complex number, where *x* is a real number and *y* is an imaginary number, the absolute value of *z* is the square root of x^{2} + y^{2}:

|*z*| = (x^{2} + y^{2})^{1/2}

When the imaginary part of the number is zero, the definition matches the usual description of an absolute value of a real number.

### References

- Bartle; Sherbert (2011).
*Introduction to Real Analysis*(4th ed.), John Wiley & Sons. ISBN 978-0-471-43331-6. - Mac Lane, Saunders; Birkhoff, Garrett (1999).
*Algebra*. American Mathematical Soc. ISBN 978-0-8218-1646-2. - Munkres, James (1991).
*Analysis on Manifolds*. Boulder, CO: Westview. ISBN 0201510359. - Rudin, Walter (1976).
*Principles of Mathematical Analysis*. New York: McGraw-Hill. ISBN 0-07-054235-X. - Stewart, James B. (2001).
*Calculus: Concepts and Contexts*. Australia: Brooks/Cole. ISBN 0-534-37718-1.