# Absolute Value

Pronunciation: /ˈæb.səˌlut ˈvæl.ju/ Explain
Abbreviation: abs

The absolute value of a number is the distance of that number from zero. For real numbers, the absolute value is also called the magnitude. In British English, absolute value is called modulus. Absolute value is written using vertical lines surrounding the values: '|x|' means the absolute value of x. In computers and calculators, absolute value is written as a function, usually abs(a) which means, 'Absolute value of a'.

The absolute value of x is written |x|. The absolute value of -7 is written |-7|.

 Click on the blue point and drag it to change the figure. The blue point labeled a represents a real number. The red point labeled |a| represents the absolute value of a. What happens to the red point if a is positive? What happens to the red point if a is negative? Manipulative 1 - Absolute Value Created with GeoGebra.

Note that absolute value is always positive or zero. It can never be negative.

### How to find the absolute value of a real number

To find the absolute value of a real number:

• If the number is positive or zero, use the number without changing it.
• If the number is negative, change the number to a positive.

#### Demonstration

Click on the blue and yellow boxes below to see the next slide.

1. Find the absolute value of a positive number. 2. Find the absolute value of a negative number. ### Formula

Absolute value can be defined using the distance formula: or a piecewise function: ### How to Graph a Linear Absolute Value Equation

 Check on one of the check boxes to select an equation to plot. Click and drag the black point on the slider to go through the steps. Plot the absolute value equations on graph paper as you go through the steps. Manipulative 2 - Plotting an Absolute Value Equation. Created with GeoGebra.

StepDiscussionExample 1:
y = |2x| - 1
Example 2:
y = |x-1| - 2
1 Find the coordinates of the vertex. The vertex is where the line changes direction. To find the x-value of the vertex, set whatever is inside the absolute value to zero and solve. Substitute that value of x back into the equation to get y. Shortcut: At vertex, everything in the absolute value equals zero. 2x = 0
x = 0
y = |2·0| - 1
y = |0| - 1
y = 0 - 1
y = -1
vertex is (x,y) = (0,-1)
x - 1 = 0
x = 1
y = |1 - 1| - 2
y = |0| - 2
y = 0 - 2
y = -2
vertex is (x,y) = (1,-2)
2 Plot a point to the right of the vertex. To do this, add 1 to the value of x at the vertex, substitute this value of x into the function, then evaluate for y. x = 0 + 1 = 1
y = |2·1| - 1
y = |2| - 1
y = 2 - 1
y = 1
point is (x,y) = (1,1).
x = 1 + 1 = 2
y = |2 - 1| - 2
y = |1| - 2
y = 1 - 2
y = -1
point is (x,y) = (2,-1).
3 Plot a point to the left of the vertex. To do this, subtract 1 from the value of x at the vertex, substitute this value of x into the function, then evaluate for y. x = 0 - 1 = -1
y = |2·(-1)| - 1
y = |-2| - 1
y = 2 - 1
y = 1
point is (x,y) = (-1,1).
x = 1 - 1 = 0
y = |0 - 1| - 2
y = |-1| - 2
y = 1 - 2
y = -1
point is (x,y) = (0,-1).
4 Draw two rays. Each ray starts at the vertex and goes through one of the two points already plotted.

### How to find the absolute value of a complex number

 Click on the blue point and drag it to change the figure. Calculate the absolute value of the complex number. Manipulative 3 - Absolute Value of a Complex Number Created with GeoGebra.
The absolute value of a complex number is the distance of that number from the origin (0,0). The distance formula is used to find the absolute value of a complex number. See manipulative 3.
Absolute Value of a Complex Number
StepEquationDiscussion
1 This is the complex number of which to find the absolute value.
2 Apply the distance formula.
3 Simplify the squares.
4 Simplify the addition.
5 Find the square root.

### Magnitude

In advanced mathematics, when referring to the absolute value of a complex number, the term magnitude is used more often. The word magnitude has a more general meaning. Vectors, which do not have a distance, have a magnitude. The magnitude of a vector is strength of the force represented by a vector. The distance formula also generalizes to a formula for magnitude of a vector. For vector <x,y>, the magnitude is .

1. McAdams, David E.. All Math Words Dictionary, absolute value. 2nd Classroom edition 20150108-4799968. pg 9. Life is a Story Problem LLC. January 8, 2015. Buy the book

### Educator Resources

McAdams, David E. Absolute Value. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/a/absolutevalue.html.

### Revision History

4/12/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Adjusted text to support How To index. Expanded discussion of absolute value of a complex number. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
3/12/2011: Increased font size on manipulative graphics. Added label 'B=abs(A)' to manipulative 1. Changed Figure 2 to Manipulative 2 and Manipulative 2 to Manipulative 3. Change section titled 'Graph' to section titled 'Graphing a Linear Absolute Value Equation' and added how to table. (McAdams, David E.)