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.
 Find the absolute value of a positive number.
 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. 

Step  Discussion  Example 1: y = 2x  1  Example 2: y = x1  2 
1 
Find the coordinates of the vertex. The vertex is where the line changes
direction. To find the xvalue 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
References
 McAdams, David E.. All Math Words Dictionary, absolute value. 2nd Classroom edition 201501084799968. pg 9. Life is a Story Problem LLC. January 8, 2015. Buy the book
Educator Resources
Cite this article as:
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.
Image Credits
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.)
6/12/2018: Removed broken links, changed Geogebra links to work with Geogebra 5, updated license, implemented new markup. (
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.)
9/30/2010: Added function notation and additional text on magnitude. (
McAdams, David E.)
12/24/2009: Added "References". (
McAdams, David E.)
12/9/2009: Added British English Modulus. (
McAdams, David E.)
11/19/2008: Added absolute value of a complex number. (
McAdams, David E.)
10/5/2008: Expanded 'More Information'. (
McAdams, David E.)
9/16/2008: Changed figure 1 to manipulative. (
McAdams, David E.)
5/29/2008: Added abs. (
McAdams, David E.)
3/3/2008: Added graph and function notation. (
McAdams, David E.)
7/12/2007: Initial version. (
McAdams, David E.)