Absolute Value
Pronunciation: /ˈæb səˌlut ˈvæl yu/ ?
The absolute value of a number is the distance of that number from zero.^{[1]} 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.

Manipulative 1  Absolute Value. Created with GeoGebra.

Click on the blue point in manipulative 1 and drag it to change the figure. The blue
point labeled A represents a value. The green point labeled B
represents the absolute value of A, or A. What happens to
the green point if A is positive? What happens to
the green point if A is negative.
Note that absolute value is always positive or zero. It can never be negative.
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:
Graphing a Linear Absolute Value Equation

Manipulative 2: Graphing a linear absolute value function.


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 = (0,1)

x  1 = 0 → x = 1 y = 1  1  2 → y = 0  2 → y = 0  2 → y = 2 vertex = (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 (1,1).

x = 1 + 1 = 2 y = 2  1  2 y = 1  2 y = 1  2 y = 1 point is (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 (1,1).

x = 1  1 = 0 y = 0  1  2 y = 1  2 y = 1  2 y = 1 point is (0,1).

4 
Draw two rays. Each ray starts at the vertex and goes through one of the two points already plotted.
 


Absolute Value of a 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.
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 .

Educator Resources
Cite this article as:
Absolute Value. 20110312. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/absolutevalue.html.
Translations
Image Credits
Revision History
20110312: 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.)
20100930: Added function notation and additional text on magnitude. (
McAdams, David.)
20091224: Added "References" (
McAdams, David.)
20091209: Added British English Modulus. (
McAdams, David.)
20081119: Added absolute value of a complex number (
McAdams, David.)
20081005: Expanded 'More Information' (
McAdams, David.)
20080916: Changed figure 1 to manipulative (
McAdams, David.)
20080529: Added abs (
McAdams, David.)
20080303: Added graph and function notation (
McAdams, David.)
20070712: Initial version (
McAdams, David.)