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|.

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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:

  1. If the number is positive or zero, use the number without changing it.
  2. 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.
    Series of images showing how to find the absolute value of a positive number
  2. Find the absolute value of a negative number.
    Series of images showing how to find the absolute value of a negative number

Formula

Absolute value can be defined using the distance formula:

|a|=square root(a^2)
or a piecewise function:
Absolute value of x is -x if x is less than zero, or x if x is greater than or equal to zero

Graphing a Linear Absolute Value Equation

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Manipulative 2: Graphing a linear absolute value function.

StepDiscussionExample 1: y = |2x| - 1Example 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 = (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

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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 D=square root of x squared plus y squared 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 |<x,y>|=square root(x<sup>2</sup> + y<sup>2</sup>).

Educator Resources

Cite this article as:


Absolute Value. 2011-03-12. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/absolutevalue.html.

Translations

Image Credits

Revision History


2011-03-12: 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.)
2010-09-30: Added function notation and additional text on magnitude. (McAdams, David.)
2009-12-24: Added "References" (McAdams, David.)
2009-12-09: Added British English Modulus. (McAdams, David.)
2008-11-19: Added absolute value of a complex number (McAdams, David.)
2008-10-05: Expanded 'More Information' (McAdams, David.)
2008-09-16: Changed figure 1 to manipulative (McAdams, David.)
2008-05-29: Added abs (McAdams, David.)
2008-03-03: Added graph and function notation (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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