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Absolute Values
By Nana, Grade 11
Algebraic Definition
• The absolute value of a number n is denoted |n|
and is defined as follows:
• If n  0, then |n| = n.
• If n < 0, then |n| = -n.
Algebraic Definition
• The absolute value of a number is always
positive.
• For example:
• Because 7 is positive, |7| = 7
• Because -5 is negative, |-5| = -(-5) = 5
Absolute Value and Distance
• If n and b are real numbers, then |n – b|
is the distance between n and b on the
number line.
Absolute Value and Distance
• For example:
• The number |3 + 4 | can be written as | 3 – (-4)|.
• Thus, represents the distance between 3 and –4
on the number line.
Absolute Value and Distance
• Special case:
• alternative definition of |c|
• When b = 0, the distance formula shows that the
distance from n to 0 is |n – 0| = |n|
Geometric Definition
• If n is a real number, then |n| is the
distance from n to 0 on the number line.
Geometric Definition
• For example:
• |5| denotes the distance from 5 to 0 on the
number line, as shown below.
Properties of Absolute Value
• Let n and b represent real numbers.
• 1. |n|  0 and |n| > 0 when n  0
• For example:
• |19| = 19  19  0
• |0| = 0
 0=0
Properties of Absolute Value
• 2. |n| = |-n|
• For example:
• Let n = 2.
Then |n| = |2| = 2 and |-n| = |-2| = 2.
Therefore, |2| = |-2|
Properties of Absolute Value
• 3. |nb| = |n|  |b|
• For example:
• Let n = 5 and b = -4
|nb| = |5 (-4)| = |-20| = 20
|n|  |b| = |5|  |-4| = 5  4 = 20
Therefore, |5 (-4)| = |5|  |-4|
Properties of Absolute Value
• 4. |n/b| = |n| / |b|, where b  0
• For example:
• Let n = -7 and b = 3
|n/b| = |-7/3| = |- 7/3| = 7/3
And |n|/|b| = |-7|/|3| = 7/3
Therefore, |-7/3| = |-7|/|3|
Sources
• Picture #1:
http://www.sosmath.com/algebra/inequalities/pictures
/pic15.gif
• Picture #2:
• http://www.showmethemath.com/Concepts_Explained
/numberLineGifs/absPos5.gif
• Pre-Calculus A Graphing Approach Book by Holt,
Rinehart, and Winston
End of presentation. Thank you.