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Home Real and Hyperreal Numbers Functions of Real Numbers Theorem and Proof: Absolute Value Function
 
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Theorem and Proof: Absolute Value Function

THEOREM 1

Let a and b be real numbers.

(i) |-a| = |a|.

(ii) |ab| = |a| · |b|.

(iii) If b ≠ 0, |a/b| = |a| / |b|

PROOF We use the equation |x| =01_real_and_hyperreal_numbers-46.gif

(i) |-a|=01_real_and_hyperreal_numbers-47.gif=01_real_and_hyperreal_numbers-48.gif=|a|

(ii) |ab| =01_real_and_hyperreal_numbers-49.gif= 01_real_and_hyperreal_numbers-50.gif= 01_real_and_hyperreal_numbers-51.gif= |a| · |b|.

(iii) The proof is similar to (ii).

Warning

The equation |a + b| = |a| + |b| is false in general. For example, |2 + (-3)| = 1, while |2| + |(-3)| = 5.
Home Real and Hyperreal Numbers Functions of Real Numbers Theorem and Proof: Absolute Value Function

Last Update: 2006-11-09