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Topic: [ap-calculus] Number System
Replies: 1   Last Post: Oct 19, 2012 7:40 PM

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Richard Sisley

Posts: 4,189
Registered: 12/6/04
Re: [ap-calculus] Number System
Posted: Oct 19, 2012 7:40 PM
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On Oct 17, 2012, at 3:17 PM, Stephanie Dumoski wrote:

> ------------------------------------------------------------------------------------------------
> I've always understood it to be the second: the complex numbers are made up of two subsets, the reals and the imaginary.
> Every real number is complex (it can be written in the form a+0i).

Here is a slight refinement of Stephanie's classification of the complex number system. First we can define the set of complex numbers as the set of all ordered pairs (a,b) where both "a" and "b" are real numbers. The set of complex numbers of the form (a,0) is identified with the set of real numbers. The set of complex numbers of the form (0,b) with b ? 0 is sometimes called the set of "pure imaginary numbers." Every non-real complex number is equal to the sum of a real number and a pure imaginary number. That is (a,b) with b ? 0 is equal to (a,0) + (0,b). The pure imaginary number (0,1) is commonly given the name "i". When a complex number is multiplied by (0,1) the product is that number rotated 90 degrees counterclockwise. Thus we can also write (a,b) as (a,0) + (b,0)*(01), or (a,0)+ (b,0)*i. Also complex numbers of the form (a,0) can be abbreviated as "a" so that (a,b) = a + b*i.

Defining complex numbers as ordered pairs of real numbers allows for a connection between addition and multiplication of complex numbers and elementary transformations of the plane. This helps make the so called imaginary numbers as "real" as the real numbers.


Richard Sisley
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