Difference between revisions of "Imaginary number"
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The complex number <math>z</math> is imaginary if and only if <math>z = i \textrm{Im}(z)</math>, where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>\textrm{Im}</math> is the [[imaginary part]] [[function]]. | The complex number <math>z</math> is imaginary if and only if <math>z = i \textrm{Im}(z)</math>, where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>\textrm{Im}</math> is the [[imaginary part]] [[function]]. | ||
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| + | The imaginary unit is <math>i</math>, which is <math>\sqrt{-1}</math>. <math>i</math> can be treated as a variable with the distinctive property that <math>i^2= -1</math>. Powers of <math>i</math> can be equal to only 4 possibilities: <math>-i, i, 1, -1</math> because the values cycle. This means that <math>i^1=i</math>, <math>i^2=-1</math>, <math>i^3=-i</math>, <math>i^4=1</math>, <math>i^5=i....</math>. | ||
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| + | Multiplying imaginary/complex numbers is not so much different from multiplying variables. In fact, the easiest way to multiply imaginary numbers is to treat <math>i</math> as a variable and then check to see if there is either <math>i^2</math> or <math>i^4</math> in which case you switch it its real value accordingly. [hide="example"]<math>(2i-1)(i+2)</math> | ||
| + | <math>= 2i^2+4i-i-2</math> (FOIL) | ||
| + | <math>= 2i^2+3i-2</math> | ||
| + | <math>= -2+3i-2</math> | ||
| + | <math>= -4+3i</math>[/hide] | ||
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==See Also== | ==See Also== | ||
Revision as of 11:49, 7 April 2011
An imaginary number is a complex number whose real part is equal to 0. In the complex plane, these numbers lie on the imaginary axis. They are sometimes also refered to as pure imaginary numbers,
The complex number 
is imaginary if and only if 
, where 
is the imaginary unit and 
is the imaginary part function.
The imaginary unit is 
, which is 
. can be treated as a variable with the distinctive property that 
. Powers of can be equal to only 4 possibilities: 
because the values cycle. This means that 
, 
, 
, 
, 
.
Multiplying imaginary/complex numbers is not so much different from multiplying variables. In fact, the easiest way to multiply imaginary numbers is to treat as a variable and then check to see if there is either 
or 
in which case you switch it its real value accordingly. [hide="example"]
(FOIL)
[/hide]
See Also
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