Difference between revisions of "Imaginary part"
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* <math>\mathrm{Im}(4e^{\frac {\pi i}6}) = \mathrm{Im}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 2</math> | * <math>\mathrm{Im}(4e^{\frac {\pi i}6}) = \mathrm{Im}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 2</math> | ||
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| + | * <math>\mathrm{Im}((1 + i)\cdot(2 + i)) = \mathrm{Im}(1 + 3i) = 3</math>. Note in particular that <math>\mathrm Im</math> is ''not'' in general a [[multiplicative function]], <math>\mathrm{Im}(w\cdot z) \neq \mathrm{Im}(w) \cdot \mathrm{Im}(z)</math> for arbitrary complex numbers <math>w, z</math>. | ||
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| + | ==Practice Problem 1== | ||
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| + | Find the conditions on <math>w</math> and <math>z</math> so that <math>\mathrm{Im}(w\cdot z) = \mathrm{Im}(w) \cdot \mathrm{Im}(z)</math>. | ||
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| + | [[Imaginary part/Practice Problem 1 | Solution]] | ||
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==See Also== | ==See Also== | ||
* [[Real part]] | * [[Real part]] | ||
Revision as of 12:47, 17 September 2006
Any complex number 
can be written in the form 
where 
is the imaginary unit and 
and 
are real numbers. Then the imaginary part of , usually denoted 
or 
, is just the value . Note in particular that the imaginary part of every complex number is real.
Geometrically, if a complex number is plotted in the complex plane, its imaginary part is its 
-coordinate (ordinate).
A complex number is real exactly when 
.
The function 
can also be defined in terms of the complex conjugate 
of : 
. (Recall that if , 
).
Examples
. Note in particular that 
is not in general a multiplicative function, 
for arbitrary complex numbers 
.
. Note in particular that 
is not in general a 
for arbitrary complex numbers 
.Practice Problem 1
Find the conditions on 
and so that 
.
