Difference between revisions of "Real part"
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* <math>\mathrm{Re}(4e^{\frac {\pi i}6}) = \mathrm{Re}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 2\sqrt 3</math> | * <math>\mathrm{Re}(4e^{\frac {\pi i}6}) = \mathrm{Re}(4(\cos \frac \pi6 + i \sin \frac\pi 6)) = 2\sqrt 3</math> | ||
− | * <math>\mathrm{Re}((1 + i)\cdot(2 + i)) = \mathrm{Re}(1 + 3i) = 1</math>. Note in particular that <math>\mathrm Re</math> is ''not'' in general a [[multiplicative function]], <math>\mathrm{Re}(w\cdot z) \neq \mathrm{Re}(w) \cdot \mathrm{Re}(z)</math> for arbitrary complex numbers <math>w, z</math>. | + | * <math>\mathrm{Re}((1 + i)\cdot(2 + i)) = \mathrm{Re}(1 + 3i) = 1</math>. Note in particular that <math>\mathrm {Re}</math> is ''not'' in general a [[multiplicative function]], <math>\mathrm{Re}(w\cdot z) \neq \mathrm{Re}(w) \cdot \mathrm{Re}(z)</math> for arbitrary complex numbers <math>w, z</math>. |
==Practice Problem 1== | ==Practice Problem 1== |
Revision as of 15:44, 21 September 2006
Any complex number can be written in the form
where
is the imaginary unit and
and
are real numbers. Then the real part of
, usually denoted
or
, is just the value
.
Geometrically, if a complex number is plotted in the complex plane, its real part is its -coordinate (abscissa).
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
.
Practice Problem 1
Find the conditions on and
so that
.