Real part: Difference between revisions
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Any [[complex number]] <math>z</math> can be written in the form <math>z = a + bi</math> where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>a</math> and <math>b</math> are [[real number]]s. Then the '''real part''' of <math>z</math>, usually denoted <math>\Re z</math> or <math>\mathrm{Re} z</math>, is just the value <math>a</math>. | Any [[complex number]] <math>z</math> can be written in the form <math>z = a + bi</math> where <math>i = \sqrt{-1}</math> is the [[imaginary unit]] and <math>a</math> and <math>b</math> are [[real number]]s. Then the '''real part''' of <math>z</math>, usually denoted <math>\Re (z)</math> or <math>\mathrm{Re} (z)</math>, is just the value <math>a</math>. | ||
Geometrically, if a complex number is plotted in the [[complex plane]], its real part is its <math>x</math>-coordinate ([[ | Geometrically, if a complex number is plotted in the [[complex plane]], its real part is its <math>x</math>-coordinate ([[abscissa]]). | ||
A complex number <math>z</math> is real exactly | A complex number <math>z</math> is real exactly when <math>z = \mathrm{Re}(z)</math>. | ||
The [[function]] <math>\mathrm{Re}</math> can also be defined in terms of the [[complex conjugate]] <math>\overline z</math> of <math>z</math>: <math>\mathrm{Re}(z) = \frac{z + \overline z}2</math>. (Recall that if <math>z = a + bi</math>, <math>\overline z = a - bi</math>). | The [[function]] <math>\mathrm{Re}</math> can also be defined in terms of the [[complex conjugate]] <math>\overline z</math> of <math>z</math>: <math>\mathrm{Re}(z) = \frac{z + \overline z}2</math>. (Recall that if <math>z = a + bi</math>, <math>\overline z = a - bi</math>). | ||
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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== | ||
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* [[Imaginary part]] | * [[Imaginary part]] | ||
[[Category:Algebra]] | |||
[[Category:Complex numbers]] | |||
Latest revision as of 14:56, 5 September 2008
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 
.
. Note in particular that 
is not in general a 
for arbitrary complex numbers 
.Practice Problem 1
Find the conditions on 
and so that 
.
