Difference between revisions of "Real part"
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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>). | ||
Revision as of 11:14, 16 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 
.
. Note in particular that 
is not in general a 
for arbitrary complex numbers 
.Practice Problem 1
Find the conditions on 
and so that 
.
