Talk:Algebra/Complex Numbers

Are we planning to add complex vector notation (polar notation) here or add that on a separate page?

(Descriptions provided for people inexperienced with complex math)

Essentially: Re^(&theta;i), where R = the absolute value of the complex number and theta is the angle from the (1,0i) in the counterclockwise direction.

To convert Re^(&theta;i) to rectangular notation or complex number format, multiply R times sin(&theta;) for the imaginary and R times cos(&theta;) for the real dimension. Ergo, from this method of calculation we can determine that e^(&pi;ri) = -1.

Additionally, what about hypercomplex numbers, such as quaternions and octonions?

Quaternion = a + bi + cj + dk where i = (-1)^(1/2); j = (-1)^(1/2); k = (-1)^(1/2) and ijk = -1; ij = k; ji = -k; et al...

-- Emperorbma 23:34, 20 Sep 2003 (UTC)

Yes yes, you'll get polar notation and a whole lot more, just wait till I add it! I'm trying to expound this in a logical manner, don't worry :) Quaternions and octonions won't be in complex numbers because they are hypercomplex. Dysprosia 01:19, 21 Sep 2003 (UTC)


 * Just seeing if you needed help, that's all. BTW, should we start a hypercomplex number article, then? -- Emperorbma 20:47, 21 Sep 2003 (UTC)
 * Sorry about the belatedness...sure, why not? :) Dysprosia 11:58, 29 Sep 2003 (UTC)
 * It looks like hypercomplex numbers would best go in Discrete mathematics so I think I'll add it there... -- Emperorbma 17:38, 29 Sep 2003 (UTC)

Introduction Section Editing
Guys, I just edited the introduction section by adding the last para which makes the concept of complex numbers more coherent. - HellKing

i
It is a common mistake to define
 * $$i=\sqrt{-1}$$.

The right definition is to introduce a new number i such that:
 * $$\,i^2=-1$$.

which is not equivalent to the first!

Complex numbers
Better than considering complex numbers as matrices, it is more didactical considering them as pairs of real numbers.

Engineering notation
A suggestion: maybe it would be interesting to add the fact that in engineering notation the imaginary part of a complex number is usually referred to as j instead of $$i$$ (because electric current is also referred to as I). Of course, I don't know about American engineering standards, but as a (European) engineering student I was told to adapt to this convention, even though I of course surely also do understand what is meant by using $$i$$ as a means to indicate the complex part of a complex number.

There's also a convention (in some old engineering texts), to use cis &theta; instead of (cos&theta; + isin&theta;). --Fephisto 19:31, 17 April 2006 (UTC)

Euler's formula
How come one prove that
 * rei&theta; = r (cos &theta; + i sin &theta;)

if rei&theta; is not even defined?

i is defined, and so is ez via Taylor series. If you want, you can define rei&theta; through a Taylor series, and see that this has the same radius of convergence of ez and equate them. --Fephisto 19:34, 17 April 2006 (UTC)

Cauchy
Is stuff about Cauchy's Line integral theorem &c. with line integrals under the Argand Plane, going to be included here? --Fephisto 19:39, 17 April 2006 (UTC)

Wow
I think this page goes way too far. Half this stuff deserves to be in Calculus, not Algebra. We all learned primary ideas about Complex Numbers in Algebra, but this definitely will confuse most readers

Forced and explained through psychology
At the opening to this material the phrases 'forced' in relation to i being equal to square root of minus 1. Also the statement suggesting explanations and ideas around root of minus 1 can be explained through psychology. Is it possible for some direction, link, or further explanation on this.