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### Real Plane, Complex Plane

```Date: 09/16/2002 at 19:20:59
From: John
Subject: Complex Analysis

When does Az + Bz + c = 0 become a straight line?
```

```
Date: 10/28/2002 at 17:05:48
From: Doctor Nitrogen
Subject: Re: Complex Analysis

Hi, John:

The complex valued equation

w = f(z) = Az + Bz + c = (A + B)z + c = 0,

is never really a straight line. Note there is a difference between

w = f(z) = (A + B)z + c, and

w = f(z) = (A + B)z + c = 0.

And a function w = f(z) in the complex plane does not have the same
meaning or nature as a real valued function y = f(x) in the real
plane. So,

y = f(x) = (A + B)x + c = 0

in the real plane, does not necessarily mean that

w = f(z) = (A + B)z + c = 0

will have the same nature or character in the complex plane. The
reason is that in the real plane you have only two points to
consider,

(x, y)

or the value of x in the domain of f, and the value of y in the range
of f. On the other hand, in the complex plane, you have four values
to consider:

(x, y) for x + yi in the complex x-y plane, and

(u(x, y), v(x, y)) for u(x, y) + v(x, y)i

in the complex u-v plane, where w = f(z) = u + vi = (A + B)z + c = 0.

Therefore, to elaborate on your question, for the complex valued
function

w = f(z) = (A + B)z + c = 0,

you would have a little "vector" with its terminus at the point x + yi
in the x-y complex plane, and you would have

w = f(z) = u(x, y) + v(x, y)i = (A + B)z + c = 0

in the second, u-v plane.

In the u-v plane, w = f(z) = (A + B)z + c = 0, meaning

(1)  z = -c/(A + B), and that

w = f(z) = u(x, y) + v(x, y)i = (A + B)z + c = 0

= Re((A + B)z + c) + Im((A + b)z + c)i = 0

= 0 + 0i = 0, because of (1).

Hence

u(x, y) = v(x, y) = 0, in the u-v plane.

But this would just give the origin in the u-v plane, meaning the
point (x + yi) in the complex x-y plane gets mapped to the origin
(0, 0) in the complex u-v plane, by w = (A + B)z + c = 0.

Incidentally, in the real plane, the equation

y = f(x) = (A + B)x + c = 0

would lead to the point (-c/(A+B), 0)), not the origin.

Keep in mind there is a difference between

w = f(z) = (A + B)z + c, and

w = f(z) = (A + B)z + c = 0, in complex analysis.

Similarly, there is a difference between

y = f(x) = (A + B)x + c, and

y = f(x) = (A + B)x + c = 0, in real analysis.

Forum/Doctor Math whenever you have any math-related questions.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Analysis
College Imaginary/Complex Numbers
High School Analysis
High School Imaginary/Complex Numbers

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