Crazily Complicated Complex Imaginary Imagination

Algebra Level pending

Let $z = a + bi$ , such that $a,b$ are real numbers and $b \neq 0$ . Which of the following statements is true?

Clarification: Signs refer to "positive" or "negative". For instance, 1 is positive, whereas $-1$ is negative. 0 is neither positive nor negative.

Image Credit: Eric Bogatin's Signal Integrity Academy Blog

It is impossible to define the sign of

z

. The sign of

z

depends on

a

and

b

. If

a

and

b

have unlike signs, then the sign of

z

is the sign of

a

. If

a

and

b

have common signs, so is

z

. If

a

and

b

have unlike signs, then the sign of

z

is the sign of

b

2 solutions

Michael Huang
Jan 6, 2017

Easy Approach

By definition of sign function, if $x$ is a number defined on the set of real numbers, $\text{sgn}(x) = \left\{ \begin{array}{rl} 1, & x > 0\\ 0, & x = 0\\ -1, & x < 0 \end{array} \right.$ If suppose $z = a + bi$ is a number that occurs in the plane, then it does not exist within the domain on the real numbers.

Mathematical Approach

Assume by contradiction that $z = a + bi = re^{i\theta}$ can be identified with a sign. Then $\text{sgn}(z) = \dfrac{z}{|z|} = \dfrac{re^{i\theta}}{r} = e^{i\theta}$ Converting into trig, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ But since $b \neq 0$ , then $\sin(\theta) \neq 0$ , so $\theta \neq \pi k$ for positive integer $k$ . So $\text{sgn}(z) = \cos(\theta) + i\sin(\theta)$ This cannot happen as the range is already defined from the previous section. There doesn't exist an imaginary unit $i$ !

Thus, it is $\boxed{\text{impossible}}$ to determine whether $z$ is negative or positive (or neither).

Note: Here is the set-theory approach .

Note: Try reading measurement with negative and positive real numbers. Other than physics and complex variable topics, where in the world can you see imaginary unit in real life? :)

Steven Chase
Jan 5, 2017

A complex number in the Cartesian plane can be described in two different ways:

1) Magnitude and angle

2) Real part and imaginary part

The notion of a "sign" can be applied to either component of a complex number, but it cannot be sensibly applied to the complex number as a whole.

Or you can consider the sign function defined on the real numbers.

This is not as surprising as you think since sign can't be defined in the complex numbers. You can also consider what @Brian Charlesworth mentioned. :)

Another way to approach this problem: Try measuring complex number with a ruler. Is it possible to determine the sign of the complex number that way? Do you see at least a thermometer showing complex number?

I hope nobody is imagining things or imagining imaginary unit in real life! XD

Michael Huang - 4 years, 5 months ago

All numbers are imaginary, but some are more imaginary than others. :) Joking aside, all numbers, real and complex, are abstractions that are useful - essential - in helping us make sense of the world. And without "imaginary", i.e., complex, numbers we would not be able to fully understand the fundamental forces of the universe, so in that sense they do provide an "order" to the seeming chaos we inhabit. But semantics aside, I can imagine a thermometer with complex units: $r$ , being positive, would represent the empirical temperature in degrees Kelvin, and the argument would represent, say, the rate of change of $r$ at a given point in time.

Brian Charlesworth - 4 years, 5 months ago

It should be noted that there is a "sign" function defined on a complex number $z = re^{i\theta}, r \ne 0$ , namely

$sgn(z) = \dfrac{z}{|z|} = e^{i\theta}$ ,

but this does not correspond to the paradigm of "ordering" outlined in the question.

Brian Charlesworth - 4 years, 5 months ago

Crazily Complicated Complex Imaginary Imagination

2 solutions

Easy Approach

Mathematical Approach

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