2
votes
4answers
54 views

What is a complex constant and how do I use it?

I have a question I am trying to understand: "Let $b$ and $c$ be complex constants such that $z^2+bz+c=0$ has two different real roots. Show that $b$ and $c$ are real." My biggest problem here is ...
4
votes
3answers
90 views

Why is this definition of complex numbers “informal”?

I'm reading the proofwiki page about complex number: https://proofwiki.org/wiki/Definition:Complex_Number According to proofwiki there is an informal and formal definitions of complex numbers. The ...
4
votes
2answers
114 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
6
votes
8answers
536 views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
1
vote
3answers
74 views

Why is $ |z|^2 = z z^* $?

I've been working with this identity but I never gave it much thought. Why is $ |z|^2 = z z^* $ ? Is this a definition or is there a formal proof?
1
vote
2answers
36 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
-3
votes
6answers
296 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
6
votes
3answers
281 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
37
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
2
votes
3answers
186 views

The negative square root of $-1$ as the value of $i$

I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i = (-1)^{1/2} $. I ...
0
votes
0answers
123 views

What is the correct definition for an imaginary number?

The following is taken from Wikipedia's definition. An imaginary number is a number whose square is less than or equal to zero. But I also heard that An imaginary number is a number whose ...
12
votes
2answers
661 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
3
votes
1answer
148 views

Complex conjugates?

Is $e^{1\over 1-ix}$ the complex conjugate of $e^{1\over 1+ix}$? Is there a simple rule to compute complex conjugates without having to find $a+ib$ form? Thanks.