42 questions linked to/from Do complex numbers really exist?
7k views

### What is a simple, physical situation where complex numbers emerge naturally? [duplicate]

I'm trying to teach middle schoolers about the emergence of complex numbers and I want to motivate this organically. By this, I mean some sort of real world problem that people were trying to solve ...
27 views

### Reference on polynomial equation that demonstrates the history of complex numbers

Pardon my ignorance and lack of thorough understanding, but I am missing a piece of the puzzle when it comes to complex numbers and can't seem to find an answer. I have been trying to understand ...
101 views

### The meaning of i [duplicate]

Can someone kind explain the mathematical quantity i to me (which is the square root of -1)? Just to be clear, I'm not actually trying to understand i per se, I'm just trying to understand how it can ...
144 views

### Two fractions in $R[\frac{1}{x}]$ are equal iff $(\exists n\ge 0)(x^n\cdot ax^i=x^n\cdot bx^j)$

Given a ring $R$ and an element $x\in R$, we can adjoint the inverse of $x$ by ring extension $R[\frac{1}{x}]=R[x,y]/\langle xy-1\rangle$, so any elements in $R[\frac{1}{x}]$ is of form $\frac{a}{x^i}$...
30 views

### Multiplication Operation in Complex Numbers (Introduction)?

I just started a book on Complex Variables and there is something I cannot grasp: The product of $z_1z_2$ is defined as follows: $z_1 = (x_1,y_1)$ and $z_2 = (x_2,y_2)$ Now how did they manage to ...
201 views

### A polynomial intersecting the x-axis while not intersecting the x-axis?(Complex Numbers) [duplicate]

I know three questions (that gained momentum) that have been posted asking a question which seems the same, but answers to none of them answer the following very well. Please jump to point 2 & 3 ...
3k views

### Help with Proof of the Associative Property of Addition of Complex Numbers

I am trying to derive a proof of the associative property of addition of complex numbers using only the properties of real numbers. I found the following answer but was hoping someone can explain why ...
107 views

### Is there a proof for existence of complex and hypercomplex numbers [duplicate]

As mathematics advanced ,mathematicians found out new type of numbers such as complex numbers and hypercomplex numbers . I had been really fascinated by this idea and the uses of these numbers. ...
375 views

### Why do we call $\sqrt{-1}$ imaginary and $-1$ real?

Looking at the numbers I can see in nature only positive real numbers. Because many problems couldn't be solved using only positive real numbers a new number set, called "negative numbers" was ...
6k views

### How did early mathematicians make it without Set theory?

It is said that Cauchy was a pioneer of rigour in calculus and a founder of complex analysis. Yet if baffles me as set theory was an invention of the 1870s, 20 years after the death of Cauchy. ...
324 views

### Are real numbers a subset of the complex numbers? [duplicate]

I am having an argument with a friend. I think that in a sense, the answer is no. My reasoning is that in linear algebra, a vector $(a, b)$ is not the same as a vector $(a, b, 0)$ because the first ...
258 views

### If $|z|=\sqrt{a^2+b^2}$, then what is $z$?

Perhaps I’m having some difficulty understanding the complex plane. Say you have a complex number $z=a+bi$, where $a$ is the real part and $b$ is the imaginary part. Why do you plot the real part on ...