43 questions linked to/from Do complex numbers really exist?
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### Quotient field operations are well-defined: fleshing out Vinberg's sketch

Let $A$ be a non-trivial integral domain. Define the relation $\sim$ on the set of pairs $A \times A\setminus\{0_A\}$ as follows: (a_1,b_1) \sim (a_2,b_2) \overset{\text{def}}{\Longleftrightarrow} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\frac{a}{x^i}$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
One of the most common manipulations performed when working with group equations is left or right multiplication, i.e. if you have a group $G$ with $a,b,c \in G$ and you have something of the form \$a =...