Linked Questions

5
votes
2answers
169 views

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} ...
44
votes
17answers
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 ...
1
vote
2answers
28 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 ...
3
votes
10answers
102 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 ...
0
votes
3answers
152 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 $\frac{a}{x^i}$...
0
votes
2answers
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 ...
-2
votes
3answers
206 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 ...
1
vote
3answers
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 ...
-2
votes
1answer
108 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. ...
6
votes
6answers
378 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 ...
47
votes
7answers
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. ...
1
vote
1answer
333 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 ...
2
votes
6answers
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 ...
1
vote
3answers
512 views

Group operation is well defined

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 =...
7
votes
8answers
1k views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
9
votes
3answers
699 views

difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
1
vote
4answers
930 views

Why is $i^2$ equal to $-1$? [duplicate]

In this KhanAcademy link at 2:25, Sal (the narrator) says that $i^2$ is negative 1 and he didn't explain why. Why is this so? What is the intuition behind it?
2
votes
2answers
167 views

Imaginary Numbers

I imagine there have been many questions about imaginary numbers, so if I am asking a question already answered my apologies. I understand that it is perfectly correct to create new number systems ...
22
votes
8answers
2k views

Refining my knowledge of the imaginary number

So I am about halfway through complex analysis (using Churchill amd Brown's book) right now. I began thinking some more about the nature and behavior of $i$ and ran into some confusion. I have seen ...
9
votes
4answers
969 views

Why does the imaginary number $i$ satisfy $i\times 0=0$?

Why does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that.
42
votes
9answers
7k views

Is complex analysis more “real” than real analysis?

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
0
votes
3answers
254 views

Isomorphism of $\mathbb{Z}/n \mathbb{Z}$ and $\mathbb{Z}_n$

Let $n\in \mathbb{Z}^+$. How do I prove that $\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}_n$? Is there any good homomorphism $\phi$ I could use that graphs $\mathbb{Z}/n\mathbb{Z}$ to $\...
0
votes
1answer
170 views

What are complex numbers, actually? [duplicate]

What are complex numbers, actually? You can prove $1=-1$ and a complex cosine function can have value greater than $1$ and so on, there are many unexpected results when we use complex numbers. So, ...
2
votes
4answers
210 views

Regarding uses of $i$ (square root of $-1$) [duplicate]

Are there any uses of 'square root of $-1$' in practical life ; like in Physics ?
53
votes
8answers
6k 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 ...
0
votes
2answers
267 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just how ...
3
votes
4answers
2k views

Understanding the quotient ring $\mathbb{R}[x]/(x^3)$.

I am having difficulty in understanding exactly the elements of the set $\mathbb{R}[x]/(x^3)$. I'll explain my thought process. The Quotient Ring is the set of additive cosets, so we have that $$\...
60
votes
15answers
7k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
5
votes
6answers
735 views

The notion of complex numbers

How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will ...
3
votes
2answers
1k views

What is a formal polynomial?

I'm starting to study Field Theory by myself, the books don't say explicitly what a polynomial is, I mean, what the $x$ of $f(x)$ in $F[x]$ is? $x\in F$? When I take $f(\alpha)$ am I taking the ...

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