Linked Questions

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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 ...
1
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3answers
507 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 =...
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3answers
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 $\frac{a}{x^i}$...
0
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3answers
251 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 $\...
-2
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3answers
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 ...
7
votes
2answers
308 views

Is it ever $i$ time?

I am asking this question as a response to reading two different questions: Is it ever Pi time? and Are complex number real? So I ask, is it ever $i$ time? Could we arbitrarily define time as ...
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 ...
2
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2answers
165 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 ...
1
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2answers
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 ...
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 ...
0
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2answers
265 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 ...
11
votes
1answer
974 views

Why is $i$ called “imaginary”?

I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition. Why, then, are they called ...
1
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1answer
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 ...
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
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1answer
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. ...

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