Linked Questions

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 ...
36
votes
5answers
6k views

An example of an easy to understand undecidable problem

I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, ...
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 ...
6
votes
4answers
840 views

Mathematical misconceptions and how to combat them

There are a lot of common misconceptions when it comes to math. A common one that has already been addressed on this site is $1 \neq .999\cdots$, as is that imaginary numbers "do not exist". Another ...
3
votes
10answers
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 ...
476
votes
21answers
78k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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 ...
4
votes
4answers
17k views

Can a circle truly exist?

Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says... A circle is a simple shape of Euclidean geometry ...
0
votes
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
votes
3answers
253 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
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
202 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
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 ...
-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. ...

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