Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
71
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Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone please explain the logic or rationale behind this?…
user64742
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52
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9 answers

Is $|1-i|$ larger than $|1|$?

I am confused about complex numbers. Does $1-i$ lie outside the unit circle? How do I show that the absolute value of $1-i$ is larger than that of $1$?
phil12
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52
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3 answers

Is $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$?

I was messing with the identity $e^{i\pi}=-1$ and I got that $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$ and on. I plugged it in to a calculator and it was infinite. It grew very fast. Does that make $i$ solvable?
42
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3 answers

Total ordering on complex numbers

Show that there doesn't exist a relation $\succ$ between complex numbers such that (i) For any two complex numbers $z,w$, one and only one of the following is true: $z\succ w,w\succ z,$ or $z=w$ (ii) For all $z_1,z_2,z_3\in\mathbb{C}$ the relation…
Mika H.
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40
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7 answers

What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known facts!) I was trying to ask the geometric…
math
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35
votes
12 answers

What allows us to use imaginary numbers?

What axiom or definition says that mathematical operations like +, -, /, and * operate on imaginary numbers? In the beginning, when there were just Reals, these operations were defined for them. Then, i was created, literally a number whose value is…
26
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6 answers

Proof that sum of complex unit roots is zero

When reading a proof of why $x^3+y^3=z^3$ has no nontrivial integer solutions I came across following identity: $$ y^3 = z^3-x^3 = (z-x)(z-\omega x)(z-\omega^2 x) \qquad \text{where } \omega = e^{2\pi i /3} \quad \text{i.e.}\quad \omega^3 =…
flawr
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25
votes
9 answers

Why did Euler use e to represent complex numbers?

From Euler we've learned that $z=re^{i\theta}$. And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$. Why must we use e to represent these numbers correctly? It seems that I could arbitrarily choose a different…
24
votes
6 answers

Can I keep adding more dimensions to complex numbers?

I know about the concept of the complex plane, but is it possible to move to the third dimension? What about arbitrary many dimensions? Edit: could you please give me some examples of 3D numbers?
jcora
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24
votes
8 answers

What's the precise meaning of imaginary number?

The same to the title,what's the precise meaning of imaginary number? And on the other hand,how can the imaginary number be reflected in Physics?
mathon
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24
votes
3 answers

Is the square root of a negative number defined?

I have been in a debate over 9gag with this new comic: "The Origins" And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$". And then, this comment cast a doubt: There is no such thing as sqrt(-1). The square root function is only …
zneak
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23
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4 answers

What is the precise definition of $i$?

This may seem like an extraordinarily trivial question and yet it has completely confounded me. The technical definition of $i$ is $$i^2=-1$$ But there are two numbers which fulfill this requirement: $$\sqrt{-1},-\sqrt{-1}$$ Wouldn't a more precise…
20
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3 answers

Why should we expect the connection between complex arithmetic and geometry?

I realized that I take it for granted that properties of complex numbers have clear geometric interpretations. Visualizing complex numbers with the help of the complex plane really helps to understand complex arithmetic better and those mysterious…
20
votes
5 answers

Proving that a complex number is real

If $$\left|\frac{Z_1 - iZ_2}{Z_1 + iZ_2}\right| = 1$$ then prove that $Z_1/Z_2$ is real . This is how I proceeded. Dividing throughout by $Z_2$ we will have $$\left|{\frac{\frac{Z_1}{Z_2} - i}{\frac{Z_1}{Z_2} + i}}\right| = 1$$ Thus…
Aditi
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18
votes
6 answers

Complex power of a complex number

Can someone explain to me, step by step, how to calculate all infinite values of, say, $(1+i)^{3+4i}$? I know how to calculate the principal value, but not how to get all infinite values...and I'm not sure how to insert the portion that gives me…
Johnny Apple
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