Mathematics Stack Exchange
Mathematics Stack Exchange

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
8
votes
7 answers

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$ Hint: solve for $b^2$ in terms of $a^2$ and then solve for $a$ I've attempted the question but I don't think I've done it correctly: $$ \begin{align*} b^2 &= 4 - a^2\\ b &=…
Daryn Wilkinson
  • 185
8
votes
6 answers

Solve $\cos{z}+\sin{z}=2$

I am trying to solve the question: $\cos{z}+\sin{z}=2$ Where $z \in \mathbb{C}$ I think I know how to solve $\cos{z}+\sin{z}=-1$: $1+2\cos^2{\frac{z}{2}}-1+2\sin…
ahorn
  • 2,236
8
votes
5 answers

What's the importance of a formula for the real and imaginary parts of a complex number?

I've learned that $$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$ And that in the number $z=a+bi$, $a$ is the real part and $b$ is the imaginary…
Red Banana
  • 23,956
  • 20
  • 91
  • 192
8
votes
5 answers

How do i find $(1+i)^{100}?$

How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times? Is there a quicker way to do this? The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ but I'm not sure what to do from here.
user160325
  • 157
8
votes
2 answers

Does it make sense to compare complex numbers in certain circumstances?

I know that $\mathbb{C}$ is an unordered field and that (strictly non-real) complex numbers cannot be 'compared' in the sense that one is less than/greater than another. However, we can compare real numbers; geometrically, this is because they lie…
beep-boop
  • 11,595
8
votes
5 answers

Defining i,j,k,l,.. for complex number in a 3D space, in 4D space, etc

Real number is often used to represent a point in a 1-dimensional number line. Real numbers are written as $a$ where $a \in \mathbb{R}$. Complex number is often used to represent a point in a 2-dimensional plane. Complex numbers are written as $a +…
Lie Ryan
  • 1,200
8
votes
1 answer

Complex Numbers....

Suppose a is a complex number such that: $$a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0$$ If m is a positive integer, find the value of: $$(a^2)^m+a^m+\frac{1}{a^m}+\frac{1}{(a^2)^m}$$ My Approach: After I could not solve it using the usual methods I tried…
user99403
8
votes
5 answers

Is there any way to represent an imaginary number?

Is there any way to represent an imaginary number? Like the square root of -1? Is there any possible way to do this? Sorry if you think this is a dumb question. I am a 7th grade student in trigonometry. And I just wanted to ask.
Andrew
  • 183
8
votes
8 answers

what is$ \sqrt{8i}$

Very simple question with an answer that I cannot understand: I have $\sqrt{8i}$, which, I suppose, is the same as $\sqrt{\sqrt{-64}}$. How come that $2+2i$ is the same as $\sqrt{8i}$? My understanding is that $\sqrt{8i}$ is the same as: (a)…
complexguest
  • 393
8
votes
3 answers

Prove equality in triangle inequality for complex numbers

We need to show that $$ |z_{1}+z_{2}+\cdots+z_{n}|=|z_{1}|+|z_{2}|+\cdots+|z_{n}|$$ if and only if $z_{1},z_{2},\dots,z_{n}$ have the same argument (i.e. $z_{j}=r_{j}e^{i\theta}$ for $j=1,\dots,n$). This way [$\Longleftarrow$] is easy, but for the…
Misunderstood Observer
  • 101
8
votes
4 answers

What does the square root of minus $i$ equal?

Can you enter the rabbit hole recursively? If the $ \sqrt{-1} = i $ then, what does $ \sqrt{-i} $ equal?
Dave Kirkby
  • 551
8
votes
1 answer

Complex Roots of Quadratics

Every red pixel corresponds to a complex root of a quadratic $ax^2 + bx + c$ with natural coefficients $0 \leq a,b,c < 100$. The 3 largest black discs are centred at $-1$, $-1/2$ and $0$, and it seems each real rational has its own disc of arbitrary…
Supware
  • 940
8
votes
3 answers

Complex solutions of a quintic equation: $z^5+4\overline z^3=0$

Solve $$z^5+4\overline z^3=0$$ This is what I did. Let $z=r(\cos\theta+i\sin\theta)$ so, the equation is: $$r^5(\cos5\theta+i\sin5\theta)+4r^3(\cos3\theta-i\sin3\theta)=0.$$ Suppose $r$ is not $0$ and divide both sides by $r^3$ to…
Omer
  • 2,490
8
votes
4 answers

Does Euler's formula give $e^{-ix}=\cos(x) -i\sin(x)$?

Does Eulers formula give $$e^{-ix}=\cos(x) -i\sin(x)$$ I know that $$e^{ix}=\cos(x)+i\sin(x)$$ But how does it work when we have a $-$ in front
Alex5207
  • 605
8
votes
2 answers

$\sum z_i^k=0$ for all $k\ge 2$ implies $z_1=\dots=z_n=0$?

Let $z_1, z_2,\dots,z_n \in \mathbb C$ such that $$z_1^k+\dots+z_n^k=0$$ for all integers $k\geq 2$. Then how to prove that $z_1=z_2=\dots=z_n=0$? My try: I got to work it for $n=2$ case, by using some brute force calulation. i.e. taking…
Mayuresh L
  • 1,830
1 2 3
…
99 100