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
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Is $33i$ a real and complex number?

I'm currently learning about complex numbers, and I was wondering: isn't $33i$ the same as $0+33i$ and therefore has a real and imaginary part. Can $33i$ be considered as real and complex?
user351345
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Finding the Inverse of an Element in a Given Group

Find the inverse of $7+3i$ in $\mathbb{C}^*$. I do not have a problem similar to this, therefore any help is appreciated.
rover2
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Real and imaginary part

I need the real and imaginary part of $\log \sin (x+iy)$. I expand $\sin(x+iy)=\sin x \cosh y+i \cos x \sinh y$. But I don't know how to do it s logarithm
Roy Robin
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evaluate complex square root

$\sqrt{i^4}$ $=\sqrt{1}=1$ or $=(\sqrt{i})^4$ $=(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})^4$ $=\exp(4\ln\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}$ $=\exp(4i\frac{\pi}{4})=\exp(\pi)i=-1$ That is in case of the principle argument of the power function I…
Hussien Mohamed
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If $z = 2 - 3i$, show that $z^2 - 4z + 13 = 0$ and hence find the value of $4z^3 - 3z^2 + 169$

This is an example given in my book, and it seems to involve division by 0, so I'm rather confused Here is how it's done in the book: We have $z = 2 - 3i $ So, $z - 2 = -3i$ Now, ${(z-2)}^2 = {(-3i)}^2$ Or, $z^2 + 4 - 4z = 9i^2$ Or $z^2 -4z + 13 =…
Nathuram
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If $w$ is the $n$_th root of unity and $|z-w^k|\le 1$ then prove $z=0$

If $w$ is the $n$_th primitive root of unity and $|z-w^k|\le 1$ then prove $z=0$.(for all $0 \le k \le n-1$) My attempt:We know that $w^k$ generates all roots of unity if $n$ is even by drawing the roots we can find two points that have the distance…
Taha Akbari
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If $z_1,z_2,z_3 \in \Bbb C$ such that $|z_1|=|z_2|= |z_3|=\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right|=1$, then$|z_1+z_2+z_3|$ is?

If $z_1, z_2, z_3$ are complex numbers such that $|z_1|=|z_2|= |z_3| $= $\left|\dfrac{1}{z_1}+\dfrac{1}{z_2}+\dfrac{1}{z_3}\right| = 1$, then$|z_1 + z_2 + z_3|$ is : (A) equal to 1(B) less than 1 (C) greater than 3 (D) equal to…
Samar Imam Zaidi
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Maximum and minimum magnitude of a complex number

For the equation $$|z-a|+|z+a|=2|c|,$$ where $|a|\le|c|$, what are the smallest and largest values of $|z|$? I am finding various bounds, but I don't know which bounds are the most precise ones. For example, I found that $|z| \le |c|$, but can this…
jimm
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Evaluate $(\sqrt{3}-3i)^6$

Evaluate $$(\sqrt{3}-3i)^6.$$ So I assume that we should write the following in polar form $r=\sqrt{(\sqrt{3})^2+(-3)^2}=\sqrt{3+9}=\sqrt{12}=2\sqrt{3},$ $\theta=\arctan{\frac{-3}{\sqrt{3}}}=-\frac{\pi}{3}.$ So…
gbox
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Solve $|\frac{z-2}{z-3}|=2$

How do we show that $|\frac{z-2}{z-3}|=2$ represents a circle, where $z\in\mathbb{C}$. $$ \frac{(x-2)^2+y^2}{(x-3)^2+y^2}=4\implies 3x^2+3y^2-20x+32=0\implies(x-10/3)^2+y^2=(2/3)^2 $$ the substitution $z=x+iy$, shows that it represent a circle with…
Sooraj S
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Complex Numbers Range of Values of Mod and Arg z

Find the range of values of $|z|$ and $\arg (z)$ for $$|z-4-4i| = 2 \sqrt{2}.$$ I'm aware that you can solve this geometrically by drawing a circle on the argand diagram and finding out the information from there. I was curious if there is an…
frog1944
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Analogous numbers to i

$i$ is defined as the square root of $-1$. I was wondering if number systems other than the complex numbers can be reached from the real numbers by a similar process. Like a number whose $\sin$ is $1.5$ or something.
Jacob Wakem
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How to show that all the complex numbers in $\mathbb{C}$ tend to the same point when their modulus tends to infinity?

I'm aware that there's only one 'infinity' in the complex numbers, and I remember one of my professors demonstrating the fact that indeed all numbers tend to the same point as their modulus goes to infinity. I do not remember exactly how he did it,…
Quantaliinuxite
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How do I show that the arc swept by $(1+\frac{i}{n})^n$ has length 1?

I'm working on a tutorial for Euler's identity, and trying to show that the sum of the lengths of the arrows in this picture converges to 1 as $n \rightarrow \infty$ The length of the bottom arrow is $\frac{1}{n}$, and each arrow gets longer by a…
jedediah
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Complex roots of polynomials, proving this particular property

Show that any root $z$ of $z^4 + z + 3 = 0$ satisfies $|z|>1$. I don't see any obvious way to show this; or any good geometrical interpretation if there is any. I tried to consider Vieta's formulae, but wasn't sure what to make use of it. I know…
Twenty-six colours
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