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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what is $\sqrt{(i^4)}$

We know that, $(a^m)^n = a^{mn} = (a^{n})^m$ So , $\sqrt {i^4} = (i^{1/2})^4 = \left(\pm\frac{1+ i}{\sqrt2} \right)^4 = -1$ $\sqrt {i^4} = (i^{2}) = -1$ $\sqrt {i^4} = \sqrt1 = 1$ I think only no $3$ is right. I must have violated some rule in…
Utshaw
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What does varying the imaginary part of a complex base do graphically?

I am trying to build up a visual understanding of how to calculate a complex number to the power of a complex number $(a+bi)^{c+di}$ on the complex plane. I have figured out what variables a,c and d do, but not b. By letting a=2 and letting c=0 we…
John Spence
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Help on solving quadratic equations with complex coefficients

Please help me out here, I'm self-studying Complex Numbers and I've gotten to a point where I'm kinda stuck. Given a general quadratic equation $az^2 + bz + c = 0$ with $a \not= 0$. Using the same algebraic manipulation as in the case of real…
Icosahedron
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Why does there exist an imaginary axis on the Argand diagram?

Complex numbers can puzzle me a bit, and I think I have some gaps in my understanding that makes it confusing for me to wrap my head around. The way I try to explain complex numbers to myself is this: If we do not include complex numbers, we can…
sangstar
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Is the following true $|z|^n=|z^n|$?

Proof attempt: $|z^n|=|z\cdot z \cdot \cdot \cdot z|=|z|\cdot|z|\cdot \cdot \cdot|z|=|z|^n$ So the property is true only for $n\in \mathbb{N}$?
gbox
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Number of values of Z(real or complex) satisying the given system of equations..?

Number of values of Z(real or complex) simultaneously satisfying the system of equations ${1+Z^2+Z^3+....+Z^{17}}$=$0$ and ${1+Z^2+Z^3+....+Z^{13}}$=$0$ is...? My attempt : $Z^{18}-1$$=$$ (Z-1)$ ${(1+Z^2+Z^3+....+Z^{17})}$ (Algebraic identity)…
Arishta
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How can I find maximum and minimum modulus of a complex number?

I have this problem. Let be given complex number $z$ such that $$|z+1|+ 4 |z-1|=25.$$ Find the greastest and the least of the modulus of $z$. I tried with minimum. Put $A(-1,0)$, $B(1,0)$ and $M(x,y)$ present of $z$. We have $O(0,0)$ is the…
minhthien_2016
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Why$\sqrt{-4}$ does not have $\pm 2i$ as its two solutions?

1) $\sqrt{-4} = 2\sqrt{-1} = 2i$ 2) $\sqrt{5+12i} = \pm (3+2i)$ where $i=\sqrt{-1}$ Why $\sqrt{-4}$ does not have $\pm 2i$ as its two solutions? Squaring both $\pm 2i$ will lead us to $-4$. Just like in (2), all the complex numbers which give $-4$…
Arishta
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Complex numbers; trigonometric identity

Use the binomial expansion to find the real and imaginary parts of $(cosθ+isinθ)^5$ Hence show that $sin5θ/sinθ=16cos^4θ-12cos^2θ+1$ I expanded this expression and I got: $cos^5θ+5icos^4θsinθ-10cos^3θsin^2θ-10icos^2θsin^3θ+5cosθsin^4θ+isin^5θ$ Then…
Markowska
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Finding the modulus of a complex number $1-e^{i\theta }$

I have an answer for this question, but I'm not very confident with it: $\left | 1-e^{i\theta } \right |^{2}$ i.e. find the square of the modulus of 1-$e^{i\theta }$ $e^{i\theta } = R(cos\theta +isin\theta )$, R=1 = $cos\theta +isin\theta$ In…
Skylineblue
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Defining the argument of a complex number at origin

For example: The locus of satisfying $\arg(z)=\frac{\pi}{3}$ is ? My Attempt: $$ z =x+iy = |z| \left[ \cos \left( \frac{\pi}{3} \right) + \sin \left( \frac{\pi}{3} \right) \right] = |z| \left[ \frac{1}{2} +i\frac{\sqrt{3}}{2} \right] \\ \implies…
Sooraj S
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Find $|w|$ without finding $w$?

$w$ is a complex number that satisfy: $$5{w^3} - 3i{\left| w \right|^2} - 2i=0$$ Can we find $|w|$ without finding $w$ or easy root?
Piseth
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How many root does $x^5+5x^3+2x^2+4x+1=0$ on the right half complex plane?

How many roots does $x^5+5x^3+2x^2+4x+1=0$ on the right half complex plane? I notice that there can't be positive real roots, so we just need to count the root on the first quadrant and then multiply by 2. We have learnt Rouche's theorem and…
JSCB
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Q6 polynomial complex

Find the equation whose roots are the fourth powers of the roots of the equation $x^3 + x + 1 = 0$. Hence find the sum of the fourth powers of the roots of the equation $x^3 + x + 1 = 0$ I am really stuck. The answer at the back says cubic is $u^3…
m.bazza
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locus of $z=w - \frac{1}{w}$

If $|w|=2$ , then set of points $z=w -\frac{1}{w}$ is equal to ? One of my friend helped me like this: $$|z| = \left| w - \frac{1}{w}\right| \leq |w| + \frac{1}{|w|} = 2 + 0.5 = 2.5 \\ \implies |z| \le 2.5$$ After that I am unable to proceed. Can…
a25bedc5-3d09-41b8-82fb-ea6c353d75ae
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