Flashcards
$$|z^4| = 16$$ What is $|z|$??
$$ 2 $$
$$\arg z^4 = \frac{\pi}{2}$$ What is $\arg z$??
$$ \frac{\pi}{8} + \frac{2\pi n}{4} $$
$$\arg z^3 = 0$$ What is $\arg z$??
$$ \frac{2\pi n}{3} $$
$$z = \sqrt[3]{4 + 4i\sqrt{3}}$$ How could you rewrite this??
$$ z = 4 + 4i\sqrt{3} $$
If the modulus of $z^3$ is $8$, what must the modulus of $z$ be??
$$ 2 $$
If the argument of $z^3$ is $\frac{\pi}{3}$, what must the argument of $z$ be??
$$ \frac{\pi}{9} $$
What does $+ \frac{2\pi n}{k}$ represent when working out the root of a complex number??
The different starting positions that would result in the same position.
In general, what do the $n$-th roots of a number form on an Argand diagram??
A regular $n$-gon.
What shape do cube roots form on an Argand diagram??
A triangle.
What letter is used to represent roots of unity??
$$ w $$
What is the sum of the roots of unity always equal to??
$$ 0 $$
What is the angle between the $n$-th roots of a number on an Argand diagram??
$$ \frac{2\pi}{n} $$
What is $1 + w + w^2 + w^3 + …$ equal to??
$$ 0 $$
What angle in radians would rotate a complex number by 30 degrees?
$$ \frac{\pi}{6} $$
What complex number will rotate a complex number by $\frac{2\pi}{3}$ radians??
$$ 1\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) $$
Given the complex number $\sqrt{3} + i$, how would you find the other two points that form an equilateral triange around the origin??
Write it in modulus-argument form and multiply by the complex number with modulus $1$ and argument $\frac{2\pi}{3}$.
If you were asked to form a regular pentagon from complex numbers that weren’t around the origin, how could you do it??
Translate the points so they are around the origin, do modulus-argument magic, translate back.
How could you rewrite $z^5 = 1$ as a 5-th degree polynomial??
$$ z^5 + 0z^4 + 0z^3 + 0z^2 + 0z - 1 = 0 $$
$$z^5 + 0z^4 + 0z^3 + 0z^2 + 0z - 1 = 0$$ Because of the rules from Roots of Polynomials, what is notable about the second coefficient being $0$??
The sum of the roots is the negative coefficient of the second term, so the sum of the roots of unity must be zero.
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date: 2021-03-04 16:06
tags:
- '@?further-maths'
- '@?school'
- '@?public'
- '@?complex-numbers'
title: Further Maths - Roots of Complex Numbers