# Further Maths - Roots of Complex Numbers

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/roots-of-complex-numbers/ · Updated: 2021-03-04 · Tags: further-maths, school, complex-numbers

## 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.

### 2022-05-15
##### If you had a complex number $$6 + 6i$$ what angle would you need to rotate it by to find the other points that form an equilateral triangle around the origin??
$$
120^\circ
$$

### 2022-05-16
##### Multiplying $6 + 6i$ by the $120^\circ$ rotation matrix gives $-(3 + \sqrt{3}) + i(3 - 3\sqrt{3})$. Instead of multiplying by the rotation matrix again to get the new number, what is simpler??
Multiplying by the $240^\circ$ rotation matrix.

##### $$\text{arg}(z^n) = 0$$ What does this mean??
$z^n$ is real _and_ positive, not just real.

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