Further Maths - Roots of Complex Numbers
Pearson Edexcel Further Mathematics 2022
Flashcards
Modulus and argument of a root
\[\arg z^4 = \frac{\pi}{2}\]
What is $\arg z$?
\[z = \sqrt[3]{4 + 4i\sqrt{3}}\]
How could you rewrite this?
If the argument of $z^3$ is $\frac{\pi}{3}$, what must the argument of $z$ be?
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.
Geometry of roots
In general, what do the $n$-th roots of a number form on an Argand diagram?
A regular $n$-gon.
What is the angle between the $n$-th roots of a number on an Argand diagram?
Rotating complex numbers
What angle in radians would rotate a complex number by 30 degrees?
What complex number will rotate a complex number by $\frac{2\pi}{3}$ radians?
Given the complex number $\sqrt{3} + i$, how would you find the other two points that form an equilateral triangle 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.
Roots of unity as polynomials
How could you rewrite $z^5 = 1$ as a 5-th degree polynomial?
\[z^5 + 0z^4 + 0z^3 + 0z^2 + 0z - 1 = 0\]
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.
Equilateral triangle from 6 plus 6i
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?
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.