Further Maths - Roots of Complex Numbers
Flashcards
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\[\vert z^4 \vert = 16\]What is $ \vert z \vert $??
\[2\]#####
\[\arg z^4 = \frac{\pi}{2}\]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??
\[z = 4 + 4i\sqrt{3}\]\[\vert z^4 \vert = 16\]
What is $ \vert z \vert $?
\[\arg z^4 = \frac{\pi}{2}\]
What is $\arg z$?
\[\arg z^3 = 0\]
What is $\arg z$?
\[z = \sqrt[3]{4 + 4i\sqrt{3}}\]
How could you rewrite this?
If the modulus of $z^3$ is $8$, what must the modulus of $z$ be?
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.
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 angle in radians would rotate a complex number by 30 degrees?
\[\frac{\pi}{6}\]What letter is used to represent roots of unity?
What is the sum of the roots of unity always equal to?
What is the angle between the $n$-th roots of a number on an Argand diagram?
What is $1 + w + w^2 + w^3 + …$ equal to?
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\[z^5 + 0z^4 + 0z^3 + 0z^2 + 0z - 1 = 0\]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
What complex number will rotate a complex number by $\frac{2\pi}{3}$ radians?
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\[\text{arg}(z^n) = 0\]What does this mean?? $z^n$ is real and positive, not just real.