@?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?

Flashcards What is $\left(z + \frac{1}{z}\right)$?? $$ 2\cos\theta $$ What is $\left(z - \frac{1}{z}\right)$?? $$ 2i\sin\theta $$ What is $\left(z - \frac{1}{z}\right)^4$?? $$ 16\sin^4\theta $$ What is $\left(z + \frac{1}{z}\right)^n$?? $$ 2^n\cos^n\theta $$ What is $\left(z^5 + \frac{1}{z^5}\right)$?? $$ 2\cos 5\theta $$ What is $\left(z^n - \frac{1}{z^n}\right)$?? $$ 2i\sin n\theta $$ If you’re trying to find out an identity for $\sin^4\theta$, what should you do?? $$ \left(z - \frac{1}{z}\right)^4 $$ If you’re trying to find out an identity for $\sin 4\theta$, what should you do?

See Also [[Further Maths - Complex Numbers]]S Sergeant further-maths/textbooks/year-2/chapter-1-complex-numbers/ex1a Flashcards What is Euler’s relation?? $$ e^{i\theta} = cos \theta + i \sin \theta $$ Why can you rewrite $e^{i\theta}$ as $\cos\theta + i\sin\theta$?? Because the Macluarin series of $\sin x$, $\cos x$ and $e^x$ match up. How can you write a complex number with argument $\theta$ and moudlus $r$ in exponential form?? $$ re^{i\theta} $$ $$e^{\pi i} = -1$$ What is this identity a special case of?

In , how could you write the distance between the two points?? $$ |z_2 - z_1| $$ This result is a case of what result for vectors?? $$ \vec{AB} = b - a $$ For two complex numbers $z_1$ and $z_2$, how could you write the distance between them?? $$ |z_2 - z_1| $$ If $z$ is a variable representing any complex number and $z_1$ is a fixed point, what is $|z - z_2|?? The distance from $z$ to $z_2$.

What is the name for $|z|$?? The modulus of $z$. What is the value of $|z|$, where $z = a + bi$?? $$ \sqrt{a^2 + b^2} $$ What is the value of $|z|^2$, where $z = a + bi$?? $$ a^2 + b^2 $$ If $|z|^2 = a^2 + b^2$, how could you also write $|z|^2$?? $$ (a + bi)(a - bi) $$ What is the word definition of the modulus of $z$?? The distance to $z$ from the origin.

Argand Diagrams Argand diagrams are a way of representing complex numbers by imagining them as points on a plane. In an Argand diagram: The $x$-axis is the “real” axis The $y$-axis is the “imaginary axis The complex number $z = x + yi$ can be represented on the diagram by the point $P(x, y)$ where $x$ and $y$ are coordinates. In other words: The horizontal position represents the real part of $z$. The vertical position represents the imaginary part of $z$.

A complex number is a number with both a real and imaginary part. See Also [[Further Maths - Polar Form]]S [[Further Maths - Argand Diagrams]]S [[Further Maths - Loci in the Argand Diagram]]S [[Further Maths - Regions in the Argand Diagram]]S [[Further Maths - Exponential Form of Complex Numbers]]S [[Further Maths - Trig Equations with Complex Numbers]]S [[Further Maths - Roots of Complex Numbers]]S Introducing $i$ First, a couple of definitions: