Complex Numbers

Main post link -> http://blog.brilliant.org/2013/02/13/complex-numbers/

Read the full explanation on the blog. Feel free to try out these test yourself questions here. If you show your work there is more to discuss.

Test Yourself

Determine the 4 roots of the equation $x^4 + 1 = 0$ , and find their real and imaginary parts.
Verify that the norm distributes over multiplication and division. Specifically, show that $N( z \times w ) = N( z ) \times N( w )$ and that $N\left( \frac {z}{w} \right) = \frac { N( z )} { N( w )}$ . Give examples to show that the norm does NOT distribute over addition and subtraction.
If $\frac { (1+2i)(2+3i)}{8+i} = a + bi$ , what is $a^2 +b^2$ ? Hint: There is no need to determine the exact values of $a$ and $b$ .
Determine the square root of $1-i$ .

Note: We can show that the square root of $z = a + bi$ is equal to $\pm \left( \sqrt{ \frac {a + \sqrt{a^2+b^2}}{2}} + sgn(b) \sqrt{ \frac {-a + \sqrt{a^2 + b^2}}{2}} \right)$ . Currently, our only way to show this is through brute force multiplication. We will be learning how to approach this problem otherwise.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Complex Numbers

Comments