1. what is complex number? (a pair of real numbers, put together, that you can multiply together)

2. why we need it? (I will leave that to you)

3. how to think about it? (Cartesian coordinate, polar coordinate. )

4. what can you do with it? (multiplication, addition.)

5. real part, imaginary part, modulus (absolute value), argument (aka phase angle)

6. complex conjugate. (hey, what does $\sqrt{-1}$ mean? $i$ or $-i$? does it matter? do we have a preference?)

(optional) other weird “numbers”? (if you walk like a number, talk like a number, operate like a number, I will call you a number!) The notion of a 'field'.

Roughly speaking, a 'ring' is a set whose elements can do addition and multiplication (among) themselves. Example: $\Z$, polynomial. (to be precise, we talk about commutative multiplication, that satisfies $x \cdot y=y\cdot x$. matrix multiplication may not be commutative.)

A 'field', is a ring where any nonzero element has a multiplicative inverse. $\Z$ is not a field. $\Q$, $\R$, $\C$ are field.

$\gdef\F{\mathbb F}$

We talked about finite field. Given a prime number $p$, we define $\F_p = \Z / p\Z$, This notation may reminds you of the quotient vector space $V/W$, indeed, $\Z / p \Z$ is the set of equivalence class, where we say two integers $n_1, n_2$ are equivalent (and write $n_1 \equiv n_2 (mod p)$), if $n_1 - n_2 \in p \Z$, i.e. the difference is a multiple of $p$. In class, we set $p=7$, and we say $1 \equiv 8 (mod 7)$. If we use $[n] = n + p \Z$ the equivalence class that $n$ belongs to, then we write $[1]=[8]$. $$\F_7 = \{ [0], [1], \cdots, [6]\}$$ We have arithematics like $$ [a] + [b] = [a+b], \quad [a] \cdot [b] = [ab]. $$ For example, $[2] \cdot [4] = [8] = [1]$. (When there is no danger of confusion, we just write $n$ for $[n]$)

Question (optional):
1. Can you write down how $\F_5$ behave? For example, what is $[2] + [4] = ?$ What is who multiply $[3]$ equasl $[1]$?

2. 'finite field version of complex number'. Take $K$ be a field. We consider $K[\sqrt{-1}] := K[x] / (x^2+1) = \{a + b x \mid a, b \in K, x^2 = -1\}$. Is this always a field? Namely, can you always define $1/(a+bx)$? We see that for $K = \R$, this works, $1/(a+bx) = (a-bx) / (a^2 + b^2)$. Does the inverse always exist for $K[\sqrt{-1}]$? Try $K = \Q, \F_5, \F_7$, see if you can find some pattern.

3. In class, we also talked about, can you define 'super complex number', that instead of using two real numbers $a,b$ to represent a complex number $a + b i$, but three real numbers? For example, we can try $$ \R[x]/(x^3-1) = \{ a+b x + c x^2 \mid a,b,c \in \R, x^3=1 \} $$ can you define multiplication on it? $ ( 1+ x + x^2) (2 + x) = ?$

Does every nonzero element has a (multiplicative) inverse? For example, $x$ has inverse, $$ 1/x = x^2. $$ $x+1$ has inverse, we have $$ \frac{1}{1+x} = \frac{1-x+x^2}{(1+x)(1-x+x^2)} = \frac{1-x+x^2}{1+x^3} = \frac{1-x+x^2}{2}. $$ Does $x-1$ has inverse?

you may ask: why we care about other 'field'? I am happy with $\C$ and $\R$. I don't have a good answer for that, maybe you will find some application some day.

(part of homework) Read Boas Ch2, section 1 - 9, find 5 interesting problems there and do it. (copy down the problem, so the grader / reader know which one you are doing).