### Appendix: Field

##### 2. Complex number

In this section, we will list important facts about the complex numbers.

Geometry We may identify **C** with the real plane **R**^{2} by `a` + `ib` ↔ (`a`, `b`). We call `a` = `Re`(`z`) the real part and `b` = `Im`(`z`) the imaginary part. Correspondingly, **C** is a real plane with specific choices of real and imaginary axis.

The geometry of **R**^{2} leads to two concepts.

geometry of **R**^{2} |
complex number |
formula |

concept |
notation |
concept |
notation |

length |
`r` |
absolute value/norm |
|`z`| |
√`a`^{2} + `b`^{2} |

angle |
`θ` |
argument |
`arg`(`z`) |
`tan`^{-1}(`b`/`a`) |

Note that angle/argument is a number well-defined only up to +2`kπ`

Clearly, the addition in **C** corresponds to the vector addition in **R**^{2}. To see the geometric meaning of multiplication, we note that the complex number is (the reason for the red equality is here)

`z` = `a` + `ib` = `r`cos`θ` + `ir`sin`θ` = `r`(cos`θ` + `i`sin`θ`) = `re`^{iθ}.

Thus the product of two complex numbers is

`z z'` = `re`^{iθ} r'e^{iθ'} = `rr' e`^{iθ}e^{iθ'} = `rr' e`^{i(θ+θ')}.

In other words, multiplying complex numbers means multiplying the lengths and adding the angles.

*Note* The exponential function `f`(`t`) = `e`^{ct} for real `t` and `c` is characterized by the property that `f'` = `cf`, `f`(0) = 1.Now for `f`(`t`) = cos`t` + `i`sin`t`, we have `f'`(`t`) = -sin`t` + `i`cos`t` = `i`(`i`sin`t` + cos`t`) = `if`(`t`) and `f`(0) = cos0 + `i`sin0 = 1. Thus `f`(`t`) = cos`t` + `i`sin`t` fits into the similar characterization with `c` replaced by `i`. This suggests cos`t` + `i`sin`t` = `e`^{it}.

Conjugation Another important thing we need to know about complex numbers is the conjugation:

`a` + `ib` = `a` - `ib`.

Geometrically, this means the reflection in the real axis.

The following lists some important facts about the the conjugation.

- addition and multiplication
`z` + `z'` = `z` + `z'`, `z z'` = `z` `z'`
- twice conjugation = identity
`w` = `z` ⇒ `z` = `w`
- absolute value
`z` `z` = |`z`|^{2}
- real and imaginary parts
- 2
`Re`(`z`) = `z` + `z`, 2`Im`(`z`) = `z` - `z`
- realness
`z` is a real number ⇔ `z` = `z`

Fundamental Theorem of Algebra The complex number is an extension of the real number. The extension was invented in order to solve problems that require taking the square root of -1. A typical such problem is to find roots (solutions of `p`(`t`) = 0) of quadratic polynomials `p`(`t`) = `a`_{0} + `a`_{1}`t` + `a`_{2}`t`^{2}. In general, a (real) quadratic polynomial may not have real roots. On the other hand, (real or complex) quadratic polynomials always have complex roots.

In fact, the existence of complex roots is not restricted to quadratic polynomials. The following is the so called fundamental theorem of algebra.

Any complex polynomial of positive degree has complex roots

For any field `F`, we may also ask whether any polynomial with coefficients in `F` always has solutions in `F`. If the answer is yes, then we say `F` is algebraically closed. Thus **C** is an algebraically closed field. It is often advantageous to develop theories (especially the ones involving polynomials) over an algebraically closed field.