Fields and complex numbers

Fields

In order to propely discuss the concept of vector spaces in linear algebra, it is necessary to develop the notion of a set of “scalars” by which we allow a vector to be multiplied. A framework within which our concept of real numbers would fit is desireable. Thus, we would like a set with two associative, commutative operations (like standard addition and multiplication) and a notion of their inverse operations (like subtraction and division). The mathematical algebraic construct that addresses this idea is the field. A field $(S,+,*)$ is a set $S$ together with two binary operations $+$ and $*$ such that the following properties are satisfied.

1. Closure of S under $+$ : For every $x,y\in S$ , $x+y\in S$ .
2. Associativity of S under $+$ : For every $x,y,z\in S$ , $(x+y)+z=x+(y+z)$ .
3. Existence of $+$ identity element: There is a $e_{+}\in S$ such that for every $x\in S$ , $e_{+}+x=x+e_{+}=x$ .
4. Existence of $+$ inverse elements: For every $x\in S$ there is a $y\in S$ such that $x+y=y+x=e_{+}$ .
5. Commutativity of S under $+$ : For every $x,y\in S$ , $x+y=y+x$ .
6. Closure of S under $*$ : For every $x,y\in S$ , $x*y\in S$ .
7. Associativity of S under $*$ : For every $x,y,z\in S$ , $(x*y)*z=x*(y*z)$ .
8. Existence of $*$ identity element: There is a $e_{*}\in S$ such that for every $x\in S$ , $e_{*}+x=x+e_{*}=x$ .
9. Existence of $*$ inverse elements: For every $x\in S$ with $x\ne e_{+}$ there is a $y\in S$ such that $x*y=y*x=e_{*}$ .
10. Commutativity of S under $*$ : For every $x,y\in S$ , $x*y=y*x$ .
11. Distributivity of $*$ over $+$ : For every $x,y,z\in S$ , $x*(y+z)=xy+xz$ .

While this definition is quite general, the two fields used most often in signal processing, at least within the scope of this course, are the real numbers and the complex numbers, each with their typical addition and multiplication operations.

The complex field

The reader is undoubtedly already sufficiently familiar with the real numbers with the typical addition and multiplication operations. However, the field of complex numbers with the typical addition and multiplication operations may be unfamiliar to some. For that reason and its importance to signal processing, it merits a brief explanation here.

Definitions

The notion of the square root of $-1$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $\sqrt{-1}$ could be defined. Euler first used $i$ for the imaginary unit but that notation did not take hold untilroughly Ampère's time. Ampère used the symbol $i$ to denote current (intensité de current).It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using $i$ for current was entrenched and electrical engineers now choose $j$ for writing complex numbers.

An imaginary number has the form $jb=\sqrt{-b^2}$ . A complex number , $z$ , consists of the ordered pair ( $a$ , $b$ ), $a$ is the real component and $b$ is the imaginary component (the $j$ is suppressed because the imaginary component of the pair is always in the second position). The imaginary number $jb$ equals ( $0$ , $b$ ). Note that $a$ and $b$ are real-valued numbers.

[link] shows that we can locate a complex number in what we call the complex plane . Here, $a$ , the real part, is the $x$ -coordinate and $b$ , the imaginary part, is the $y$ -coordinate.

The complex plane