## Fields

Fields denoted by **{F,+,x}** are a set of elements which consists of 2 binary operations and must have the following properties:
* **Closure**: For any `a,b` in F, the result of (a+b) or (a.b) will also be a member of the Field F i.e if (a.b)=c, then c belongs to the set F as well.
* **Associative**: For any `a,b,c` in F:
$$a+(b+c)=(a+b)+c$$ $$a.(b.c)=(a.b).c$$
* **Commutative**: For any `a,b` in F:
$$a+b=b+a$$ $$a.b=b.a$$
* **Additive Identity**: There exists an element `e` in F such that for all `a` in F:
$$a + e = e + a = a$$
* **Additive Inverse**: For every `a` in F, there exists an `a'` such that:
$$a + a' = a' + a = e$$
* **Multiplicative Identity**: There exists an element `e` in F such that for all `a` in F:
$$a.e = e.a = a$$
* **Multiplicative Inverse**: For every `a` in F except 0, there exists an $a^{-1}$ such that:
$$a.a^{-1} = a^{-1}.a = 1$$

## Examples

We'll go over some of the known Number Systems to check whether they qualify to be a Field.

1. **Natural Numbers(N)**: Comprises of (1,2,3,4...)
* Closure:✅
* For any (a,b) lets say:
* 4 + 5 = 9 which belongs to N
* 9.10 = 90 which belongs to N
* Associativity:✅
* For any (a,b,c) lets say:
* 4 + (5 + 3)= (5 + 4) + 3 = 12 which belongs to N
* (9.10).3 = 9.(10.3) = 270 which belongs to N
* Commutative:✅
* For any (a,b) lets say:
* 4 + 5 = 5 + 4 = 9 which belongs to N
* 9.10 = 10.9 = 90 which belongs to N
* Additive Identity: ❌
* From intuition, we can say 0 is the additive identity element since:
* 3 + 0 = 0 + 3 = 3
* BUT '0' is generally not included in the set of Natural Numbers.

Hence, Natural Numbers are not a Field.

2. **Whole Numbers(W)**: Comprises of (0,1,2,3,4...)
* Closure:✅
* Associativity:✅
* Commutative:✅
* Additive Identity:✅
* 0 is included in this set:
* 3 + 0 = 0 + 3 = 3
* Additive Inverse: ❌
* Negative numbers are allowed in this set

Hence, Whole Numbers are not a Field.

3. **Real Numbers(W)**: Comprises of (...,-4,-3,-2,-1,0,1,2,3,4...)
* Closure:✅
* Associativity:✅
* Commutative:✅
* Additive Identity:✅
* Additive Inverse: ✅
* Negative numbers are allowed:
* 2 + (-2) = 0 (which is the Additive identity for R)
* Mulitplicative Identity: ✅
* By inuition, this is the number 1:
* 5.1 = 5
* Multiplicative Inverse: ✅
* Since R allows floating point numbers:
* 6.($1/6$) = 1 (which is the mulitplicative inverse for R)

Hence, Reak Numbers are a Field.

4. **Complex Numbers (C)**: This is a field as well.

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#### Note: The notations used in this page may be different than the ones you will find in research papers. This is done on purpose to improve readability. I like to use relatively easy-to-remember symbols.