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.