A Field is a set whose members follow several rules. Any set that follows all the rules is a set. The most commonly known fields are the set of Integers and the set of Real Numbers.
The set has two operations defined on it called Addition and Multiplication. Several definitions and math statements known as Field Axioms must all be true.
Associativity is true for both addition and multiplication:
- (a+b)+c = a+(b+c)
- (a*b)*c = a*(b*c)
Commutativity is true for both addition and multiplication:
- a+b = b+a
- a*b = b*a
Distributivity of Multiplication over Addition:
a*(b+c) = a*b + a*c
An inverse element exists for addition and an inverse element exists for multiplication.
Identity elements exist for addition and multiplication.
Appendix A
One of the above statement to the rules to be as brief as possible. Therefore things we want you to know have been put here to be separate from those explanations.
The addition of an element and its additive inverse equal the additive identity element.
example:
The multiplication of an element and its multiplicative inverse equal the multiplicative identity element.
example:
Appendix B
Someone might ask, what about the distributivity of addition over multiplication?
If it were true (it actually is true in Boolean Algebra), then the following would be true:
- a+(b*c) = (a+b) * (a+c)
Observe what happens in an example:
3+(4*5) = (3+4)*(3+5)
3 + 20 = 7 * 8
23 = 56
False
Vectoreverything 