What is a field?
Any set of numbers that obeys all 11 axioms is a field
a + b = b + a
Commutative for Addition
ab = bc
Commutative for Multiplication
x = x
Reflexive Property
Using variables to stand for numbers, write an example of the following axiom:
Commutative for Addition
Commutative for Multiplication
Addition: a + b = b + a
Multiplication: ab = ba
List all 11 Field Axioms
1/2. Closure for Addition and Multiplication
3/4. Commutative for Addition and Multiplication
5/6. Associative for Addition and Multiplication
7. Distributve
8/9 Multiplicative and Additive Identity
10/11 Multiplicative and Additive Inverse
a + (b+ c) = (a + b) + c
Associative for Addition
a(bc) = (ab) c
Associative for Multiplication
If x = y, then y = x
Symmetry
Using variables to stand for numbers, write an example of the following axiom:
Associative for Addition
Associative for Multiplication
Addition: a + (b + c) = (a+ b) + c
Multiplication: a(bc) = (ab)c
What is the Additive Identity Element and the Multiplicative Identity Element
Additive Identity Element is 0
Multiplicative Identity Element is 1
a(b +c) = ab + ac
Distributive
a * 1 = a
Multiplicative Identity
If x = y and y = z, then x = z
Transitivity for Equality
Using variables to stand for numbers, write an example of the following axiom:
Distributive
a (b + c) = ab + ac
x + (-x) = 0
For every real number, x, you can add the opposite to that number (-x) and the result is zero
a + 0 = a
Additive Identity
a * 1/a = 1
Multiplicative Inverse
If x > y. and y > z, then x > z
If x < y and y < z, then x < z
Transitivity for Order
Using variables to stand for numbers, write an example of the following axiom:
Identity Element for Addition
a + 0 = a
Explain the multiplicative inverse axiom.
x * 1/x = 1
For every real number, x, you can multiply that number by the reciprocal, 1/x, and the result is 1
a + (-a) = 0
Additive Inverse
x *(y+z) + ( - [x * (y+z)])
Additive Identity
If x ad y are real numbers, then one of the following is true
y < x OR y > x OR y = x
Trichotomy
Using variables to stand for numbers, write an example of the following axiom:
Identity Element for Multiplication
a * 1 = a