Mathematical Axioms for the Fundamental Operations Addition and Multiplication

Axioms of numbers are mathematical statements that are the starting points for other statements. The axioms cannot be proved mathematically, because they are the foundation from which we get mathematical theorems. These axioms are considered self-evident. Even though we don't have mathematical proof, we take it for granted that they are true.

The following is axioms of the fundamental operations for the real number system. (Real numbers are every number except imaginary numbers.) These axioms lay the ground work for the properties of addition and multiplication. There is some mathematical proof for these laws, but the validity of it is controversial enough to still call them axioms. The vast majority of mathematicians however, stick with these laws whole heartedly as theorems.

We have the commutative laws, the closure laws, the associative laws, the distributive laws, the identity laws, and the inverse laws.

Commutative Laws:

The commutative laws state that the order in addition and multiplication does not matter. Thus x+y=y+x and xy = yx.

We can try out the commutative laws using two real numbers. 4+5 = 5+4 = 9, and 4*5 = 5*4 = 45.

Closure Laws:

From the Closure Laws we get that the sum of x+y, and the product x*y are unique numbers.

Associative Laws:

The associative laws states that in repeated multiplication or addition, grouping does not matter. We can look at an example of the associative laws:

x+(y+z) = (x+y) +z = x+y+z 3+(4+5) = (3+4)+5 = 3+4+5 = 12

x(yz) = (xy)z = xyz 3(4*5) = (3*4)5 = 60

Distributive Laws:

The distributive laws states that multiplication is, as the axiom implies, distributed over addition. An example of the Distribute Laws is:

x(y+z) = xy+yz 3(4+5) = 3*4+3*5 = 12+15 = 27

Identity Laws

From the identity laws we get two things.

First, there is a unique number 0 with the property that 0 plus any given number equals the number itself. 0+x = x 0+4 = 4

Secondly, the identity laws also say that there is a unique number 1 with the property that 1 multiplied with any given number equals the number itself. 1*x = x 1*4 = 4

Inverse Laws

The last axiom of operations is the inverse laws. The inverse laws state first that for any real number, there is a negative of that number (the additive inverse), that when added together equals zero. An example would be:

x + (-x) = (-x)+x = 0 4 + (-4) = (-4) + 4 = 0

The inverse laws also states that for any real number except 0, there is the same number to the power of -1 (the multiplicative inverse, or reciprocal), that when multiplied equals 1. An example of the last part of the inverse laws is:

x * x^(-1) = (x^(-1))*x = 1 4*4(^-1) = (4^(-1))*4 = 1

With that being said, to conclude here is a summary of the axioms of operation.

Commutative laws say that order does not matter in multiplication and addition. The closure laws say that the sum and product of two real numbers are unique real numbers. The associative laws say that in repeated addition and multiplication, grouping does not matter. The distributive laws say that multiplication is distributed over addition. The identity laws say that zero plus a given real number equals that number, and that one multiplied with any given real number equals that number. Finally we have the inverse law that says that adding a real number to its negative equals zero, and that multiplying a real number with its reciprocal equals 1.