The Axioms of Real Numbers

When this blog began, I discussed the Five Postulates of Euclid, statements which are assumed to be true, where the entire Geometry that you know in elementary school and high school was built upon. From these 5 statements and several definitions came theorems and theorems that needed proofs. These theorems built the entire Euclidean Geometry.

I have also mentioned that there are other axioms such as these. One example are the axioms or real numbers. They are also assumptions about real numbers that do not need proofs.

I am sure that you are familiar with these axioms and have been using them since you learned Algebra. In middle school and high school, you call them properties of real numbers. In this post, we formalize a little what you have learned and we will use them to prove some theorems later. 

Axiom 1A: Closure Property for Addition

For any a and b in R, a + b is in \mathbb{R}.

Axiom 1M: Closure Property for Multiplication

For any a and b in \mathbb{R}, ab is in \mathbb{R}.

Note: This means that if any two real numbers, their sum is also a real number and their product is also a real number.

Axiom 2A: Associativity for Addition

For any a, b, and c in \mathbb{R}, (a + b) + c = a + (b + c).

Axiom 2M: Associativity for Multiplication

For any a, b, and c in \mathbb{R}, (ab) + c = a(bc).

Axiom 3A: Commutativity for Addition For any a, b in \mathbb{R}, a + b = b + a.

Axiom 3M: Commutativity for Multiplication

For any a, b in \mathbb{R}, ab = ba.

Axiom 4: Distributivity of Multiplication over Addition

For any a, b, c in \mathbb{R}, (+ b) = ca + cb and  (a + b)c = ac + bc.

Axiom 5A: Existence of an Additive Identity

There exists a unique number 0 such that a + 0 = a for any real number a.

Axiom 5M: Existence of a Multiplicative Identity

There is a unique number 1, such that a(1) = a and 1(a) = a. 

Axiom 6A: Existence of Additive Inverses

For any number a, there is a unique number –a such that a + (-a) = 0.

Axiom 6M: Existence of Multiplicative Inverses

For every number a not equal to 0, there exists a unique number 1/a such that a(1/a) = 1.

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