# Construction of number systems

If we want fully understand numbers and number sets, we must learn how to construct each sets of numbers.

## Peano Axioms

The set of natural numbers is axiomatic introduced by Italian mathematician Giuseppe Peano. These axioms are known as Peano’s axioms.

Peano’s Axioms for natural numbers:

is a set with the following properties:

(1) has a distinguished element which we call

(2) There exists

(3) is one-to-one function (injective function)

(4) There does not exists an element such that  (not surjective function)

(5) If such that: (Principal of Induction)

(a),

(b) if ,

then set .

A set is the set of natural numbers and its element are natural numbers. A function is called a successor function.

Lemma 1. If and , then there exists such that .

One of the main features of a set is that each element in it, except number , has an immediate predecessor and each element has an immediate follower. There is the smallest natural number, number ,  however, there is no the largest natural number.

Now, we can define operation of addition and multiplaying by recursive rules.

Addition of natural numbers

There is a unique function , , with the following properties:

,

.

Theorem 1. (Associativity) For all three natural numbers and the following is valid:

Proof.

Let be a set of all natural numbers for which is valid , that is:

We must prove that . Proof we conduct by using the principle of induction by .

Firstly, we must show that , that is :

It follows that .

Now, suppose that , that is . We must show that is also in the set , that is

We have

Therefore, .

Since and from the assumption that follows , by the principle of induction we conclude . The statement is true for all natural numbers and .

Theorem 2. (Commutativity) For any two natural numbers and the following is valid:

For proof the commutativity of natural numbers we need to impose two minor lemma.

Lemma 2. For any two natural numbers and is valid:

Lemma 3. For all natural number is valid:

Proof. ( Theorem 2.)

Proof we conduct by using the principle of induction by . We define a set as:

By the 5. Peano’s axiom we need to prove:

(1) :

Therefore, .

(2) Suppose that . Then , that is :

We have proven that .

Since and from the assumption that follows , by the principle of induction we conclude . The statement is true for all natural numbers and .

### Multiplication of natural numbers

There is a unique function , , with the following properties:

, ,

, .

The properties of multiplication of natural numbers.

Associativity. For all three natural numbers and is valid:

Commutativity. For all two natural numbers and is valid:

Distributive law. For all natural numbers and is valid:

Ordering on

Definition. Let . We say that if there exists a such that x + z = y.

## Construction of number systems: Integers

Now, we can construct integers. Of course, we will use natural number to construct integers.

Consider the set , and relations if is valid . Let the set be equivalence classes under this relation. Now we can define on set  like:

If sets and , then sets and are non-empty subsets of set and thus we may pick elements form set , and elements (c,d) from set .

Now we can define an operation (addition) denoted by .

Now, we want to define multiplication of integers denoted by like:

.

For natural numbers and integers we can write down corresponding properties.

• Associativity of addition:
• Commutativity od addition:
• Additive inverse:
• Existing of neutral element for addition:
• Distributivity:
• Associative of multiplication:
• Commutativity of multiplication:
• Existing of neutral element for multiplication:

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