## Construction of number sxstems: Natural numbers

## Peano Axioms

For *constraction of number systems* we need *Peano Axioms*. If we want fully understand numbers and number sets, we must learn how to constract each sets od numbers.

Now, we can write down **Peano Axioms **for **natural numbers**:

is a set with the following properties:

- has a distinguished element which we call
- There exists a distinguished set map
- is one-to-one function (injective function)
- There does not exit an element such that (not surjective function)
- Let set such that: (Principal of Induction)
- if , then .

- Then set is equal to set .

We call such a set, set of natural number

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

*Addition*

(1) For all ,

(2) For any ,

*Multiplication*

(1) For all ,

(2) For any ,

## Construction of number sxstems: 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:

### For Those Who Want To Learn More:

- Mathematical reasoning
- Naming decimal places
- Rounding numbers
- Fractions
- Subtraction of natural numbers
- Understanding of Remainder theorem through divisibility…
- Numerals, the tally system, numerical systems
- What is an exponent; addition, subtraction, multiplication…
- Factoring and Prime factors
- Inverse function