Natural Numbers

Natural numbers are numbers that follows Peano axioms.

Peano Axiom

1. 1 is a natural number.
2. If \(n\) is a natural number then \(n++\) is also a natural number.
3. 1 is not a successor of any natural number. Hence, \(n++\neq 1\) for every natural number \(n\).
4. For two natural numbers \(m\) and \(n\): if \(m\neq n\) then \(m++\neq n++\) or if \(m=n\) then \(m++=n++\).
5. Principle of mathematical induction: Let \(P(n)\) be any property pertaining to natural number \(n\). Suppose that \(P(1)\) is true and suppose that whenever \(P(n)\) is true, \(P(n++)\) is also true. Then \(P(n)\) is true for every natural number \(n\).

Operations of Natural numbers

From Peano Axioms only one operation (i.e. increment) is defined on natural number. The repeated application of this operation can used to define other operations on natural numbers like addition and multiplication.

Addition

Definition: For a natural number \(m\), we define \(m++ = m+1\) and by induction if this addition operation can be defined for \(n++\) as \(m+n++ = (m+n)++ = m+(n+1)\)

Based on this definition of addition, following properties of addition can be established:

1. Addition is commutative. i.e. \(n+m = m+n\)
2. Addition is associative. i.e. \((a+b)+c = a+(b+c)\)
3. Cancellation law: if \(a+b=a+c\), then \(b=c\).

These properties can be proved by using principle of mathematical induction (the last Peano axiom). Moreover, the operation of addition also lead to ordering in natural number.

Definition of zero and whole number

Zero (written as 0) is a number whose successor is 1 and for any natural number \(a\), \(a+0=a\). If we include 0 in the set of natural number then the set is called whole number.

Ordering of natural numbers

Definition: Let \(n\) and \(m\) are natural numbers. We say that \(n\) is equal or greater than \(m\) and write it \(n\geq m\) (or \(m\leq n\)), iff we have \(n=m+a\) for some whole number \(a\). We say that \(n\) is strictly greater than \(m\) (written as \(n>m\)) iff \(n\geq m\) and \(n\neq m\).

Using mathematical induction, following properties of ordering in whole numbers can be established (\(a\), \(b\), and \(c\) are whole numbers):

1. Order is reflexive. \(a\geq a\)
2. Order is transitive. If \(a\geq b\) and \(b\geq c\), then \(a\geq c\).
3. Order is anti-symmetric. If \(a\geq b\) and \(b\geq a\), then \(a=b\).
4. Addition preserves order. If \(a\geq b\), then \(a+c \geq b+c\).
5. \(a<b\) iff \(a++\geq b\).
6. \(a<b\) iff \(b = a + d\) for some natural number d.

Proposition 1: Trichotomy of order for whole numbers: If \(a\) and \(b\) are whole numbers then exatly one of the following statements is true: \(a<b\), \(a=b\), or \(a>b\).

Proposition 2: Strong principle of mathematical induction: let \(m_0\) be a whole number, and let \(P(m)\) be a property pertaining to an arbitrary whole number \(m\). Suppose that for each \(m\geq m_0\), we have the following implication: if \(P(m')\) is true for all natural numbers \(m_0\geq m' \geq m\), then \(P(m)\) is also true. (In particular, this means that \(P(m_0)\) is true, since in this case the hypothesis is vacuous.) Then we can conclude that \(P(m)\) is true for all natural numbers \(m\geq m_0\).