The Natural Numbers

Judy Ann P. JandulongBSEd III - Mathematics

The Peano Postulates

are properties or axioms set to develop the system of natural numbers.

by Giuseppe Peano.(1858 – 1932, Italian)

System of Natural numbers “ natural number

(denoted by N),” “successor,” and “1”

The Peano Postulates

Postulate 1. (P5.1):1 is a natural number, i.e. 1 Є N.

Postulate 2 (P5.2):For each natural number n, there exists, corresponding to it, another natural number n* called the successor of n.

Postulate 2 (P5.2):For each natural number n, there

exists, corresponding to it, another natural number n* called the successor of n.

Definition 5.1. Successor of a number n.

The successor of n, where n is a natural number , is the number n* = n + 1, where “+” is the ordinary addition.

Postulate 3 (5.3):1 is not the successor of any natural number, i.e. for each n Є N, n* ≠ 1.

Postulate 4 (5.4):If m, n Є N and m* = n*, then m =

n.

Axiom of Mathematical Induction

Postulate 5 (P5.5): If S is any subset of N which is known

to have the following two properties:

i. The natural number 1 is in the set S.ii. If any natural number k is in the set S,

then the successor k* of k must also be in the set S.

Then S is equal to N.

Addition on N

Addition on NDefinition 5.2. Addition on N

Since m* = m + 1, then we define addition on N by n + m* = (n + m)* whenever n + m, called the sum, is

defined.

Axiom 1 (A5.1)Closure Law for

Addition

Given any pair of natural numbers, m and n, in the stated order , there exists one and only one natural number, denoted by m + n, called the sum of m and n. The numbers m and n are called terms of the sum.

For all m, n Є N, n + m Є N.

Axiom 2 (A5.2)The Commutative Law for

Addition

If a and b are any natural numbers, then

a + b = b + a

Axiom 3 (A5.3) Associative Law for Addition

If a, b, c are any natural number, then(a + b) + c = a + (b + c)

Multiplication on N

Multiplication on NDefinition 5.3 Product; Factor

If a and b are natural numbers, the product of a and b shall mean the number b +b +b +… +b where there are a number of b’s in the sum.

In symbols,ab = b +b +b +… +b

where the number of b terms on the right of the equals sign is a. The numbers a and b are called the factors of the product.

Axiom 4 (M5.1)Postulate of Closure for Multiplication

If a and b are natural numbers, given in the stated order, there exists one and only one natural number denoted by ab or (a)(b) called the product of a and b.

Axiom 5 (M5.2)Commutative Law for

Multiplication

If a and b are any natural numbers, then

ab = ba

Axiom 6 (M5.3)Associative Law for Multiplication

If a, b, c are any natural numbers, then

(ab)c = a(bc)

Axiom 7 (D5.1) Distributive Law of Multiplication over Addition

If a, b, c are natural numbers, then

a (b + c) = a • b + a • c

or (b + c) a = b • a + c • a = a • b + a • c

Definition 5.4 Similar Terms

Two terms are called similar terms if they have a common factor.

Subtraction and Division on N

Subtraction on N

Definition 5.5 Difference• If a and b are natural numbers, the

difference of a and b is a natural number x such that b + x = a, provided such number exists.

• In symbos,a – b = x iff b + x = a

Definition 5.6 (Greater Than)

If a and b are natural numbers, then we say that a is greater than b if there exists a natural number x such that b + x = a.

In symbols,a > b if there is x such that b + x = a

Definition 5.7 (Less Than)

A natural number x is less than another number y if and only if y is greater than x.

In symbols, x < y iff y > x

Division on N Definition 5.8

Quotient; Multiple; Divisible; Factor

• If a and b are natural numbers, the quotient in dividing a by b is a natural number x such that bx = a, provided such a number exists.

• In symbols, a ÷ b = x if bx = a

• In the statement bx = a, we say that a is a multiple of b or a is divisible by b or b is a factor of a.

Theorem 5.1.

If p and q are natural numbers, then

(p + q) – p = q

Theorem 5.2.

If p and q are any natural numbers, then

(pq) ÷ p = q