Natural numbers

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THE CONCEPT OF NATURAL NUMBERS

2010

Definition

The set {0,1, 2, 3, 4, ...} is called the set of natural numbers and denoted by N, i.e.

N = {0,1, 2, 3, 4, ...}

Betweenness in N

If a and b are natural numbers with a > b, then there are (a – b) – 1 natural numbers between a and b.

Definition

Two different natural numbers are called consecutive natural numbers if there is no natural number between them.

ADDITION OF NATURAL NUMBERS

If a, b, and c are natural numbers, where a + b = c, then a and b are called the addends and c is called the sum.

Properties of Addition in N

If a, b ∈ N, then (a + b) ∈ N. We say that N is closed under addition.

Closure Property

Properties of Addition in N

If a, b ∈ N, then a + b = b + a . Therefore, addition is commutative in N.

Commutative Property

Properties of Addition in N

If a, b, c ∈ N, then (a+b)+c = a+(b+c). Therefore, addition is associative in N.

Associative Property

Properties of Addition in N

If a ∈ N, then a+0 = 0+a = a. Therefore, 0 is the additive identity or the identity element for addition in N.

Identity Element

SUBTRACTION OF NATURAL NUMBERS

If a, b, c ∈ N and a – b = c, then a is called the minuend, b is called the subtrahend, and c is called the difference.

Property

If a, b, c ∈ N where a – b = c, then a – c = b.

Properties of Subtraction in N

The set of natural numbers is not closed under subtraction.

The set of natural numbers is not commutative under subtraction.

The set of natural numbers is not associative under subtraction.

There is no identity element for N under subtraction.

MULTIPLICATION OF NATURAL NUMBERS

If a, b, c ∈ N, where a ⋅ b = c, then a and b are called the factors and c is called the product.

Properties of Multiplication in N

If a, b ∈ N, then a⋅ b ∈ N. Therefore, N is closed under multiplication.

Closure Property

Properties of Multiplication in N

If a, b ∈ N, then a⋅ b = b⋅ a. Therefore, multiplication is commutative in N.

Commutative Property

Properties of Multiplication in N

If a, b, c ∈ N, then a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c. Therefore, multiplication is associative in N.

Associative Property

Properties of Multiplication in N

If a ∈ N, then a⋅ 1 = 1⋅ a = a. Therefore, 1 is the multiplicative identity or the identity element for multiplication in N.

Identity Element

Distributive Property of Multiplication Over

Addition and Subtraction

For any natural numbers, a, b, and c:

a⋅(b + c) = (a⋅b) + (a⋅c) and (b + c)⋅a = (b⋅a) + (c⋅a)

and

a⋅(b – c) = (a⋅b) – (a⋅c) and (b – c)⋅a = (b⋅a) – (c⋅a)

In other words, multiplication is distributive over addition and subtraction.

DIVISION OF NATURAL NUMBERS

If a, b, c ∈ N, and a ÷ b = c, then a is called the dividend, b is called the divisor and c is called the quotient.

Division with Remainder

dividend = (divisor ⋅ quotient) + remainder

Zero in Division

If a ∈ N then 0 ÷ a = 0. However, a ÷ 0 and 0 ÷ 0 are undefined.

Properties of Division in N

The set of natural numbers is not closed under division.

The set of natural numbers is not commutative under division.

The set of natural numbers is not associative under division.

Division is not distributive over addition and subtraction in N.