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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.
