1. THE CONCEPT OF NATURAL NUMBERS
2010
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2. Definition
S 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, ...}

3. 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.

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

5. ADDITION OF NATURAL
NUMBERS
S 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.

6. Properties of Addition in N
Closure Property
If a, b ∈ N, then (a + b) ∈ N. We
say that N is closed under addition.

7. Properties of Addition in N
Commutative Property
If a, b ∈ N, then a + b = b + a .
Therefore, addition is commutative
in N.

8. Properties of Addition in N
Associative Property
If a, b, c ∈ N, then (a+b)+c = a+(b+c).
Therefore, addition is associative in N.

9. Properties of Addition in N
Identity Element
S If a ∈ N, then a+0 = 0+a = a.
Therefore, 0 is the additive
identity or the identity element for
addition in N.

10. SUBTRACTION OF NATURAL
NUMBERS
S 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.

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

12. Properties of Subtraction in N
S The set of natural numbers is not closed under
subtraction.
S The set of natural numbers is not commutative under
subtraction.
S The set of natural numbers is not associative under
subtraction.
S There is no identity element for N under subtraction.

13. MULTIPLICATION OF
NATURAL NUMBERS
S If a, b, c ∈ N, where a ⋅ b = c,
then a and b are called
thefactors and c is called
theproduct.

14. Properties of Multiplication in
N
Closure Property
S If a, b ∈ N, then a⋅ b ∈ N.
Therefore, N is closed under
multiplication.

15. Properties of Multiplication in
N
Commutative Property
S If a, b ∈ N, then a⋅ b = b⋅ a.
Therefore, multiplication is
commutative in N.

16. Properties of Multiplication in
N
Associative Property
S If a, b, c ∈ N, then
a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c. Therefore,
multiplication is associative in N.

17. Properties of Multiplication in
N
Identity Element
S If a ∈ N, then a⋅ 1 = 1⋅ a = a.
Therefore, 1 is the multiplicative
identity or the identity element for
multiplication in N.

18. Distributive Property of
Multiplication Over Addition
and Subtraction
S 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.

19. DIVISION OF NATURAL
NUMBERS
S If a, b, c ∈ N, and a ÷ b = c,
then a is called the dividend,
b is called the divisor and c is
calledthe quotient.

20. Division with Remainder
S dividend = (divisor ⋅ quotient) + remainder

21. Zero in Division
S If a ∈ N then 0 ÷ a = 0. However,
a ÷ 0 and 0 ÷ 0 are undefined.

22. Properties of Division in N
S The set of natural numbers is not closed under division.
S The set of natural numbers is not commutative under
division.
S The set of natural numbers is not associative under
division.
S Division is not distributive over addition and subtraction
in N.