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# Natural numbers

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### Natural numbers

1. 1. THE CONCEPT OF NATURAL NUMBERS2010 S
2. 2. DefinitionS 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. 3. Betweenness in NIf a and b are natural numbers witha >b, then there are (a – b) – 1natural numbers between a and b.
4. 4. DefinitionTwo different natural numbers are calledconsecutive natural numbers if thereis no natural number between them.
5. 5. ADDITION OF NATURAL NUMBERSS 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. 6. Properties of Addition in NClosure PropertyIf a, b ∈ N, then (a + b) ∈ N. Wesay that N is closed under addition.
7. 7. Properties of Addition in NCommutative PropertyIf a, b ∈ N, then a + b = b + a .Therefore, addition is commutativein N.
8. 8. Properties of Addition in N Associative PropertyIf a, b, c ∈ N, then (a+b)+c = a+(b+c).Therefore, addition is associative in N.
9. 9. Properties of Addition in NIdentity ElementS If a ∈ N, then a+0 = 0+a = a. Therefore, 0 is the additive identity or the identity element for addition in N.
10. 10. SUBTRACTION OF NATURAL NUMBERSS 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. 11. PropertyS If a, b, c ∈ N where a – b = c, then a – c = b.
12. 12. Properties of Subtraction in NS 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. 13. MULTIPLICATION OF NATURAL NUMBERSS If a, b, c ∈ N, where a ⋅ b = c, then a and b are called thefactors and c is called theproduct.
14. 14. Properties of Multiplication in NClosure PropertyS If a, b ∈ N, then a⋅ b ∈ N. Therefore, N is closed under multiplication.
15. 15. Properties of Multiplication in NCommutative PropertyS If a, b ∈ N, then a⋅ b = b⋅ a. Therefore, multiplication is commutative in N.
16. 16. Properties of Multiplication in NAssociative PropertyS If a, b, c ∈ N, then a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c. Therefore, multiplication is associative in N.
17. 17. Properties of Multiplication in NIdentity ElementS If a ∈ N, then a⋅ 1 = 1⋅ a = a. Therefore, 1 is the multiplicative identity or the identity element for multiplication in N.
18. 18. Distributive Property of Multiplication Over Addition and SubtractionS For any natural numbers, a, b, and c:a⋅(b + c) = (a⋅b) + (a⋅c) and (b + c)⋅a = (b⋅a) + (c⋅a) anda⋅(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. 19. DIVISION OF NATURAL NUMBERSS 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. 20. Division with RemainderS dividend = (divisor ⋅ quotient) + remainder
21. 21. Zero in DivisionS If a ∈ N then 0 ÷ a = 0. However, a ÷ 0 and 0 ÷ 0 are undefined.
22. 22. Properties of Division in NS 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.
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