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The natural numbers

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  • 1. Judy Ann P. Jandulong BSEd III - Mathematics
  • 2.  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”
  • 3. 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.
  • 4. Postulate 2 (P5.2): For each natural number n, there exists, corresponding to it, another natural number n* called the successor of n. . The successor of n, where n is a natural number , is the number n* = n + 1, where “+” is the ordinary addition.
  • 5. 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.
  • 6.  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.
  • 7. Addition on N
  • 8. Addition on N Definition 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.
  • 9. 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.
  • 10. Axiom 2 (A5.2) The Commutative Law for Addition  If a and b are any natural numbers, then a+b=b+a
  • 11. Axiom 3 (A5.3) Associative Law for Addition If a, b, c are any natural number, then (a + b) + c = a + (b + c)
  • 12. Multiplication on N
  • 13. Multiplication on N Definition 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.
  • 14. 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.
  • 15. Axiom 5 (M5.2) Commutative Law for Multiplication  If a and b are any natural numbers, then ab = ba
  • 16. Axiom 6 (M5.3) Associative Law for Multiplication  If a, b, c are any natural numbers, then (ab)c = a(bc)
  • 17. 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
  • 18. Definition 5.4 Similar Terms  Two terms are called similar terms if they have a common factor.
  • 19. Subtraction and Division on N
  • 20. 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 •
  • 21. 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
  • 22. 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
  • 23. 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.
  • 24. Theorem 5.1.  If p and q are natural numbers, then (p + q) – p = q
  • 25. Theorem 5.2.  If p and q are any natural numbers, then (pq) ÷ p = q