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WHOLE NUMBERS; INTEGERS PowerPoint PPT Presentation


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WHOLE NUMBERS; INTEGERS. Whole numbers: Z 0,+ = the natural numbers  {0}. Integers:. Properties:. Z 0,+ is closed under addition and multiplication. Z 0,+ is not closed under subtraction and division. Z is closed under addition, subtraction, and multiplication.

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WHOLE NUMBERS; INTEGERS

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Whole numbers integers l.jpg

WHOLE NUMBERS; INTEGERS

Whole numbers:

Z0,+ = the natural numbers  {0}.

Integers:


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Properties:

  • Z0,+is closed under addition and multiplication.

  • Z0,+ is not closed under subtraction and division.

  • Z is closed under addition, subtraction, and multiplication.

  • Z is not closed under division.


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Multiples; Divisors (Factors)

Let m, n  Z.

m is a multiple of n if and only if there is an integer k such that

m = k n.

n is a divisor of m if and only if m is a multiple of n.


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Greatest Common Divisor; Least Common Multiple

Let a, b  Z, with a, b  0. The greatest common divisor of a and b, denoted gcd(a,b), is the largest integer that divides both a and b.

The least common multiple of a and b, denoted lcm(a,b), is the smallest positive multiple of both a and b.


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THEOREM: Let a, b  Z, a, b  0. Then

and


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THE RATIONAL NUMBERS

The set of rational numbers, Q, is given by

NOTE:


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“CLOSURE”

The set of rational numbers, Q, is closed under addition, subtraction, and multiplication; Q {0} is closed under division.


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ARITHMETIC


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Field Axioms

Addition (+): Let a, b, c be rational numbers.

A1. a + b = b + a (commutative)

A2. a + (b + c) = (a + b) + c (associative)

A3. a + 0 = 0 + a = a (additive identity)

A4. There exists a unique number ã such that

a + ã = ã +a = 0 (additive inverse)

ã is denoted by – a


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  • Multiplication (·): Let a, b, c be rational numbers:

  • M1. a b = b a (commutative)

  • M2. a (b c) = (a b) c (associative)

  • M3. a 1 = 1 a = a (multiplicative identity)

  • M4. If a  0, then there exists a unique â such

  • that

  • a â = â a = 1 (multiplicative inverse)

  • â is denoted by a-1 or by 1/a.


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D. a(b + c) = ab + ac

(a + b)c = ac + bc

Distributive laws (the connection between addition and multiplication).


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DECIMAL REPRESENTATIONS

Let be a rational number.

Use long division to divide p by q. The result is the decimal representation of r.


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Examples


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ALTERNATIVE DEFINITION.

The set of rational numbers Q is the set of all terminating or (eventually) repeating decimals.


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Repeating versus Terminating Decimals

Problem: Given a rational number

The decimal expansion of r either terminates

or repeats. Give a condition that will imply

that the decimal expansion:

a. Terminates;

b. repeats.


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Answer:

The decimal expansion of r terminates if and only if the prime factorization of q has the form


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Converting decimal expansions to fractions

Problems: Write

in the form


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Locating a rational number as a point on the real number line.


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Distribution of the rational numbers on the real line.

Let a and b be any two distinct real numbers with a < b. Then there is rational number r such that

a < r < b.

That is, there is a rational number between any two real numbers. The rational numbers are dense on the real line.


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