WHOLE NUMBERS INTEGERS

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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 lt b. Then there is rational number r
such that a lt r lt b. That is, there is a
rational number between any two real numbers.
The rational numbers are dense on the real line.