The Real Number System, p'192241

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The Real Number System, p.192-241

• OBJECTIVES (or GOALS)
• How to identify and classify real numbers
• How to use real numbers and their properties

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Number Theory, p. 193

• Natural numbers 1, 2, 3, 4, , p.193
• Divisibility- if dividing a by b leaves a
  remainder of 0, then b is a divisor of a, 194.
• Prime number- a natural number grater than 1 that
  has only itself and 1 as factors, p. 195.
• Composite number- a natural number greater than 1
  that is divisible by a number other than itself
  and 1, p. 195.

3
Fundamental Theorem of Arithmetic, p. 196

• Every composite number can be expressed as a
  product of prime numbers in one and only one way
  (if the order of the factors is disregarded).

4
Natural numbers 1, 2, 3, 4, , p.193 Whole
numbers 0, 1, 2, 3, 4, , p.
203 Integers -3, -2, -1, 0, 1, 2, 3, ,
p. 203
-1
1
3
-4
-3
-2
-5
0
2
4
5
Positive direction
Negative direction
ORIGIN
Nonnegative describes a number that is either
positive or zero.
5
-1
1
3
-4
-3
-2
-5
0
2
4
5
Order on the Real Number Line, p.203 The order of
a and b is denoted by the inequality a lt b
. Note a lt b if and only if a lies to the left
of b. b is greater than a b gt a
a is less than or equal to b
b is greater than or equal to a
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Absolute value the distance between from 0 to
the point representing the real number on the
real number line The distance is also called
the magnitude, p. 205.
Definition of subtraction a b a (-
b) add the opposite, p. 207 Additive
Identity a 0 a Additive Inverse a (- a)
0 p. 206
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Natural numbers 1, 2, 3, 4, , p.193 Whole
numbers 0, 1, 2, 3, 4, , p.
203 Integers -3, -2, -1, 0, 1, 2, 3, ,
p. 203 p. 214
Rational numbers p/q p and q are integers,

Fundamental Principle of Rational Numbers, p.
215 If is rational number and c is any number
other than 0,
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Rational numbers and decimals, p. 218

• Any rational number can be expressed as a
  decimal. The resulting decimal will either
  terminate (stop), or it will have a digit that
  repeats or a block of digits that repeat.

Multiplying Rational Numbers, p. 221. The product
of two rational numbers is the product of their
numerators divided by the product of the
denominators. If and are rational numbers,
then
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Dividing Rational Numbers, p. 221. The quotient
of two rational numbers is the product of the
first number and the reciprocal of the second
number. If and are rational numbers and ,
then Multiplicative Inverse
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Density of the Rational Numbers, p. 224

• If r and s represent rational numbers, with r lt
  t, then there is a rational number s such that s
  is between r and t r lt s lt t.

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Natural numbers 1, 2, 3, 4, , p.193 Whole
numbers 0, 1, 2, 3, 4, , p.
203 Integers -3, -2, -1, 0, 1, 2, 3, ,
p. 203 p. 214
Rational numbers p/q p and q are integers,

Irrational numbers nonrepeating or
nonterminating decimals, p. 229.
REAL NUMBERS the set of numbers used to measure
or count things in everyday life, p.237.
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Basic Rules of Algebra, p.239 Let a, b,
c be real numbers.
Commutativity a b b a ab
ba Associativity (a b) c a (b
c) (ab) c a (bc) Distributive a(b c) ab
ac (a b) c ac bc
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