Transcript chapter 6

Chapter 6:
The Real Numbers and Their
Representations
Chapter 6: The Reals and Their
Representations
• 6.1: Real Numbers, Order and Absolute Value
• 6.2: Operations, Properties and Applications
• 6.3: Rational Numbers and Decimal
Representations
• 6.4: Irrational Numbers and Decimal
Representations
6.1
Sets of Numbers
• Naturals {1,2,3,…}
• Whole Numbers {0,1,2,3,…}
• Integers {…,-2,-1,0,1,2,…}
6.1
• Rationals = {x | x is a quotient
of two integers p/q with q not
equal to 0}
6.1
• Irrationals = {x | x is not
rational}
• Reals = {x | x can be
represented by a point on the
number line}
Order
6.1
• Two real numbers can be compared, or
ordered, on the real number line.
• If they represent the same point then they
are equal.
• If a is to the left of b, then a is less than b.
a<b
• If a is to the right of b, then a is greater
than b.
a>b
Additive Inverses
6.1
• For any real x (except 0), there is exactly
one number on the number line that is the
same distance from 0 but on the other side
of x. This is the additive inverse, or
opposite, of x.
• The additive inverse of x is -x
Double Negative Rule
• For any real number x,
-(-x) = x
6.1
6.1
Absolute Values
| x | = x if x ≥ 0, -x if x < 0
6.2
Operations on Reals
• Addition
• Subtraction
• Multiplication
• Division
What happens to the sign?
6.2
Order of Operations (BEDMAS)
1. Work separately above and below any
fraction bar
2. Use the rules within each set of brackets
(work from the inside out)
3. Apply any exponents
4. Do any multiplications or divisions in the
order they occur, from left to right
5. Do any additions or subtractions in the
order they occur, from left to right
Properties of Addition and
Multiplication
• Closure: a + b, ab are defined
• Commutative:
a+b=b+a
ab = ba
• Associative:
a+(b+c)=(a+b)+c
a(bc)=(ab)c
6.2
Properties Continued
• Identity:
a+0=a=0+a
a(1) = a
• Inverse:
a + (-a) = 0
a(1/a) =1
• Distributive Property:
a(b + c) = ab + ac
(b + c)a = ba + ca
Fractions
6.3
Rational Numbers
Operations on Fractions
6.3
6.3
Density Property of Rationals
If r and t are distinct rational
numbers, with r < t, then
there exists a rational number
s such that
r<s<t
6.3
Decimal Representation of Rationals
Any rational number can be expressed as
either a terminating decimal or a repeating
decimal. Suppose a/b is in lowest terms.
Find the prime factors of the denominator b.
factors are 2s and/or 5s ↔
terminating decimal
•Prime factors include a prime
other than 2 or 5 ↔ repeating
decimal
• Prime
Converting Between Decimal and
Fraction
• Fraction → Decimal: decide if decimal is
terminating or repeating.
• Terminating: Do long division of fraction
until remainder is 0.
• Repeating: Do long division until you
repeat a remainder so that you know what
the repeating part is.
6.3
6.3
Converting cont’d
• Decimal → Fraction: decide if decimal is
terminating or repeating.
• Terminating: write decimal as a fraction with the
numerator being the terminating part and the
denominator a power of 10. Simplify to get in
lowest terms.
• Repeating: determine how many digits are
repeated, then use the same power of 10 to
multiply the decimal. Let x be your number and
solve an equation for x
6.3
Proof that 0.9999… = 1
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Let x = 0.9999…
Then 10x = 9.999…
10x – x = 9.999… - 0.999… = 9
Thus 9x = 9.
Solve for x to get x = 1 (!!!!)
6.4
Irrational Numbers