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

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}

6.1

Order• 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

6.1

Additive Inverses• 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

6.1

Double Negative Rule

• For any real number x,

-(-x) = x

6.1

Absolute Values

| x | = x if x ≥ 0, -x if x < 0

6.1

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 exponents4. Do any multiplications or divisions in the

order they occur, from left to right5. Do any additions or subtractions in the

order they occur, from left to right

6.2

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

Rational Numbers6.3

Operations on Fractions6.3

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 RationalsAny 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.

• Prime factors are 2s and/or 5s ↔ terminating decimal

•Prime factors include a prime other than 2 or 5 ↔ repeating decimal

6.3

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

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

• 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 (!!!!)

6.3

Irrational Numbers6.4