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 6.5: Applications of Decimals and Percents

Sets of Numbers Naturals {1,2,3,…} Whole Numbers {0,1,2,3,…} 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}

Irrationals = {x | x is not rational} 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 6.1 For any real number x, -(-x) = x

6.1 Absolute Values | x | = x if x ≥ 0, -x if x < 0

Operations on Reals Addition Subtraction Multiplication Division 6.2 Addition Subtraction Multiplication Division What happens to the sign?

Order of Operations (BEDMAS) 6.2 Work separately above and below any fraction bar Use the rules within each set of brackets (work from the inside out) Apply any exponents Do any multiplications or divisions in the order they occur, from left to right Do any additions or subtractions in the order they occur, from left to right

Properties of Addition and Multiplication 6.2 Closure: a + b, ab are defined Commutative: a + b = b + a ab = ba Associative: a+(b+c)=(a+b)+c a(bc)=(ab)c

Properties Continued Identity: a + 0 = a = 0 + a a(1) = a Inverse: Distributive Property: a(b + c) = ab + ac (b + c)a = ba + ca

Fractions

Rational Numbers 6.3  

Operations on Fractions 6.3  

Density Property of Rationals 6.3 If r and t are distinct rational numbers, with r < t, then there exists a rational number s such that r < s < t

Decimal Representation of Rationals 6.3 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. Prime factors are 2s and/or 5s ↔ terminating decimal Prime factors include a prime other than 2 or 5 ↔ repeating decimal

Converting Between Decimal and Fraction 6.3 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.

Converting cont’d 6.3 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

Proof that 0.9999… = 1 Let x = 0.9999… Then 10x = 9.999… 6.3 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 (!!!!)

Irrational Numbers 6.4