Understanding Real Numbers and Their Operations

1. Chapter 6: The Real Numbers and Their Representations

2. 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

3. 6.1 Sets of Numbers • Naturals {1,2,3,…} • Whole Numbers {0,1,2,3,…} • Integers {…,-2,-1,0,1,2,…}

4. 6.1 • Rationals = {x | x is a quotient of two integers p/q with q not equal to 0}

5. 6.1 • Irrationals = {x | x is not rational} • Reals = {x | x can be represented by a point on the number line}

6. 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 thanb. a < b • If a is to the right of b, then a is greater thanb. a > b

7. 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

8. 6.1 Double Negative Rule • For any real number x, -(-x) = x

9. 6.1 Absolute Values | x | = x if x≥ 0, -x if x < 0

10. 6.2 Operations on Reals • Addition • Subtraction • Multiplication • Division What happens to the sign?

11. 6.2 Order of Operations (BEDMAS) • 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

12. 6.2 Properties of Addition and Multiplication • Closure • Commutative • Associative • Identity • Inverse • Distributive

13. 6.3 Rational Numbers • Rationals = { x | x is a quotient of integers with denominator not 0} • Fundamental property of rationals: If a,b and k are integers with b and k not 0, then ak/bk = a/b • Cross-product test: a/b = c/d if and only if ad = bc

14. 6.3 Operations on Fractions • Adding/Subtracting: Need to find common denominator. One way is to cross-multiply: a/b ± c/d = ad/bd ± bc/bd • Multiplying: a/b × c/d = ac/bd • Dividing: a/b ÷ c/d = a/b × d/c (invert and multiply)

15. 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

16. 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. • Prime factors are 2s and/or 5s ↔ terminating decimal • Prime factors include a prime other than 2 or 5 ↔ repeating decimal

17. 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.

18. 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

19. 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!