Chapter 9

1. Chapter 9 Rational and Real numbers

2. Introduction Fractions Integers Counting numbers Whole numbers is a subset of is a subset of

3. Definition A rational number is a quotient of the form where a and b are integers with b≠ 0. Fractions Integers Whole numbers Rational numbers

4. Closure properties of rational numbers • The set of rational numbers is closed under all 4 operations +, - , ×, ÷. • In other words, • The sum of any two rational numbers is still a rational number. • The difference of any two rational numbers is still a rational number. • The product of any two rational numbers is still a rational number. • The quotient of any rational number by any non-zero rational number is still a rational number.

5. Decimal representations Every rational number has either a terminating or a repeating decimal representation. However, the representation may not be unique. For example, ¼ = 0.25, but it is also equal to 0.24999999…

6. Real numbers A real number is a number that either has a finite decimal representation or has an infinite decimal representation. In particular, the set of real numbers include those numbers with non-terminating and non-repeating decimal representations such as π= 3.141592653589793 … These number are called irrational numbers.

7. Is there any useful irrational number at all? is irrational and it is the length of any diagonal in a unit square. 1 1 never terminates and never repeats.

8. Theorem The square root of any whole number is either a whole number or an irrational number Example Since we know that 23 is not a perfect square, will not be a whole number. The above theorem tells us that is not even a rational number, that means its decimal representation will not be repeating or terminating.

9. All decimals Irrational numbers Rational numbers (non-repeating, non-terminating decimals) Terminating decimals Repeating decimals

10. 3 17 is not terminating is terminating 11 625 21 7 7 is terminating = = 1920 640 7 1 2 5 Converting Fractions to Decimals Fact: The decimal expansion of any fraction a/b is either terminating or repeating. • Theorem: • If the fraction a/b is in its reduced form, then its decimal expansion is terminating if and only if b is one of the following forms. • a product of 2’s only, • a product of 5’s only • a product of 2’s and 5’s only. Examples:

11. Example: The decimal expansion of 13 13 will have exactly 3 decimal places. = 3 1 40 2 5 Converting Fractions to Decimals Now we know what kind of fractions will have terminating decimal expansions, but can we predict how many decimal places there will be in the expansion? Theorem: If the fraction a/b is in its reduced form, and b = 2m5n then the decimal expansion of a/b is terminating with number of decimal places exactly equal to max{m, n}

12. Converting Fractions to Decimals One more question: If we know that a certain fraction has repeating decimal expansion, can we predict its cycle length? Unfortunately there is no formula to calculate the precise cycle length. All we know is an upper bound and a small (not too helpful) property. Theorem If p is a prime number other than 2 and 5, then the cycle length of 1/p is at most (p – 1), and the cycle length must divide (p – 1). Example: The cycle length of 1/31 is at most 30, and it must divide 30. In fact, the cycle length of 1/31 is 15.

13. Converting Fractions to Decimals More examples There is no obvious pattern on the cycle length, and a large denominator can have a small cycle length.

14. Converting Fractions to Decimals • More facts (optional) • If p is a prime other than 2 or 5, then the cycle length of 1/(p2) is at mostp(p – 1) and the cycle length must divide p(p – 1). Example: cycle length of 1/7 is 6, cycle length of 1/49 is 42 (= 7×6). 2. If p and q are different primes other than 2 and 5, then the cycle length of 1/pq will be at most (p – 1)(q – 1) and divides (p – 1)(q – 1). Example: Cycle length of 1/(7×11) is less than 6×10 = 60, and must divide 60. It turns out that the cycle length of 1/77 is only 6.

15. Converting Decimals to Fractions From the previous theorem, we see that only repeating or terminating decimals can be converted to a fraction. • Procedures: • terminating decimal, eg. 0.35742 = 35742/100000The number of 0’s in the denominator is equal to the number of decimal places. • repeating decimals of type I, eg • 0.2222 ··· = 2/9 • 0.47474747··· = 47/99 • 0.528528528··· = 528/999

16. Converting Decimals to Fractions • Procedures: • repeating decimals of type I, eg. 0.2222 ··· = 2/9 0.47474747··· = 47/99 0.528528528··· = 528/999 • repeating decimals of type II, eg. • 0.0626262··· = 62/990 • 0.00626262··· = 62/9900 • 0.000344934493449··· = 3449/9999000

17. Converting Decimals to Fractions • Procedures: • repeating decimals of type II, eg. • 0.0626262··· = 62/990 • 0.00626262··· = 62/9900 • 0.000344934493449··· = 3449/9999000 • 4) repeating decimals of type III, eg. 0.576666··· = 0.57 + 0.006666···

18. Converting Decimals to Fractions • Procedures: • repeating decimals of type II, eg. • 0.0626262··· = 62/990 • 0.00626262··· = 62/9900 • 0.000344934493449··· = 3449/9999000 • 4) repeating decimals of type III, eg. 0.576666··· = 0.57 + 0.006666···

19. Irrational numbers Even though most irrational numbers have unpredictable decimal expansions, some do have certain patterns, except that the patterns are not repeating. Examples: 1. 0.12345678910111213141516 ··· 2. 0.1010010001000010000010000001··· We can therefore deduce that there must be many irrational nmbers. In fact, there are more irrational numbers than rational numbers. In addition, there is at least one irrational number between any two given rational numbers. (see lab 10)