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Irrational Numbers and Decimal Representation

Convert each terminating decimal into a quotient of integers. Write each in lowest terms. 83. .4

84. .9

85. .85

86. .105

87. .934

88. .7984

Use the method of Example 6 to decide whether each of the following rational numbers would yield a repeating or a terminating decimal. (Hint: Write in lowest terms before trying to decide.) 89.

8 15

90.

8 35

13 125

91.

92.

95. Follow through on each part of this exercise in order. (a) Find the decimal for 13. (b) Find the decimal for 23. (c) By adding the decimal expressions obtained in parts (a) and (b), obtain a decimal expression for 13  23  33  1. (d) Does your result seem bothersome? Read the

6.4

3 24

93.

22 55

94.

24 75

margin note on terminating and repeating decimals in this section, which refers to this idea. 96. It is a fact that 13  .333 . . . . Multiply both sides of this equation by 3. Does your answer bother you? See the margin note on terminating and repeating decimals in this section.

Irrational Numbers and Decimal Representation Basic Concepts

In the previous section we saw that any rational number has a decimal form that terminates or repeats. Also, every repeating or terminating decimal represents a rational number. Some decimals, however, neither repeat nor terminate. For example, the decimal .102001000200001000002…

 



un



√2



1

90°

its

√2



c

a



The irrational number 2 was discovered by the Pythagoreans in about 500 B.C. This discovery was a great setback to their philosophy that everything is based upon the whole numbers. The Pythagoreans kept their findings secret, and legend has it that members of the group who divulged this discovery were sent out to sea, and, according to Proclus (410–485), “perished in a shipwreck, to a man.”

b In a right triangle a2 + b2 = c2.

90° 1

   1 unit  

    

Tsu Ch’ung-chih (about 500 A.D.), the Chinese mathematician honored on the above stamp, investigated the digits of . Aryabhata, his Indian contemporary, gave 3.1416 as the value.

does not terminate and does not repeat. (It is true that there is a pattern in this decimal, but no single block of digits repeats indefinitely.)* A number represented by a nonrepeating, nonterminating decimal is called an irrational number. As the name implies, it cannot be represented as a quotient of integers. The decimal number just mentioned is an irrational number. Other irrational numbers include 2, 7, and  (the ratio of the circumference of a circle to its diameter). There are infinitely many irrational numbers. Figure 12 illustrates how a point with coordinate 2 can be located on a number line.

0

1 unit

1

√2

2

FIGURE 12

*In this section we will assume that the digits of a number such as this continue indefinitely in the pattern established. The next few digits would be 000000100000002, and so on.

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The proof that 2 is irrational is a classic example of a proof by contradiction. We begin by assuming that 2 is rational, which leads to a contradiction, or absurdity. The method is also called reductio ad absurdum (Latin for “reduce to the absurd”). In order to understand the proof, we consider three preliminary facts: 1. When a rational number is written in lowest terms, the greatest common factor of the numerator and denominator is 1. 2. If an integer is even, then it has 2 as a factor and may be written in the form 2k, where k is an integer. 3. If a perfect square is even, then its square root is even. 1

1 1 1 1

_ √6 _ √7 _ √8

1 1

_ √4

_ √5

Theorem

1 _ √3 _ √2

Proof:

__ √10

Assume that 2 is a rational number. Then by definition,

1

p 2  q ,

1 _ √9

2 is an irrational number.

for some integers p and q.

Furthermore, assume that pq is the form of 2 that is written in lowest terms, so the greatest common factor of p and q is 1. Squaring both sides of the equation gives __ √11

__ √12

1 1

__ √13

2

p2 q2

1

and multiplying through by q2 gives An interesting way to represent the lengths corresponding to 2, 3, 4, 5, and so on, is shown in the figure. Use the Pythagorean theorem to verify the lengths in the figure.

2q2  p2. This last equation indicates that 2 is a factor of p2. So p2 is even, and thus p is even. Since p is even, it may be written in the form 2k, where k is an integer. Now, substitute 2k for p in the last equation and simplify: 2q2  2k2 2q2  4k 2 q2  2k 2. Since 2 is a factor of q2, q2 must be even, and thus, q must be even. This leads to a contradiction: p and q cannot both be even because they would then have a common factor of 2, although it was assumed that their greatest common factor is 1. Therefore, since the original assumption that 2 is rational has led to a contra diction, it must follow that 2 is irrational.

These are calculator approximations of irrational numbers.

A calculator with a square root key can give approximations of square roots of numbers that are not perfect squares. To show that they are approximations, we use the  symbol to indicate “is approximately equal to.” Some such calculator approximations are as follows: 2  1.414213562,

6  2.449489743,

and

1949  44.14748011 .

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6.4

The number e is a fundamental number in our universe. For this reason, e, like , is called a universal constant. If there are intelligent beings elsewhere, they too will have to use e to do higher mathematics. The letter e is used to honor Leonhard Euler, who published extensive results on the number in 1748. The first few digits of the decimal value of e are 2.7182818. Since it is an irrational number, its decimal expansion never terminates and never repeats. The properties of e are used in calculus and in higher mathematics extensively.

The symbol for  in use today is a Greek letter. It was first used in England in the 1700s. In 1859 the symbol for  shown above was proposed by Professor Benjamin Peirce of Harvard.

Irrational Numbers and Decimal Representation

285

Not all square roots are irrational. For example, 4  2,

36  6,

and

100  10

are all rational numbers. However, if n is a positive integer that is not the square of an integer, then n is an irrational number. The chart below shows some examples of rational numbers and irrational numbers.

Rational Numbers

Irrational Numbers

3 4 .64

2 .23233233323333…

.74

5

16



1.618

1  5 2

2.718

e

The exact value of the golden ratio An important number in higher mathematics— see the margin note.

One of the most important irrational numbers is , the ratio of the circumference to the diameter of a circle. (Many formulas from geometry involve , such as the formula for area of a circle, A   r 2.) For some 4000 years mathematicians have been finding better and better approximations for . The ancient Egyptians used a method for finding the area of a circle that is equivalent to a value of 3.1605 for . 1 The Babylonians used numbers that give 3 8 for . In the Bible (I Kings 7:23), a verse describes a circular pool at King Solomon’s temple, about 1000 B.C. The pool is said to be ten cubits across, “and a line of 30 cubits did compass it round about.” This implies a value of 3 for  (to the nearest whole number).

FOR FURTHER THOUGHT In 2002, the computer scientist Yasumasa Kanada and his colleagues of the Information Techology Center at Tokyo University announced the computation of 1,241,100,000,000 decimal digits of pi, over six times their own previous record of 206,158,430,000 digits, set in 1999. According to “The  Pages” at the web site

http://www.cecm.sfu.ca, “The computation of Pi is virtually the only topic from the most ancient stratum of mathematics that is still of serious interest to modern mathematical research.” A visit to http://www.joyofpi.com will lead to interesting links about this fascinating number.

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Source: David Blatner, The Joy of Pi, www.joyofpi.com.

One of the reasons for computing so many digits is to determine how often each digit appears and to identify any interesting patterns among the digits. Two American mathematicians, Gregory and David Chudnovsky, have spent a great deal of time and effort looking for patterns in the digits. For example, six 9s in a row appear relatively early in the decimal, within the first 800 decimal places. And past the half-billion mark appears the sequence 123456789. One conjecture about  deals with what mathematicians term “normality.” The normality conjecture says that all digits appear with the same average frequency. According to Gregory, “There is absolutely no doubt that  is a ‘normal’ number. Yet we can’t prove it. We don’t even know how to try to prove it.” The computation of  has fascinated mathematicians and laymen for centuries. In

the nineteenth century the British mathematician William Shanks spent many years of his life calculating  to 707 decimal places. It turned out that only the first 527 were correct. In 1767 J. H. Lambert proved that  is irrational (and thus its decimal will never terminate and never repeat). Nevertheless, the 1897 Indiana state legislature considered a bill that would have legislated the value of . In one part of the bill, the value was stated to be 4, and in another part, 3.2. Amazingly, the bill passed the House, but the Senate postponed action on the bill indefinitely! The following expressions are some which may be used to compute  to more and more decimal places.

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6.4

Irrational Numbers and Decimal Representation

 2  2  4  4  6  6  8…  2 1  3  3  5  5  7  7…  1 1 1 1    4 3 5 7 2 2 2  2 2  2  2      2 2 2 The fascinating history of  has been chronicled by Petr Beckman in the book A History of Pi.

For Group Discussion 1. Have each class member ask someone outside of class “What is ?” Then as a class, discuss the various responses obtained. 2. As with Mount Everest, some people enjoy climbing the mountain of  simply because it is there. Have you ever tackled a project for no reason other than to simply say “I did it”? Share any such experiences with the class. 3. Divide the class into three groups, and, armed with calculators, spend a few minutes calculating the expressions given

287

previously to approximate . Compare your results to see which one of the expressions converges toward  the fastest. 4. A mnemonic device is a scheme whereby one is able to recall facts by memorizing something completely unrelated to the facts. One way of learning the first few digits of the decimal for  is to memorize a sentence (or several sentences) and count the letters in each word of the sentence. For example, “See, I know a digit,” will give the first 5 digits of  : “See” has 3 letters, “I” has 1 letter, “know” has 4 letters, “a” has 1 letter, and “digit” has 5 letters. So the first five digits are 3.1415. Verify that the following mnemonic devices work. (See the margin note on the next page for 31 decimal digits of  .) (a) “May I have a large container of coffee?” (b) “See, I have a rhyme assisting my feeble brain, its tasks ofttimes resisting.” (c) “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”

Square Roots

In everyday mathematical work, nearly all of our calculations deal with rational numbers, usually in decimal form. In our study of mathematics, however, we must sometimes perform operations with irrational numbers, and in many instances the irrational numbers are square roots. Recall that a, for a  0, is the nonnegative number whose square is a; that is,  a 2  a. We will now look at some simple operations with square roots. Notice that 4  9  2  3  6 4  9  36  6 . and Thus, 4  9  4  9. This is a particular case of the following product rule.

Product Rule for Square Roots For nonnegative real numbers a and b, a  b  a  b .

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This poem, dedicated to Archimedes (“the immortal Syracusan”), allows us to learn the first 31 digits of the decimal representation of . By replacing each word with the number of letters it contains, with a decimal point following the initial 3, the decimal is found. The poem was written by A. C. Orr, and appeared in the Literary Digest in 1906. Now I, even I, would celebrate In rhymes unapt, the great Immortal Syracusan, rivaled nevermore, Who in his wondrous lore Passed on before, Left men his guidance How to circles mensurate. From this poem, we can determine these digits of  : 3.14159265358979323846264338 3279. Archimedes was able to use circles inscribed and circumscribed by polygons to find that the value of  is somewhere between 22371 and 227.

The Real Numbers and Their Representations

Just as every rational number ab can be written in simplest (lowest) terms (by using the fundamental property of rational numbers), every square root radical has a simplest form. A square root radical is in simplified form if the following three conditions are met.

Simplified Form of a Square Root Radical 1. 2. 3.

The number under the radical (radicand) has no factor (except 1) that is a perfect square. The radicand has no fractions. No denominator contains a radical.

EXAMPLE 1 Simplify 27. Since 9 is a factor of 27 and 9 is a perfect square, 27 is not in simplified form. The first condition in the box above is not met. We simplify as follows. 27  9  3  9  3

Use the product rule.

 33

9  3, since 32  9.



Expressions such as 27 and 33 are exact values of the square root of 27. If we use the square root key of a calculator, we find 27  5.196152423. If we find 3 and then multiply the result by 3, we get 33  31.732050808  5.196152423.

The 1 after the first line indicates that the equality is true. The calculator also shows the same approximations for 27 and 33 in the second and third answers. (See Example 1.)

Notice that these approximations are the same, as we would expect. (Due to various methods of calculating, there may be a discrepancy in the final digit of the calculation.) Understand, however, that the calculator approximations do not actually prove that the two numbers are equal, but only strongly suggest equality. The work done in Example 1 actually provides the mathematical justification that they are indeed equal. A rule similar to the product rule exists for quotients.

Quotient Rule for Square Roots For nonnegative real numbers a and positive real numbers b, a  b



a . b

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6.4

EXAMPLE (a)

The radical symbol above comes from the Latin word for root, radix. It was first used by Leonardo da Pisa (Fibonacci) in 1220. The sixteenth-century German symbol we use today probably is also derived from the letter r.

(b) (c)



2

Irrational Numbers and Decimal Representation

289

Simplify each radical.

25 9 Because the radicand contains a fraction, the radical expression is not simplified. (See condition 2 in the box preceding Example 1.) Use the quotient rule as follows.

 

3 3 3   4 2 4



25 25 5   9 3 9

1 1 1   2 2 2 This expression is not in simplified form, since condition 3 is not met. To give an equivalent expression with no radical in the denominator, we use a procedure called rationalizing the denominator. Multiply 12 by 22, which is a form of 1, the identity element for multiplication. 1 1 2 2    2 2 2 2

2  2  2



The simplified form of 12 is 22.

Is 4  9  4  9 true? Simple computation shows that the answer is “no,” since 4  9  2  3  5, while 4  9  13, and 5  13. Square root radicals may be added, however, if they have the same radicand. Such radicals are like radicals. We add (and subtract) like radicals with the distributive property. EXAMPLE

3

Add or subtract as indicated.

(a) 36  46 Since both terms contain 6, they are like radicals, and may be combined. 36  46  3  46  76

Distributive property Add.

(b) 18  32 At first glance it seems that we cannot subtract these terms. However, if we first simplify 18 and 32, then it can be done. 18  32  9  2  16  2  9  2  16  2  32  42  3  42  12  2

Product rule Take square roots. Distributive property Subtract. 1  a  a



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From Example 3, we see that like radicals may be added or subtracted by adding or subtracting their coefficients (the numbers by which they are multiplied) and keeping the same radical. For example, 97  87  177 since 9  8  17 43  123  83 , since 4  12  8 and so on. In the statements of the product and quotient rules for square roots, the radicands could not be negative. While 2 is a real number, for example, 2 is not: there is no real number whose square is 2. The same may be said for any negative radicand. In order to handle this situation, mathematicians have extended our number system to include complex numbers, discussed in the Extension at the end of this chapter.

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9 .41

10

14

.32

 7

15. (a) Find the following sum.

16. (a) Find the following sum.

.272772777277772…

.010110111011110…

.616116111611116…

.252552555255552…

(b) Based on the result of part (a), we can conclude that the sum of two numbers may be a(n) number.

(b) Based on the result of part (a), we can conclude that the sum of two numbers may be a(n) number.

Use a calculator to find a rational decimal approximation for each of the following irrational numbers. Give as many places as your calculator shows. 17. 39

18. 44

19. 15.1

21. 884

22. 643

23.



20. 33.6

9 8

24.



6 5

Use the methods of Examples 1 and 2 to simplify each of the following expressions. Then, use a calculator to approximate both the given expression and the simplified expression. (Both should be the same.) 25. 50 31.

5 6

26. 32 32.

3 2

27. 75 33.



7 4

28. 150 34.



8 9

29. 288 35.



7 3

30. 200 36.



14 5

Use the method of Example 3 to perform the indicated operations. 37. 17  217

38. 319  19

39. 57  7

40. 327  27

41. 318  2

42. 248  3

43. 12  75

44. 227  300