Download 6.4 Irrational Numbers and Decimal Representation

6.4
283
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.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
284
CHAPTER 6
The Real Numbers and Their Representations
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 .
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
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.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
(continued)
286
CHAPTER 6
The Real Numbers and Their Representations
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.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
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 .
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
288
CHAPTER 6
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
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
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
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
290
CHAPTER 6
The Real Numbers and Their Representations
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.
An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
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