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DECIMAL REPRESENTATION OF REAL NUMBERS

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Abstract: Irrational numbers such as 2, π and e are represented by decimal expan ... real numbers expressed in nondecimal bases such as binary. History. The regular use of the decimal point appears to have been introduced about 1585, ...

CS131 Part III, Sequences and Series
CS131 Mathematics for Computer Scientists II Note 18
DECIMAL REPRESENTATION OF REAL
NUMBERS
Recall that if r < 1 then the geometric series (Note 17) n≥0 rn has sum
1/(1 − r), i.e.,
1
1 + r + r2 + r3 + · · · = .
1−r
If a1 , a2 , · · · is a sequence of decimal digits, so that each ai belongs to
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, then .a1 a2 a3 · · · denotes the real number
a1 a2 a3
+ 2 + 3 + ··· .
10 10 10
To see why this sum exists, note that we have an /10n ≤ 9/10n for each n.
Now the series 1/10n converges (r =1/10 above), so 9/10n converges,
and it follows from the Comparison Test (Note 17) that an /10n converges.
It is clear that a terminating decimal .a1 a2 · · · an represents a rational num
ber since
a1 a2 an
.a1 a2 · · · an = + 2 + · · · + n.
10 10 10
The general form of a repeating decimal is
.a1 a2 · · · am b1 b2 · · · bn b1 b2 · · · bn · · · ,
˙ ˙
which is denoted by .a1 a2 · · · am b1 b2 · · · bn .
Repeating decimal as rational number. A repeating decimal repre
sents a rational number and can be expressed as a quotient of two integers
by summing an appropriate geometric series as the following problem illus
trates.
˙ ˙
Problem. Express the repeating decimal 0.59102 as the quotient of two
integers.
18–1
Solution. We have
˙ ˙
0.59102 = 0.59102102 · · · = 59
+ 102
+ 102
+ 102
+ ···
100 105 108 1011
59 102 1 1 59 102 1
= 100 + 105 1+ 103 + 106 + ··· = 100 + 105 1−1/103
59 102 1 59 102 1 59×999+102
= 100 + 100 103 −1 = 100 + 100 999 = 99900
= 59043/99900.
A nonzero number with a terminating decimal representation can also be
written as a nonterminating decimal since we can introduce an inﬁnity of
˙ ˙
repeating 9’s. For example 0.5 = 0.49 and 1 = 0.9 .
˙
Problem. Show that 0.49 = 1/2 .
Solution. From the deﬁnition of a repeating decimal we have
˙
0.49 = 4
+ 9
+ 9
+ ··· = 4
+ 9 1
= 4
+ 9
= 4
+ 1
= 1.
10 102 103 10 102 1−1/10 10 100−10 10 10 2
√
Irrational numbers such as 2, π and e are represented by decimal expan
sions which do not terminate or repeat.
The decimal expansion of an arbitrary real number, pictured at x on the
real line, can be computed to any desired accuracy in the following way.
If x is not an integer then it lies between two consecutive integers a0 and
a0 + 1. Now divide the interval between these two integers into 10 equal
parts. If x is not on one of the subdivision points then it must lie between
two consecutive subdivision points. Hence we have
a1 a1 + 1
a0 + < x < a0 + ,
10 10
where a1 ∈ {0, 1, . . . , 9}. Dividing this new interval containing x into 10
equal subintervals (each of length 1/102 ) gives an inequality of the form
a1 a2 a1 a2 + 1
a0 + + 2 < x < a0 + + .
10 10 10 102
Continuing in this way, we obtain two sequences of ﬁnite decimals each
converging to x, one increasing from below and the other decreasing from
above. By repeating this procedure suﬃciently many times, the decimal
expansion of x can be obtained to any desired degree of accuracy.
18–2
Rational numbers as repeating decimals. We have seen that terminat
ing and repeating decimal expansions represent rational numbers. Now we
show that, conversely, every rational number has a terminating or repeating
expansion.
The decimal expansion of a rational number r/s ∈ (0, 1) can be computed
using the division algorithm (generations of schoolchildren used to call this
process long division).
Dividing 10r by s we have
10r = sq1 + r1 where 0 ≤ r1 < s.
Repeatedly multiplying remainders by 10 and dividing by s produces se
quences q1 , q2 , . . . and r1 , r2 , . . . with
10ri = sqi+1 + ri+1 where 0 ≤ ri < s.
Hence
ri qi+1 1 ri+1
= +
s 10 10 s
for each i.
Now for any n we have
r q1 1 r1
= +
s 10 10 s
q1 1 q2 1 r2 q1 q2 1 r2
= + + = + 2+ 2
10 10 10 10 s 10 10 10 s
.
. .
.
. .
q1 q2 qn 1 rn
= + 2 + ··· + n + n .
10 10 10 10 s
Also each qi is one of the digits 0, 1, . . . , 9 since
0 ≤ 10ri < 10s =⇒ 0 ≤ sqi+1 + ri+1 < 10s
=⇒ 0 ≤ sqi+1 < 10s − ri+1 ≤ 10s − s
=⇒ 0 ≤ qi+1 ≤ 9.
If at some stage in the division we obtain a remainder rn = 0 then
r q1 q2 qn
= + 2 + · · · + n = 0.q1 q2 · · · qn .
s 10 10 10
18–3
Otherwise, since there are only ﬁnitely many values for the remainders
(which must lie between 0 and s − 1), the sequence r1 , r2 , . . . must contain
repetitions.
Suppose that rn is the ﬁrst repeated remainder, so that rn = rm for some
m < n. Dividing 10rn by s is the same as dividing 10rm by s, so the quotient
and remainder are the same as before.
Hence the process repeats and we have qn+1 = qm+1 , qn+2 = qm+2 · · · , which
gives
r/s = 0.q1 q2 · · · qm · · · qn .
˙ ˙
Problem. Find the decimal expansion of 5/14.
Solution.
0·35714285
145·00000000
4 2
80
70
100
98
20
14
60
56
40
28
120
112
80
70
˙ ˙
Hence 5/14 = 0.3571428 since the remainder 8 is repeated.
Nondenary bases. We can also represent real numbers in bases other
than ten. In base two, if b1 , b2 , b3 , . . . are binary digits (0 or 1), then
.b1 b2 b3 · · · represents the number
b1 b2 b3
+ 2 + 3 + ··· .
2 2 2
For example we have
1/2 = 0.5ten = 0.1two and 1/4 = 0.25ten = 0.01two .
18–4
Problem. Find the binary representation of the baseten fraction 1/3.
Solution. We divide 3 into 1 in base two i.e. divide 11 into 1.
0·0101
111·0000
11
100
11
100
˙˙
so (1/3)ten = 0.010101 · · · = 0.01 since the remainder 1 is repeated.
ABSTRACT
Content terminating decimals, repeating decimal as rational number, rational numbers as repeating deci
mals, nondenary bases
In this Note we consider decimal numbers in terms of an associated geometric series. We also study the
relationship between repeating decimals and rational numbers and viceversa. We ﬁnish by considering
real numbers expressed in nondecimal bases such as binary.
History
The regular use of the decimal point appears to have been introduced about 1585, but the occasional use
of decimal fractions can be traced back as far as the 12th century.
Muhammad ibnMusa Khwarizmi, [c. 780c. 850] was a Persian mathematician who wrote a book on
algebra, from part of whose title ( aljabr ) comes the word ‘algebra’, and a book in which he intro
duced to the West the HinduArabic decimal number system. The word ‘algorithm’ is a corruption of his
name. He compiled astronomical tables and was responsible for introducing the concept of zero into Arab
mathematics.
Simon Stevinus [c. 15481620] was a Flemish scientist who, in physics, developed statics and hydrodynam
ics; he also introduced decimal notation into Western mathematics.
John Napier [15501617], 8th Laird of Merchiston, was a Scottish mathematician who invented logarithms
in 1614 and ‘Napier’s bones’, an early mechanical calculating device for multiplication and division.
Napier arranged his logarithmic calculations in convenient tables which evolved into what generations of
schoolchildren came to know as ‘log tables’. These have now been superseded by the use of hand calculators
and digital computers.
It was Napier who ﬁrst used and then popularised the decimal point to separate the whole part from the
fractional part of a number.
18–5
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