Example: Convert

92.322

Base conversion for expansions and fractions

Fall 2013

Here are some examples of conversions between expansions and fractions in different bases.

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First, a decimal example. To convert a fraction to a decimal expansion, use long division. If a remainder of zero occurs, the expansion is complete (and finite). If a remainder is repeated, then the expansion is infinite and repeating.

Example:

Convert

3 8

to

a

decimal

expansion.

Solution:

0. 3 7 5 8 3. 0 0 0

24 60 56 40 40 0

The procedure terminates when a remainder of zero appears, so we con-

clude

that

3 8

=

0.375.

The reverse procedure is also quite easy, since

0.375 = 375 = ? ? ? reducing to lowest terms ? ? ? = 3.

1000

8

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Of course, many fractions do not have terminating decimal expansions. Indeed, fractions will only have finite decimal expansions when the denominator, in lowest terms, is a product of powers of 2 and 5 only. (Can you figure out why?)

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Example:

Convert

39 74

to

a

decimal

expansion.

Solution:

0. 5 2 7 0 74 3 9. 0 0 0 0

370 2 00 1 48 520 518 20 0 20

at which point the procedure terminates, since the remainder of 20 has ap-

peared for the second time. We when a remainder of zero appears, so we

conclude that

39 = 0.5270270 . . . = 0.5270, 74

an infinite repeating decimal.

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Suppose instead we were given the repeating decimal 0.5270 and asked for its fractional form. To perform this conversion, we observe that a repeating decimal is also a geometric series:

0.5270 = 0.5 + (0.1)(0.270270270270 . . .)

= 0.5 + (0.1)(270)(0.001001001001 . . .)

= 1 + 27 2

1 1000

+

1 10002

+

1 10003

+

???

= 1 + 27 1

2

999

= 1 + 27 2 999

= 1 + 1 = 39. 2 37 74

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Here we are using the formula

x + x2 + x3 + ? ? ? = x when |x| < 1, 1-x

so that

1 + 1 + 1 +??? = n n2 n3

1

n

1

-

1 n

=

1 n-1

when |n| < 1.

(1)

This trick tells us immediately, for example, that

0.44444 . . . = 4 9

0.429342934293 . . . = 4293 9999

0.20178787878 . . . = 0.201 + (0.001)(0.78) = 201 + 78 1000 99000

and so on. These fractional expressions can then be simplified and reduced to lowest terms in the usual manner.

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The presence in the previous example of the 9s, 99s, and so on is due to the n - 1 appearing in the denominator of the identity (??). Here n is a power of 10, because we are working with decimal notation, which expresses all numbers as (limits of) sums of powers of 10. Suppose that, instead, we are working in base 8. Now the relevant denominators would, for the same reason, be numbers such as

(7)8 = (10)8 - 1 (77)8 = (100)8 - 1 (777)8 = (1000)8 - 1

and so on. [In decimal, these numbers would be

7 = 8 - 1 63 = 82 - 1 511 = 83 - 1

and so on. Be careful and consistent with your notation!]

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Example: (a) Convert the octal expansion

(0.42424242 . . .)8 = (0.42)8

to an octal fraction. (b) What is this fraction in decimal notation? (c) What is the decimal expansion of this number?

Solution: For part (a), the geometric series argument earlier tells us that

(0.42)8 =

42 77 8

in octal.

For part (b), simply convert the numerator and denominators to decimal notation:

(42)8 = 2 + 4(8) = 34 and (77)8 = 7 + 7(8) = 63,

so that

42 = 34

(2)

77 8 63

in decimal notation.

For

part

(c),

use

long

division (or

your

calculator)

to

show

that

34 63

=

0.539682.

Notice something interesting about the identity (??). It is clear from the right-hand side that the fraction is in lowest terms, because GCD(34, 63) = 1. This means that the octal fraction

42

77 8

is in lowest terms as well. "Wait a minute," you say, "isn't 42 divisible by 7?" No! It is not divisible by 7. This is octal notation, so the symbol `42' is not the number 6 ? 7. The number 6 ? 7 is represented in octal as `52'. Examples like this demonstrate the extra care needed when using bases other than 10.

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All of the methods demonstrated above for decimal expansions and fractions generalize to other bases. Here is another example in octal (base 8).

Example: Convert the octal fraction

43 200

to its octal expansion.

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Solution: Let's try long division in base 8. It works the same way as in before, except that we must remember that 7 + 1 = 8 and so on.

0. 2 1 4 200 4 3. 0 0 0

400 3 00 2 00 1 000 1 000 0

The procedure terminates when the remainder is zero, so we conclude that

43 200

8

= (0.214)8.

Let's check our work. To convert (0.214)8 into a fraction, let's avoid confusion by doing our computations in decimal:

(0.214)8

=

2 8

+

1 82

+

4 83

=

2(64)

+ 1(8) 512

+4

=

140 512

=

35 128

in decimal. Since 35 = (43)8 and 128 = (200)8, we have

(0.214)8 =

35 = 128 10

43 , 200 8

as expected.

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