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### Fraction or Decimal?

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Date: 06/23/98 at 15:49:36
From: Jessica Burton
Subject: Precision of fractions

Dr. Math,

In general, which is more precise, a fraction or a decimal (for
instance, 1/3 vs. 0.33)?

Thanks,
Jessica
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Date: 06/24/98 at 09:00:58
From: Doctor Jerry
Subject: Re: Precision of fractions

Hi Jessica,

I'm guessing that you are thinking of the decimal representation of
fractions. For example 1/4 = 0.25. In this case, both 1/4 and 0.25
have equal precision.

I wrote 1/4 = 0.25 because these two things represent exactly the same
number. The fraction 1/3, however, is different in that if you divide
1 by 3 you will get 0.3333333....  The threes never stop. If you
divide 1 by 4, you get 0.25 and that's it. So, I can say that
1/3 = 0.3333.... (the dots mean that the 3 is repeated indefinitely)
but 1/3 is not equal to 0.33. In this case, 1/3 is more precise. In
fact, 1/3 - 0.33 = 1/3 - 33/100 = 1/300, which is the error you would
commit if you were to use 0.33 in place of 1/3.

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
```

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Date: 06/24/98 at 11:55:03
From: Doctor Peterson
Subject: Re: Precision of fractions

Hi, Jessica -

This is a fascinating question, because it leads into some ideas worth

My first answer is that fractions are unquestionably more precise,
in at least two ways. First, any rational number can be exactly
represented by a fraction (that's what a rational number is, in the
first place), while most rational numbers can't be exactly represented
by a decimal.

Your example of 1/3 makes this very clear. It may take a huge
numerator and denominator to represent some numbers, but even a simple
little number like 1/3 can't be represented exactly by any number of
decimal places (unless you use a notation to indicate a repeating
decimal, in which case it is just as exact as a fraction). In fact,
*most* rational numbers do not produce terminating decimals; only
those whose denominators contain only factors of 2 and 5 can be
represented exactly by a finite decimal. So fractions mean exactly
what they say, while decimals are usually just approximations.

Secondly, when you work with fractions, you don't lose any of that
precision, as long as you are only adding, subtracting, multiplying,
dividing, and taking (integer) powers. You have probably had the
experience of doing a series of calculations on a calculator and
finding that the answer was .99999998 when you expected 1.0; that's
because a calculator can only store a limited number of decimal
places, and calculations can increase the error caused by rounding
until it becomes noticeable. With decimals, that is unavoidable,
because you can never store all the digits; with fractions, it will
only happen when the numerator or denominator gets too big to handle.

On the other hand, sometimes a number can be very precise, but not
really accurate. How can that be? I can think of two cases where
fractions are inaccurate. First, there is the mathematical problem of
"real" numbers: not all numbers are rational. If you take the square
root of 1/2, the result can't be represented by any fraction, so you
would have to approximate it by some fraction, such as 29/41. Then
doesn't accurately represent the truth! In fact, since most real
numbers are irrational, most numbers can't be represented accurately
by a fraction!

Second, there is the scientific problem of "real" numbers: nothing we
can measure in the real world is exact, so the precision of a fraction
doesn't accurately represent our knowledge. If I measure something as
1/2 inch, it may really be 1001/2000 inch. Again, the precision of my
fraction is misleading. I don't really know that it is exactly 1/2
inch; the fraction is just an approximation.

A benefit of decimals is that they provide an easy way to indicate how
precise your measurement is. If I read the length off a ruler, I can
say it's 0.5 inch; if I use a laser to measure it, I might say 0.50000
inch, because I know that my measurement was no more than 0.000005
away from the correct value. That way, the precision of my number
reflects the accuracy of my measurement, and I am not implying more
precision than I really have. To put it another way, decimals give me
a way to control my level of precision, and in that way can be said to
be more precise than fractions!

To sum this up: Fractions are technically more precise, but either one
is only as accurate as you make it; both can be used either as an
approximation or as an exact value. Working with rational numbers, a
decimal will usually be only an approximation; but with real numbers
(in either sense), you usually can do no better than an approximation
anyway.

You should have known you wouldn't get a simple yes or no answer that
would settle your bet. You'll have to decide which of you is right,
based on how you are defining precision!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Fractions
Elementary Measurement
Middle School Fractions
Middle School Measurement

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