Fraction or Decimal?Date: 06/23/98 at 15:49:36 From: Jessica Burton Subject: Precision of fractions Dr. Math, Please settle a bet... In general, which is more precise, a fraction or a decimal (for instance, 1/3 vs. 0.33)? Thanks, Jessica 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/ 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 thinking about. 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 your answer looks precise, but the precision is misleading, because it 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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