Non-terminating Decimal Representations of Fractions
Date: 11/10/97 at 19:07:48 From: MRS.SWIZ Subject: Non-terminating decimal representations of fractions Dear Dr. Math, Why is it that you can take a perfectly finite, limited quantity like one-third (you take a pie, divide it into three simple, equal pieces, and you get three and only three pieces - no more - no less) but when you turn it into a decimal you get .333... on into infinity? It seems intuitively that something is inherently wrong with the decimal representation of that quantity. I am an Education grad student with a BA in Fine Arts and no clue (obviously) about math. Thanks. Diane
Date: 11/10/97 at 20:08:07 From: Doctor Tom Subject: Re: Non-terminating decimal representations of fractions Hi Diane, Unfortunately, the short answer is, "That's the way it is." Since we happen to have ten fingers and thus use base 10 (decimal) arithmetic, we're stuck with it. I'm not sure if you know anything about other number bases, but the choice of 10 is totally arbitrary. Base 2 and base 16 are heavily used in computers, but base ten (decimal) is so heavily entrenched that the only other example I can think of where another base is commonly used is in old-style counting: dozen = 12, gross = 144 = 12*12, great gross = 1768 = 12*12*12. It's a sort of a start of a base 12 system. In case you don't know about base systems, think about counting in decimal. Everything is regular in the units digit until you get to nine, but the next step "overflows," and drives the 9 to zero, but pumps up the next digit over (which may overflow itself - 99 => 100 -- and so on). In base 5, for example, counting would go like this: 0,1,2,3,4,10,11,12,13,14,20,21,22,23,24,30,31,32,33,34,40,41 42,43,44,100,101,102,103,104,110,... Base 3 is like this: 0,1,2,10,11,12,20,21,22,100,101,102,110,111,112,120,121,122, 200,201,202,210,211,212,220,221,222,1000,1001,1002,... Stare at these until you get the pattern. It's just like the decimal system, but the overflow occurs earlier. In decimal, if you look at a number like 123.45, the "1" represents the number of 10*10s, the "2", the number of 10s, the "3" the number of 1s, the "4", the number of 1/10s, and the 5, the number of 1/(10*10)s. In base 5, the number 12.34 represents 1 of 5, 2 of 1, 3 of 1/5 and 4 of 1/(5*5) = 1/25, so to convert 12.34 (in base 5) to base ten, you get: 1*5 + 2*1 + 3*(1/5) + 4*(1/25) = 7 19/25 = 7.76 In base 3, the fraction 1/3 has a great expression: 0.1, but then you have trouble with other fractions. For example, 1/2 (one-half) is expressed by .111111... (base 3). No matter what base you choose, some fractions will have a clean expression in a "decimal-like" form, and others won't. I put "decimal- like" in quotes, since "decimal" refers to 10, and if you're working in base 3, it's technically called a ternary system, the fraction is not a "decimal fraction" but a "ternary fraction," and it's not a "decimal point" but a "ternary point." It's an interesting mathematical question to find the relation between the base and the numbers that come out "nicely." It has to do with what numbers divide evenly into the base. For 10, only 1, 2, 5, and 10 do. For base 3, only 1 and 3 do. Base ten will thus tend to have more numbers that work out with clean decimal expansions. But if only we used base 360, we'd be in heaven! Look what goes evenly into it: 1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180, and 360. Of course it would be a giant pain to have 360 different characters to memorize instead of our current set of 10: "0", "1", ..., "9" :^) I hope this helps. -Doctor Tom, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 11/10/97 at 22:08:05 From: MRS.SWIZ Subject: Re: Non-terminating decimal representations of fractions Dear Dr. Math, Please disregard my question to you about non-terminating decimals. I have found the answer and it is so simple that I feel somewhat foolish. Decimals are based on tens and three will never divide ten evenly! Never mind. Sorry. Diane
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.