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Thread: What's More Exact: Decimals or Fractions?

  1. #1 What's More Exact: Decimals or Fractions? 
    Forum Radioactive Isotope zinjanthropos's Avatar
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    Perhaps our resident mathematicians can help settle an argument at work on the accuracy of fractions versus decimals. I'm saying that fractions are more exact. I say this because there are certain fractions that one cannot write in decimal form so that it is as accurate as the fraction it represents, such as .33333 repeating versus 1/3. People against me say although it appears as if fractions are more accurate, they are not in all cases. They say that in certain calculations decimals work more accurately then fractions. One guy said that calculating square roots is more exact using decimals although I couldn't follow his reasoning.

    I work with some smart people who do a lot of calculating in a day so I have a feeling I'm wrong. What is the opinions of our experts here?


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    A fraction is an exact ratio of 2 numbers, and if those 2 numbers are integers, or at least rational numbers, then the fraction can more appropriately be called a rational number. An irrational number can be represented as an approximation to a rational number to an extremely high degree of accuracy. Depending on how "exact" you want to be when expressing an irrational in rational terms, determines the more appropriate method.
    equal to 3 significant figures
    equal to 6 significant figures

    Rationals can be more exact, but just leaving the square root alone is the most exact method.

    Bear in mind, a decimal is also a fraction.
    Last edited by Arcane_Mathematician; October 22nd, 2013 at 08:40 PM. Reason: added information
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    It depends on the specific number you want to represent.
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    Quote Originally Posted by zinjanthropos View Post
    Perhaps our resident mathematicians can help settle an argument at work on the accuracy of fractions versus decimals. I'm saying that fractions are more exact. I say this because there are certain fractions that one cannot write in decimal form so that it is as accurate as the fraction it represents, such as .33333 repeating versus 1/3. People against me say although it appears as if fractions are more accurate, they are not in all cases. They say that in certain calculations decimals work more accurately then fractions. One guy said that calculating square roots is more exact using decimals although I couldn't follow his reasoning.

    I work with some smart people who do a lot of calculating in a day so I have a feeling I'm wrong. What is the opinions of our experts here?
    I think I can shed some light on this for you.

    A rational number is a number that can be expressed as the ratio of two whole numbers. 1/2, 2/3, and 47 (= 47/1) are three familiar examples of rational numbers.

    Every rational number can be expressed as a repeating or terminating decimal. For example 1/2 = .5 and 1/3 = .333...

    where the '...' means that the 3's go on "forever," whatever that means. We can make mathematical sense of "forever" by inventing the theory of convergent infinite series, which is taught in the first year of calculus.

    Mathematically, .333... is every bit as precise as 1/3. But the idea of the "..." part implies an infinite process: something we can do mathematically, but could never do in the real world. So perhaps you might say that 1/3 is intuitively more appealing to the average person.

    Now, here's the big problem. Not every number is rational! For example there is no fraction for sqrt(2). That is, no matter what whole numbers m and n you pick, m/n is not the square root of 2. Euclid wrote down a beautiful proof of this fact around 2300 years ago.

    So now we're stuck. There are real numbers that are not rational numbers; that is, they're not the ratio of any two whole numbers. For those numbers we can either write down a bunch of their digits and some dots: 1.414213... which isn't very exact at all! Or we can write down a shorthand like "sqrt(2)" and give an algorithm for computing all the digits.

    The most exact way of specifying an irrational number like sqrt(2) is to give an algorithm for its digits. The next best description is to give a few digits and then toss in some dots "..." to stand for all the numbers we didn't write down. But there is no way to write sqrt(2) as a fraction or rational number, and that's the sad truth. No matter what whole numbers m and n we try, m/n is never going to be the square root of 2.

    By the way, with regard to your friend saying that "calculating square roots is more exact using decimals," I'm not sure I agree. For any decimal approximation to sqrt(2) like 1.414213... we can write down the corresponding fraction 1,414,213/1,000,000. They're both the same. Of course the fraction isn't exactly sqrt(2), but neither is 1.414213... until you say what all the rest of the digits are. But you can never do that, because not only are there infinitely many of them, but there is no pattern to them. So unlike .333... where you can say "the dots mean there are all 3's," with sqrt(2) you have to say that the dots stand for an endless list of numbers with no pattern at all.
    Last edited by someguy1; October 23rd, 2013 at 12:01 AM.
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    Any rational number can be written exactly with either fractions or decimals so there is no question of "accuracy".

    Given any irrational number, x, and a fraction, y, approximating x, there exist a decimal, z, approximating x better than y.

    Given any irrational number, x, and a decimal, y, approximating x, there exist a fraction, z, approximating x better than y.

    But, in any case "a represents b better than c" is a question of numbers not how the numbers are represented. And the difference between fractions and decimals is one of how numbers are represented, not the numbers themselves.
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    Okay, I hope you're not one of those people who denies 0.999... = 1, because it sounds like you're saying 0.333... doesn't exactly equal 1/3. If there are only a finite amount of 3s, I wouldn't argue a bit that they're not equal, but with an infinite amount, they are.
    In my sense, "fraction form" is not just fractions. sqrt(2), for instance, cannot be exactly represented by a fraction or decimal (at least, a finite one), but in "fraction form" it is 2 under a radical sign. "Fraction form" also includes numbers in terms of pi and e.
    Fractions are much easier to work with than decimals in most cases, regardless of what middle and high schoolers tend to think.
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    Quote Originally Posted by zinjanthropos View Post
    I'm saying that fractions are more exact. I say this because there are certain fractions that one cannot write in decimal form so that it is as accurate as the fraction it represents, such as .33333 repeating versus 1/3.
    I agree with your reasoning. It's quite clear that there are fractions which cannot be expressed in finite decimal form. Indeed, if the prime factorisation of the denominator contains any other prime number than 2 or 5, then the fraction cannot be expressed in finite decimal form. Given that every finite decimal is a fraction whose denominator is a power of 10, but not every fraction is a finite decimal, it is clear that there is greater freedom to approximate a number compactly by choosing fractions. It turns out that the notion of continued fractions provides a way to establish the most accurate fraction form of a number.
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    Quote Originally Posted by adelady View Post
    Different horses for different courses.
    W ♪ ♪ Dixi...
    From a life history lesson. Working as a builders hand I can argue the metric system is much simpler and very accurate.
    I draw a very simple case of the division of equal parts of a space. If I am required to convert to inches and divide a large space I can see that use of metrics makes that task easy.. That weights and torque, and horse power.. gallons and foot pounds..
    While working with oscilloscopes and electric harmonics I use the metric. BUT, I also understand that for some the use of fractions other than base 10 can be useful. Where in the world you are is part of this question.. I can not think of any country other than the USA that stays so far from the metric.. I do not understand the stubborn stance I perceive. A little like 'the tower of babel' ...
    and that Douglas Adams saw it so clear I find amusing still.
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