4 ms·

>But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Fractions are great until you actu

>But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Fractions are great until you actually need to evaluate them to use their value. Saying that any number can be perfectly represented with fractions is much like saying that any integral calculation can be performed by simply writing a tall S in front of an algebraic expression. You still have to calculate the final result. >Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational. That's just it though: we have no closed form way of calculating the values of trigonometric functions. We only have algorithms that describe how to approximate their values with acceptable amounts of error.

jdmichal 9y ago
> Fractions are great until you actually need to evaluate them to use their value. What does this even mean? Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰.
- chroem- 9y ago
  >Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰. That evaluates exactly to 25. Now try to pour me a glass containing exactly 1/7 gallons of milk without making an approximation.
  - yongjik 9y ago
    I'll pour you exactly 1/7 gallon of milk if you pour me exactly 25 gallons. * Hell, wait, what am I going to do with 24 6/7 gallons of milk?
  - jdmichal 9y ago
    Is such a task meant to be impossible? I don't even... I can assure you that it is, in fact, physically possible for a glass of milk to contain exactly 1/7 a gallon of milk. I assume your "cleverness" is in the assertion that I cannot "prove" 1/7 of a gallon using existent tools because such tools would report the result in a base-10 decimal, which would be necessity have to be approximated because 1/7 is non-terminating in base-10. Of course, I could instead prove it by weight. Or by combining 7 such equal portions and resulting in a single gallon. The latter being an example of exactly why fractional representation of rational numbers is useful. EDIT: As an aside, by any chance are you one of the set of people who argue that .99999999... repeating infinitely is not the same value as 1? EDIT2: This just reminded me a lot of the barometer question, as I think of various ways I could demonstrate 1/7 a gallon. Of course, some things are off the table due to being a heterogeneous substance at the molecular level. Some additional options would be on the table if it were homogeneous, like water. https://en.wikipedia.org/wiki/Barometer_question https://en.wikipedia.org/wiki/Barometer_question
  - deleted 9y ago
    [deleted]
  - Lazare 9y ago
    > Now try to pour me a glass containing exactly 1/7 gallons of milk without making an approximation. This is an exceptionally silly argument. Now, I happen to live somewhere where the metric system in use, so allow me to translate your challenge. So you're saying that it's harder to pour 540.773 ml of milk (ie, 1/7th of a gallon) than 757.082 ml of milk (ie, 1/5th of a gallon)? (Notice how both of those are numerical approximations? How strange...) There's nothing harder or easier about pouring 757.082 ml compared to 540.773 ml, nor do we have some magical pouring tools which allow one number to be poured with greater or lesser accuracy than the other. Nor does it suddenly become possible to pour it with more accuracy if you use a measuring cup with ounces on it. :) You seem to be under the hilarious misapprehension that it's somehow easier to pour something if you happen to be measuring it in units that happen to have a non-repeating decimal representation in whatever base you happen to be working in. That's not really how reality works. (And I'm not even going to start on what "exact" means when dealing with a non-molecularly homogenous fluid.) Real world examples are not going to help your argument here, because the real world can only be interacted with via approximations. 1/7th of a gallon of milk is every bit as much of an approximation as 25 gallons.
  - AnimalMuppet 9y ago
    Pour me a glass containing exactly 1/6 of a gallon of milk. Now pour me a glass containing exactly 1/6 of a gallon of milk, but using base 60. How does that help? (Yes, what helps is having a measuring device calibrated with markings for the amount you're trying to pour. But I have several measuring cups that have markings for 1/3 cup, even though there is no exact base-10 representation of that fraction.)
  - mcguire 9y ago
    It is indeed much easier to pour 8/60 + 34/3600 gallons of milk.
  - deleted 9y ago
    [deleted]

jdmichal 9y ago

> Fractions are great until you actually need to evaluate them to use their value. What does this even mean? Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰.

chroem- 9y ago
>Fractions are already a value, or rather they are a representation of a value. The same as the decimal number 25 is a representation of 2×10¹ + 5×10⁰. That evaluates exactly to 25. Now try to pour me a glass containing exactly 1/7 gallons of milk without making an approximation.
- yongjik 9y ago
  I'll pour you exactly 1/7 gallon of milk if you pour me exactly 25 gallons. * Hell, wait, what am I going to do with 24 6/7 gallons of milk?
- jdmichal 9y ago
  Is such a task meant to be impossible? I don't even... I can assure you that it is, in fact, physically possible for a glass of milk to contain exactly 1/7 a gallon of milk. I assume your "cleverness" is in the assertion that I cannot "prove" 1/7 of a gallon using existent tools because such tools would report the result in a base-10 decimal, which would be necessity have to be approximated because 1/7 is non-terminating in base-10. Of course, I could instead prove it by weight. Or by combining 7 such equal portions and resulting in a single gallon. The latter being an example of exactly why fractional representation of rational numbers is useful. EDIT: As an aside, by any chance are you one of the set of people who argue that .99999999... repeating infinitely is not the same value as 1? EDIT2: This just reminded me a lot of the barometer question, as I think of various ways I could demonstrate 1/7 a gallon. Of course, some things are off the table due to being a heterogeneous substance at the molecular level. Some additional options would be on the table if it were homogeneous, like water. https://en.wikipedia.org/wiki/Barometer_question https://en.wikipedia.org/wiki/Barometer_question
- deleted 9y ago
  [deleted]
- Lazare 9y ago
  > Now try to pour me a glass containing exactly 1/7 gallons of milk without making an approximation. This is an exceptionally silly argument. Now, I happen to live somewhere where the metric system in use, so allow me to translate your challenge. So you're saying that it's harder to pour 540.773 ml of milk (ie, 1/7th of a gallon) than 757.082 ml of milk (ie, 1/5th of a gallon)? (Notice how both of those are numerical approximations? How strange...) There's nothing harder or easier about pouring 757.082 ml compared to 540.773 ml, nor do we have some magical pouring tools which allow one number to be poured with greater or lesser accuracy than the other. Nor does it suddenly become possible to pour it with more accuracy if you use a measuring cup with ounces on it. :) You seem to be under the hilarious misapprehension that it's somehow easier to pour something if you happen to be measuring it in units that happen to have a non-repeating decimal representation in whatever base you happen to be working in. That's not really how reality works. (And I'm not even going to start on what "exact" means when dealing with a non-molecularly homogenous fluid.) Real world examples are not going to help your argument here, because the real world can only be interacted with via approximations. 1/7th of a gallon of milk is every bit as much of an approximation as 25 gallons.
- AnimalMuppet 9y ago
  Pour me a glass containing exactly 1/6 of a gallon of milk. Now pour me a glass containing exactly 1/6 of a gallon of milk, but using base 60. How does that help? (Yes, what helps is having a measuring device calibrated with markings for the amount you're trying to pour. But I have several measuring cups that have markings for 1/3 cup, even though there is no exact base-10 representation of that fraction.)
- mcguire 9y ago
  It is indeed much easier to pour 8/60 + 34/3600 gallons of milk.
- deleted 9y ago
  [deleted]