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I disagree. Mansfeld and Wildberger made a number of trivially provable false claims. I read the SciAm article as 1) focusing on their false claims and 2) not

by Lazare 9y ago

I disagree. Mansfeld and Wildberger made a number of trivially provable false claims. I read the SciAm article as 1) focusing on their false claims and 2) not trying to "rally the public" against rational trigonometry at all, much less on emotional grounds. In particular, Mansfeld and Wildberger (especially as reported in other outlets) are making the claim that "rational trigonometry" is more accurate, and that's a concrete claim with is 1) easily verifiable and 2) wrong. And that's not even touching on the bizarre claim that "we count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so much wrong with that, it's hard to know where to start.

chroem- 9y ago
>that's a concrete claim with is 1) easily verifiable and 2) wrong I fail to see how it's wrong. If you call Math.sin(x) it will return an iterative numerical approximation of sine. Would you care to elaborate rather than casually dismissing it? > There's so much wrong with that, it's hard to know where to start. Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that.
- AnimalMuppet 9y ago
  OK, as a purely useless concession, I will grant you that base 60 gives you more rational numbers that you can express as terminating decimals. (Uh, not decimal decimals, but... whatever the right term is.) But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational.
  - chroem- 9y ago
    >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
- Lazare 9y ago
  > I fail to see how it's wrong. If you call Math.sin(x) it will return a numerical approximation of sin(). Right. Now what's being argued here is that instead of measuring an angle X, calculating the sine of X, and then (probably) rounding the result to a convenient decimal approximation, we should measure an angle X, round it to an angle Y with a convenient non-repeating decimal representation, and calculate the sine of Y. (That's the whole point of Mansfield's and Wildberger's analysis of the tablet; that it was a trig table that only gave answers for angles with convenient answers, rather than, as modern trig tables do, giving it for all angles.) So instead of measuring an angle as 29.88 degrees, calculating the sine of 29.88 as 0.49818510533, then rounding to 0.4982 or whatever, we should instead calculate the sine of 30 degrees, and get the wonderful totally accurate "exact fraction" 1/2. 1/2 is totally accurate representation of the sine of 30 in a way that 0.4982 is not a totally accurate representation of the sine of 29.88, yes... But 0.4982 is a much more accurate representation of the angle you've actually measured. From an engineering point of view (which offhand seems like the only one where accuracy matters), normal trig is obviously, obviously better than rational trig, because normal trig is more accurate in the ways that matter. The more sigfigs I have in my measurement of my angle, the more decimal places I can leave in my numerical approximation of the answer. It may always be inexact in a way that 1/2 is not, but it'll also always be exact in a much more important way. :) > Such as? 60 has more prime factorizations than 10 So it does, but that's not the claim being made. Rather, the claim was: "We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." To which one might note that: 1. All fractions are exact, by definition. 1/3 is an exact fraction; it's exactly 1/3, in exactly the way 1/2 is exactly 1/2. The fact it's decimal representation is repeating does not impact this. 2. Mansfeld apparently means "fractions with terminating decimal representations" when he says "exact", which is a weird and non-standard usage. But even so, you can construct tons of fractions in base 10 that have non-repeating decimal representations, like 2/5ths, or 5/8th. 3. Possibly he means "...which only allows you to construct two fractions using a numerator of 1 and a denominator consisting of a number smaller than the size of the base which has non-repeating decimal representations..." (which would be a pretty weird thing to focus on) it's still wrong; it's missing, eg, 1/4th. 4. Finally, he reckons that in base 60, 7/60 + 30/3600 aka 1/8 aka 0.125 is an "exact fraction". That sort of factorization is legitimate, but obviously if that's an exact fraction so is 2/10 + 5/100 aka 1/4th. Or come to that, 6/10 aka 0.6. In short, the claim is gibberish. It's using a phrase with no clear definition in a way which is internally inconsistent. There is no way that 1/8th is an "exact fraction" in base 60, but only 1/2 and 1/5 are "exact fractions" in base 10. (Also, you're using the word "irrational" wrong.)
- jdmichal 9y ago
  > Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that. Actually, I can argue with that. Irrationality is a property of the number, not the representation. An irrational number cannot be written as a terminating "decimal" in any base. And, of course, a rational number can be written as an integer fraction in any base. https://math.stackexchange.com/a/625481 https://math.stackexchange.com/a/625481 EDIT: Ignoring, of course, doing silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory. Also, I wonder if every number except pi and its multiples are non-terminating in base-pi...
  - sillysaurus3 9y ago
    silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory Hmmm... Anybody know any resources related to that? I'm interested in the properties of base-pi. I found this: https://math.stackexchange.com/a/2188852 https://math.stackexchange.com/a/2188852 Strange bases are pretty cool. Any integer can be converted into base-sqrt(2) by interleaving zeoes into its binary representation. E.g. 5 in base-sqrt(2) is 010001, 21 (0b10101) is 0b0100010001, etc. Possibly useless, but neat. Also: https://www.quora.com/Can-we-have-a-number-system-with-the-base-pi https://www.quora.com/Can-we-have-a-number-system-with-the-b... Knuth considers transcendental bases (ee is a transcendental number) in The Art of Computer Programming – in the section called Positional Number Systems. One of the consequences of using irrational bases is unfortunately that numbers need not have a unique representation. EDIT: Wow, this is awesome: https://www.quora.com/What-is-10-in-base-pi https://www.quora.com/What-is-10-in-base-pi Specifically the answer from Dave Williamson. It's ridiculously comprehensive. He shows how to calculate it using pure math, using a calculator, by writing a program, and demonstrates a partial proof on some of the properties.
    - Someone 9y ago
      Neither mentions base π, but you may be interested in https://en.wikipedia.org/wiki/Non-integer_representation https://en.wikipedia.org/wiki/Non-integer_representation, which even claims applications: "There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998)." and https://en.wikipedia.org/wiki/Quater-imaginary_base https://en.wikipedia.org/wiki/Quater-imaginary_base. (Surprisingly, there seems to be less content on this on mathworld)
ordu 9y ago
> "rational trigonometry" is more accurate, and that's a concrete claim with is 1) easily verifiable and 2) wrong. I do not see how it is easily verifiable. Rational trigonometry, as I understand, does not use angles, so it is not clear for me how to use it in a lot of cases. I belive there are ways to do it, but it needs some research or reading Wildberger... So, its not seems for me as "easily". > the bizarre claim that "we count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so much wrong with that, it's hard to know where to start. To my mind, it is just imprecise wording from Wildberger side. Or maybe it precise in his rational way of thinking, but we, accepting irrational numbers, see it imprecise. I belive, that he meant to say, that 1/2 and 1/5 is some kind of "base" fractions and all fractions that can be written presice is `(1/2)^m*(1/5)^n`, where m and n are non-negative intergers. And to write precise is means to write in decimal representation. I cannot see any other explanation of Wildberger words which would not imply that Wildberger just plain stupid, which is improbable. Really, I like the idea: we know how to solve a lot problems with irrational numbers, but does it possible to solve them without irrational? How far mathematics can go without irrational numbers?

by Lazare 9y ago

chroem- 9y ago

>that's a concrete claim with is 1) easily verifiable and 2) wrong I fail to see how it's wrong. If you call Math.sin(x) it will return an iterative numerical approximation of sine. Would you care to elaborate rather than casually dismissing it? > There's so much wrong with that, it's hard to know where to start. Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that.

AnimalMuppet 9y ago
OK, as a purely useless concession, I will grant you that base 60 gives you more rational numbers that you can express as terminating decimals. (Uh, not decimal decimals, but... whatever the right term is.) But, first, every rational number is exactly representable in any base (as a fraction, rather than as a decimal expansion). Second, this doesn't help you at all if you're taking trig functions of almost all angles, where the result is not rational.
- chroem- 9y ago
  >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
Lazare 9y ago
> I fail to see how it's wrong. If you call Math.sin(x) it will return a numerical approximation of sin(). Right. Now what's being argued here is that instead of measuring an angle X, calculating the sine of X, and then (probably) rounding the result to a convenient decimal approximation, we should measure an angle X, round it to an angle Y with a convenient non-repeating decimal representation, and calculate the sine of Y. (That's the whole point of Mansfield's and Wildberger's analysis of the tablet; that it was a trig table that only gave answers for angles with convenient answers, rather than, as modern trig tables do, giving it for all angles.) So instead of measuring an angle as 29.88 degrees, calculating the sine of 29.88 as 0.49818510533, then rounding to 0.4982 or whatever, we should instead calculate the sine of 30 degrees, and get the wonderful totally accurate "exact fraction" 1/2. 1/2 is totally accurate representation of the sine of 30 in a way that 0.4982 is not a totally accurate representation of the sine of 29.88, yes... But 0.4982 is a much more accurate representation of the angle you've actually measured. From an engineering point of view (which offhand seems like the only one where accuracy matters), normal trig is obviously, obviously better than rational trig, because normal trig is more accurate in the ways that matter. The more sigfigs I have in my measurement of my angle, the more decimal places I can leave in my numerical approximation of the answer. It may always be inexact in a way that 1/2 is not, but it'll also always be exact in a much more important way. :) > Such as? 60 has more prime factorizations than 10 So it does, but that's not the claim being made. Rather, the claim was: "We count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." To which one might note that: 1. All fractions are exact, by definition. 1/3 is an exact fraction; it's exactly 1/3, in exactly the way 1/2 is exactly 1/2. The fact it's decimal representation is repeating does not impact this. 2. Mansfeld apparently means "fractions with terminating decimal representations" when he says "exact", which is a weird and non-standard usage. But even so, you can construct tons of fractions in base 10 that have non-repeating decimal representations, like 2/5ths, or 5/8th. 3. Possibly he means "...which only allows you to construct two fractions using a numerator of 1 and a denominator consisting of a number smaller than the size of the base which has non-repeating decimal representations..." (which would be a pretty weird thing to focus on) it's still wrong; it's missing, eg, 1/4th. 4. Finally, he reckons that in base 60, 7/60 + 30/3600 aka 1/8 aka 0.125 is an "exact fraction". That sort of factorization is legitimate, but obviously if that's an exact fraction so is 2/10 + 5/100 aka 1/4th. Or come to that, 6/10 aka 0.6. In short, the claim is gibberish. It's using a phrase with no clear definition in a way which is internally inconsistent. There is no way that 1/8th is an "exact fraction" in base 60, but only 1/2 and 1/5 are "exact fractions" in base 10. (Also, you're using the word "irrational" wrong.)
jdmichal 9y ago
> Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that. Actually, I can argue with that. Irrationality is a property of the number, not the representation. An irrational number cannot be written as a terminating "decimal" in any base. And, of course, a rational number can be written as an integer fraction in any base. https://math.stackexchange.com/a/625481 https://math.stackexchange.com/a/625481 EDIT: Ignoring, of course, doing silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory. Also, I wonder if every number except pi and its multiples are non-terminating in base-pi...
- sillysaurus3 9y ago
  silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory Hmmm... Anybody know any resources related to that? I'm interested in the properties of base-pi. I found this: https://math.stackexchange.com/a/2188852 https://math.stackexchange.com/a/2188852 Strange bases are pretty cool. Any integer can be converted into base-sqrt(2) by interleaving zeoes into its binary representation. E.g. 5 in base-sqrt(2) is 010001, 21 (0b10101) is 0b0100010001, etc. Possibly useless, but neat. Also: https://www.quora.com/Can-we-have-a-number-system-with-the-base-pi https://www.quora.com/Can-we-have-a-number-system-with-the-b... Knuth considers transcendental bases (ee is a transcendental number) in The Art of Computer Programming – in the section called Positional Number Systems. One of the consequences of using irrational bases is unfortunately that numbers need not have a unique representation. EDIT: Wow, this is awesome: https://www.quora.com/What-is-10-in-base-pi https://www.quora.com/What-is-10-in-base-pi Specifically the answer from Dave Williamson. It's ridiculously comprehensive. He shows how to calculate it using pure math, using a calculator, by writing a program, and demonstrates a partial proof on some of the properties.
  - Someone 9y ago
    Neither mentions base π, but you may be interested in https://en.wikipedia.org/wiki/Non-integer_representation https://en.wikipedia.org/wiki/Non-integer_representation, which even claims applications: "There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998)." and https://en.wikipedia.org/wiki/Quater-imaginary_base https://en.wikipedia.org/wiki/Quater-imaginary_base. (Surprisingly, there seems to be less content on this on mathworld)

ordu 9y ago

> "rational trigonometry" is more accurate, and that's a concrete claim with is 1) easily verifiable and 2) wrong. I do not see how it is easily verifiable. Rational trigonometry, as I understand, does not use angles, so it is not clear for me how to use it in a lot of cases. I belive there are ways to do it, but it needs some research or reading Wildberger... So, its not seems for me as "easily". > the bizarre claim that "we count in base 10, which only has two exact fractions: 1/2, which is 0.5, and 1/5." There's so much wrong with that, it's hard to know where to start. To my mind, it is just imprecise wording from Wildberger side. Or maybe it precise in his rational way of thinking, but we, accepting irrational numbers, see it imprecise. I belive, that he meant to say, that 1/2 and 1/5 is some kind of "base" fractions and all fractions that can be written presice is `(1/2)^m*(1/5)^n`, where m and n are non-negative intergers. And to write precise is means to write in decimal representation. I cannot see any other explanation of Wildberger words which would not imply that Wildberger just plain stupid, which is improbable. Really, I like the idea: we know how to solve a lot problems with irrational numbers, but does it possible to solve them without irrational? How far mathematics can go without irrational numbers?