5 ms·

This article seems to boil down to "stop liking what I don't like." Is Wildberger's distaste for real numbers eccentric? Yes. Is there anything fundamentally

This article seems to boil down to "stop liking what I don't like." Is Wildberger's distaste for real numbers eccentric? Yes. Is there anything fundamentally wrong with what he is proposing? Absolutely not. As long as rational trigonometry is logically consistent then you really can't argue against it except on aesthetic grounds. I don't see why popsci outlets like Scientific American should feel the need to rally the public against rational trigonometry on emotional grounds.

AnimalMuppet 9y ago
I didn't read it that way. They are rallying against the claim that rational trigonometry is "superior" or "more accurate", and not on emotional grounds.
- deleted 9y ago
  [deleted]
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.
- 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?
agjacobson 9y ago
It's not rational trigonometry. The angles are irrational. In fact, it's useless, and was probably a decorative demonstration of Pythagorean triples.

AnimalMuppet 9y ago

I didn't read it that way. They are rallying against the claim that rational trigonometry is "superior" or "more accurate", and not on emotional grounds.

deleted 9y ago
[deleted]

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.
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?

agjacobson 9y ago

It's not rational trigonometry. The angles are irrational. In fact, it's useless, and was probably a decorative demonstration of Pythagorean triples.