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.
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.
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?
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.
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
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...
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
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-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
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.
(Surprisingly, there seems to be less content on this on mathworld)
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.)
Let's start with an apparent nitpick. You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample. For modern tables, the samples tend to be regularly spaced, while the Babylonians apparently chose to sample the rational points. Which is better likely depends on the use case.
It's not obvious that "normal trig" is better from an engineering point of view. At least not for all engineers. If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. That is, it's fairly easy to make rational ratios between the opposite. It's then handy to know what mathematical properties the resulting angle has. Or else to be able to look at your table and pick ratios that have some useful property. You could not do this with a "normal" trig table, or even a modern a hand calculator.
That's true, and my wording was poor. But during the heyday of trig tables, they were very thorough, giving you results for angles (typically) very, very close to whatever you were trying to calculate, and to usefully high number of decimal places. (I'm old enough - just - that I actually used them in school.)
But more to the point, we don't use trig tables any more, we calculate the number on the fly to any arbitrary desired degree of accuracy. That's always going to be the best choice; there's no tradeoff here.
> If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many.
Also quite true, but of course, no temple wall is exactly N cubits high; the measured height is at best an approximation, so the resulting ratio of height to width (or whatever) is at best an approximation. All we need from our trig tables (or calculator) is enough accuracy. And in practice, that's not hard.
The reality is that, as the linked article says, "it seems like a solution to a problem that doesn’t exist". When we actually used trig tables to build bridges, we didn't have issues with trig tables, and now we have a much better solution. Trig tables that, as you put it, sample the rational points just doesn't seem like it's better for any use case.
If you happen to need an angle measure for whatever reason (e.g. interacting with legacy systems) you can then compute it to however many binary digits are appropriate for your needs.
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.
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.
That evaluates exactly to 25. 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.
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.
(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.)
* Hell, wait, what am I going to do with 24 6/7 gallons of milk?