Absurd Line Numbers
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Clive Semmens (2335) 3276 posts |
Escaping from Status of Zap in Announcements…
I don’t know of a floating point representation that explicitly deals with surds, only with their approximations to whatever precision the particular representation achieves – at which point they’re not really different in any significant way from any other floating-point number. Using these beasts for line numbers seems eccentric shall we say. Interesting idea though – a system that does represent surds precisely, storing not only mantissa and exponent*, but a fractional power (as a ratio of two integers? Floating point?) it’s raised to… mathematically nice, not so sure it’s practically useful… * Or perhaps not mantissa and exponent, but a ratio of two integers (again)… |
Clive Semmens (2335) 3276 posts |
The most pleasing version, from my (loony) pov, would be: (a/b)^(1/c) where a, b and c are integers – preferably at least 64 bits (although b and c don’t need to have quite so many, perhaps…) It could be: (a/b)^(d/c) and then I might be happy to allow fewer bits in a, which I originally wanted to be have enough bits that d was unnecessary. (Mathematical “elegance” preferred over practical utility, you understand.) Using these beasts for line numbers would be lunatic – keeping the little buggers in order would be a nightmare. At least normal FP numbers are easy to sort. |
Steve Pampling (1551) 8155 posts |
Thank [insert deity of choice] for that. |
Clive Semmens (2335) 3276 posts |
Or, indeed, Absurd. |
Rick Murray (539) 13806 posts |
Is this all theoretical or do you have a proof of concept? ;-) |
Chris Hall (132) 3554 posts |
not so sure it’s practically useful… The suggestion of surds was intended to be humourous. The suggestion was to use real numbers (rather than rational real numbers, which excludes surds) and, I suspect, may have been just one step better than using imaginary numbers as StrongEd does. After all trying to squeeze a line in between 2.2361 and 2.2362 and using √5 to do so would be ridiculous. |
Clive Semmens (2335) 3276 posts |
I must admit I rather assumed that. It led me into musings about how one might represent surds exactly in digital computers. |
Chris Hall (132) 3554 posts |
What about Pi or e? They can’t be expressed as integer fractions raised to a power although e raised to the i PI is -1. |
Clive Semmens (2335) 3276 posts |
Yes. |
Clive Semmens (2335) 3276 posts |
Indeed, or any other non-surd irrational numbers. I did only consider surds. Increasing generalization becomes increasingly difficult! As for e^(i pi) – this is related to my username on some other sites: coshipi (which is also -1, of course) |
Clive Semmens (2335) 3276 posts |
Indeed. Using 2.23615 would seem far more reasonable, albeit all these numbers are only achievable approximately using standard floating point representations. |
GavinWraith (26) 1563 posts |
I have probably mentioned this before, but Abbas Edalat developed a representation for real numbers using lazy streams of rational approximations. This allows the standard rules of arithmetic e.g. associativity of addition, to hold precisely. BBC Basic’s approach to transcendental functions is to use continued fractions of linear functions as approximants; an approach which could be developed in the same direction. The whole point of floating point numbers, though, is to use a constant amount of storage for each number. That cannot work if a number is understood actually to be a stream of data that never ends. A lot of computer science is built around faking infinity . Take stacks, for example. Stack overflow has to be confronted by the implementor, who hopes that the everyday user will not have to be bothered by it in reasonable circumstances. We are very finite beings, who may well be inhabiting a finite cosmos, but infinity, whatever its ontological status, is so convenient that we cannot do without it, even in computers, not just in our fantasy. I may have mentioned Yoshindo Suzuki, a colleague of mine, who offered a cask of beer as a prize for the first student to give a correct definition ( definition mind, not evaluation ) of two to the power of the square root of two. |
Chris Hall (132) 3554 posts |
Do you get a consolation prize if you can define the square root of 2 raised to the power of 2? |
Clive Semmens (2335) 3276 posts |
My representation of surds also could use a constant amount of storage for each number; but of course as with the standard ways of storing floating point numbers, it’s limited in the range of surds it can store. This is obviously inevitable if you use a fixed amount of storage…
Despite first class honours in maths, I’ve no confidence in my ability to define anything. |
Rick Murray (539) 13806 posts |
Uh, so okay. I’ve just looked at https://en.wikipedia.org/wiki/Surd_(mathematics). The only part I sort of understood was root extraction (in bold, for some reason) but even I know that maths only applies to teeth when trying to work out if the tooth fairy is honest. Could somebody explain a surd in a way that uses simple language? |
Clive Semmens (2335) 3276 posts |
A surd is a real number that can be expressed as a rational number raised to a rational power, but not as a rational number. (A rational number is a number that is the ratio between two integers, otherwise known as a fraction.) So the square root of two is a surd, as is the ninth root of seventeen and a third, or the 3/4th root of 2. That last one is also the 4th root of 8, hence (generalizing that) my (a/b)^(1/c) version, since all surds can be expressed in that form. Note that a, b and c are non-zero integers – and c is not allowed to be one or minus one, or the number would be rational. |
Clive Semmens (2335) 3276 posts |
If c is negative, that just turns a/b upside-down. If a and b are both negative, there’s no problem. If a is negative but b is not, or vice-versa, the number will be an imaginary number if c is even…and this way of storing surds doesn’t work for imaginary roots like these, because it can’t distinguish between the c solutions. Which are far nastier than just the +/- of square roots, lying as they do on the vertices of a c-sided polygon centred on zero in the complex plane… |
Jeffrey Lee (213) 6048 posts |
Real programmers use line vectors, with program flow controlled by parametric equations. |
Clive Semmens (2335) 3276 posts |
I’m not even sure what that means, Jeffrey! 8~) |
GavinWraith (26) 1563 posts |
In 1957 I met John Conway who showed me this result If a proper triangle (i.e. that is not squashed flat along a line) has sides in rational proportion to each other also has its angles in rational proportion to each other, then it is equilateral, i.e. its sides are equal in length.The proof goes like this: we can scale the triangle so that one if its sides has length 1 (in whatever units you want to use) and the other two are rational numbers, say a and b . Since the interior angles add up to 180 degrees, the angles have to be multiples of (360/N) degrees for some whole number N. Call the complex number of unit length with argument (360/N) x . If we put one vertex of the triangle at 0, and another at 1, in the complex plane, then following round the triangle we get an equation where a, b, m and n are whole numbers. This is an equation in the so called cyclotomic field of N-th roots of unity, which is the smallest collection of complex numbers containing x and all the rational numbers, and has been much studied. Its only automorphisms (functions that preserve addition, subtraction and multiplication and 1) are got by replacing x by x^k for some k less than, and having no common factors with, N. So we get another triangle from the equation having the same lengths of sides. But ruler and compass construction shows that only two triangles can be constructed on the same base (the side of length 1) with the same lengths of sides. Hence k can only take two values. That means N must be 6. So the angles must be (360/6) = 60 degrees.
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Clive Semmens (2335) 3276 posts |
I think there’s a slip in the explanation of that proof there:
How would you scale a 3,4,5 triangle so one side is length 1 and the other two are whole numbers? Its side lengths are all in rational relationship… (its angles aren’t, of course, but that’s what you’re trying to prove) |
GavinWraith (26) 1563 posts |
Beg pardon – I meant rational numbers. Corrected now. |
Clive Semmens (2335) 3276 posts |
👍 |
Steve Drain (222) 1620 posts |
In 1964 I first met John Conway in Sidney Sussex … That’s as far as it goes, apart from the fun with games. ;-) |
GavinWraith (26) 1563 posts |
John Conway was manning a stall for the Archimedeans at the fair in the Guildhall for the 1957 intake. One of the things he was demonstrating was a sort of computer built with string and meccano. You poured a cup of marbles into a funnel, … . I signed up to join the Archimedeans and the Trinity Mathematical Society. At that time he was a student of Davenport, and was supposed to be writing a thesis on number theory. He used to carry around a string of poppet beads to demonstrate some clever tricks with knots. |
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