Absurd Line Numbers

Escaping from Status of Zap in Announcements…

floating point line numbers?

Including surds?

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)…

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.

where a, b and c are integers

Thank [insert deity of choice] for that.
When I first looked at that I was thinking “c” – you know as in e=mc^2
Now that would make your line numbers “interesting”

Now that would make your line numbers “interesting”

Or, indeed, Absurd.

Is this all theoretical or do you have a proof of concept?

;-)

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.

The suggestion of surds was intended to be humourous.

I must admit I rather assumed that.

It led me into musings about how one might represent surds exactly in digital computers.

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.

Is this all theoretical or do you have a proof of concept?

Yes.

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.

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)

After all trying to squeeze a line in between 2.2361 and 2.2362 and using √5 to do so would be ridiculous.

Indeed. Using 2.23615 would seem far more reasonable, albeit all these numbers are only achievable approximately using standard floating point representations.

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.

Do you get a consolation prize if you can define the square root of 2 raised to the power of 2?

The whole point of floating point numbers, though, is to use a constant amount of storage for each number.

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…

give a correct definition ( definition mind, not evaluation )

Despite first class honours in maths, I’ve no confidence in my ability to define anything.

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?

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.

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…

Real programmers use line vectors, with program flow controlled by parametric equations.

I’m not even sure what that means, Jeffrey! 8~)

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.

1 + a(x^m) + b(x^n) = 0

1 + a(x^mk) + b(x^nk) = 0

I think there’s a slip in the explanation of that proof there:

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 whole numbers, say a and b.

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)

Beg pardon – I meant rational numbers. Corrected now.

👍

In 1957 I met John Conway …

In 1964 I first met John Conway in Sidney Sussex …

That’s as far as it goes, apart from the fun with games. ;-)

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.