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Hellbent

Thinking about different base number systems

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Would a base 12 or 16 number system produce the sAme irrational niumbers as our base 10? or would, say, pi maybe not be irrational? Are there any good pop math books that explore hypothetical implications of different base number systems?

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Hellbent said:

Would a base 12 or 16 number system produce the sAme irrational niumbers as our base 10? or would, say, pi maybe not be irrational?

Yes.

An irrational number is defined as a number that cannot be expressed as a fraction. For example, 1/3 is a rational number: even though it is infinitely long when written in decimal base (0.3333333333333333333333333333333333333... ad nauseam), it can be expressed as a simple fraction, as I actually just did. See? If you use a base 6, for example, then 1/3 becomes 0.2. You can see that in the base 60 we use for hours and minutes: one third of an hour is twenty minutes. (60 is a good number for a base if you want simple maths, because it can be divided easily by 2, 3, 4 and 5. 10 can only be divided by 2 and 5. 16, dear to the heart of any self-respecting computer nerd, can only be divided neatly by powers of two.)

With irrational number, though, you can't design such niceties. No matter what base you are, they'll remain annoying.

Basically, since rational numbers can be expressed as fractions, that means that for any rational number r, you can find two integers a and b so that r = a/b. As you can see, the base matters not. Let's say your rational number is 27/35 (in decimal base). In hex, that's 1B/23. In trinary, that's 1000/1022. See? The base doesn't change anything. It's still the same number represented by the same two integers, the writing change but the values remain the same.

So, if you can express a number as the fraction of two integers in a base, you can express it as the fraction of two integers in any base, because you can express any integer in any base.

Therefore, if you can't express a number as the fraction of two integers in a base, you can't express it this way in any base. Being irrational makes you irrational regardless of representation.

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Hellbent said:

Would a base 12 or 16 number system produce the sAme irrational niumbers as our base 10? or would, say, pi maybe not be irrational? Are there any good pop math books that explore hypothetical implications of different base number systems?

It's not irrational vs rational that's affected by base change; it's periodic vs nonperiodic fraction that's affected. For example, with a base-9 system you could easily express numbers like 0.444444... exactly as 0.4. Unfortunately computers by using base-2 are quite disadvantaged when dealing with decimals. Basically any fractional number that isn't a sum of negative powers of 2 will result in a periodic binary fraction (which the computer has to truncate!) If you tell the computer to remember numbers like 0.1, 0.3, 0.4 (but not 0.5, 0.25, 0.375, 0.0625), it will decay them!

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thanks guys. That's really fascinating to me. I thought for sure irrational was just a biproduct of the base and that you could devise a base system that would allow pi to EDIT not go on forever. But if I understand you all right, pi will go on forever regardless the base you express it with!? I didn't follow your post Printz :-/... but periodic vs non-periodic intrigues me, though I have no idea what they are.

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I'm afraid Hellbent will end up as a wannabe numerologist/kabbalist nutcase.

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Hellbent said:

I didn't follow your post Printz :-/... but periodic vs non-periodic intrigues me, though I have no idea what they are.

What you supposed was that weird numbers like "pi" or "e" become regular-looking if you change the base. They don't. There are other kinds of numbers that can become simple: it's those occurences such as 1/3 = 0.333333... or 1/7 = 0.142857142857... that start to look tidy on base-n systems. Conversely, nice decimal numbers like 0.1 are badly represented in other bases. See below:

For example

Decimal       Ternary (base 3)      Binary (base 2)
0              0                      0
1              1                      1
2              2                     10
3             10                     11
4             11                    100
5             12                    101
6             20                    110
1/3=0.333...   0.1  (yeah!)           0.010101010101... (w/e)
2/3=0.666...   0.2  (yeah!)           0.101010101010... (w/e)
1/9=0.111...   0.01 (yeah!)           0.000111000111... (w/e)
1/2=0.5        0.11111111...(no!)     0.1 (ok)
1/4=0.25       0.02020202...(no!)     0.01 (ok)
1/10=0.1       0.00220022...(no!)     0.0001100110011... (suck!)
In some systems certain numbers are periodic while in others they're not. That (suck!) part does have a point. If you tell the computer to memorize decimal 0.1, it will try to remember that infinite sequence of 0.0001100110011... But since it can't store infinity, it will stop after a few digits, so instead of memorizing 0.1, it will remember something like 0.0999334325 or whatever.

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Hellbent said:

thanks guys. That's really fascinating to me. I thought for sure irrational was just a biproduct of the base and that you could devise a base system that would allow pi to go on forever. But if I understand you all right, pi will go on forever regardless the base you express it with!?

Well, you can use an irrational base to represent an irrational number using a finite number of digits. This way you could represent π as 10 in base π. OK, now try to represent an arbitrary integer. :P

You can read more about it here.

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What if you use base pi? :P



(Yeah, I know that's not how it works, but I'm sure some nut out there is crazy enough to try and figure it out anyway.)

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Xtroose said:

Well, you can use an irrational base to represent an irrational number using a finite number of digits. This way you could represent π as 10 in base π. OK, now try to represent an arbitrary integer. :P

And how do you define the digits? The idea behind a base of n is that it has integer digits defined between 0 and n-1, and they're factors for powers of n. We get the number n here -- it's PI. But which are the available digits?

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Xaser said:

What if you use base pi? :P



(Yeah, I know that's not how it works, but I'm sure some nut out there is crazy enough to try and figure it out anyway.)


If you follow Xtroose's link just above your post, you'll see that they actually have.

What I like is using the square root of two as the base. It's super easy easy to convert a number representation from binary to squarerootoftwoery, which is quite remarkable.

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I wish I had smart. :(

But you guys answered my question that numbers like e and pi never cease being irrational regardless of the base used, which I think is pretty damn interesting.

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exp(x) said:

Changing base only affects the representation of a number, not any of its properties.

I didn't realize numbers like pi and e actually had the properties of being inherently irrational--I thought irrational was merely an effect of the number system; but now I understand there is a real, concrete, (universally/always true) relationship between what pi is and whole numbers. As long as your base is not an irrational number, pi will always be non-terminating, non-repeating. I never really had an appreciation for that. I used to think people were reading into it too much--attributing it too much meaning, but it really is rather phenomalogical². I mean, I thought numbers were just called irrational because they never terminate and never repeat, but never really thought about the implications of that-it's like right up there, on par with bending space/time. So it seems to me there is more to a number being irrational than its non-repeating-non-terminating properties. Maybe someone can describe what more there is to an irrational number than non-repeating-not-terminating? (or am I reading into this too much now?)

btw, thinking about irrational numbers, particularly pi since it's easy to conceptualize where the value comes from, makes the head hurt in a 'concept slipping away' kind of way. I mean, you can't ever really grasp the irrationality of pi.. it's like.. on the tip of your conceptualization but you'll never be able to conceptualize it. A total mind fuck and tease. (Maes just might end up being right). Thinking about this is making me afraid of math and numbers.

I imagine anytime you divide the value of a curve by a straight line whose value is a whole number you'll get an irrational number. It's funny, so easily are we able to produce the value of pi, and yet.. it is impossible to actually know the numerical value of pi. To the 5th trillionth digit is probably accurate enough for most applications, but it's still not as precise as zero or 1. :p

EDIT: loop this method to calculate pi to ever increasing precision?

pi/(3.14159265358979 + .00000000000000103084 + .00000000000000070270811235 + .00000000000000047902917308953614 + .0000000000000003265494515246904 + .00000000000000022260553277648688 + .00000000000000015174799097452123 + .00000000000000010344510523880247 + .000000000000000070517505563969)

each number added is the quotient of pi/3.14159265358979 - 1, then that number divided into pi, the quotient of that minus 1 then added to the last quotient repeat indefinitely for more accurate values of pi? I didn't explain that well, but it's a simple process.

each number added is the quotient of pi divided by the sum of the numerical number for pi given by a calculator and the last remainder. I don't know how to explain it clearly. :-/ I used http://web2.0calc.com/ and just copied the formula into notepad and just added the new quotient values into notepad.

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Hellbent said:

To the 5th trillionth digit is probably accurate enough for most applications...


"Most applications" actually make do with a great deal less digits than that. In fact, search the math library of your favourite programming language or math environment, and you'll see that you're limited by the precision afforded by IEEE754 floating points. In layman terms, that means maybe 15-16 digits, and that's actually good enough for "most applications" including mechanical and electrical engineering, signal processing etc. aka, what matters most today.

Those computations to discover the n-th digit of pi are more of an exercise in futility or benchmarks for the latest generation of supercomputers, as no sane production system would ever use arbitrary precision arithmetic in order to be able to use even 100 digits of pi for common engineering tasks.

The only place where you might need more precise pi are super-zoomed fractals and other similar -extraordinary- applications, which however suffer a great performance penalty to be able to use arbitrary-precision arithmetics: no modern mass-produced CPU or FPU is able to accelerate that like they can with IEER754 floating points.

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Hellbent said:

So it seems to me there is more to a number being irrational than its non-repeating-non-terminating properties. Maybe someone can describe what more there is to an irrational number than non-repeating-not-terminating? (or am I reading into this too much now?)


Non-repeating, non-terminating is not the definition of an irrational number. It's just a side-effect of its nature.

The nature of an irrational number is that it cannot be written as a fraction.



Take any number you want. For example, 59944329.326467. A fine number it is. Well, I can write it as a fraction, the laziest of which is 59944329326467/1000000. There you go: two integers let me define my number as a fraction, therefore my number is rational. That's all folks.

An irrational number cannot be written as a fraction. All a fraction does is approximate it. So pi, delicious pi, can be approximated as 10/3, but it's not 10/3. It can be approximated as 22/7, but it's not 22/7. You can increase the precision of the approximation as much as you want, but it's never going to be actually pi -- just a rational number that's quite close, but no cigar.

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printz said:

And how do you define the digits? The idea behind a base of n is that it has integer digits defined between 0 and n-1, and they're factors for powers of n. We get the number n here -- it's PI. But which are the available digits?

The digits are all non-negative integers smaller than the base. In base π, these are 0, 1, 2 and 3. This also demonstrates that you can make such representations only for non-integer bases greater than 1.

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Gez said:
The nature of an irrational number is that it cannot be written as a fraction of integers.

Fixed, since (for example) 1/√2 is a fraction but also irrational.


Besides, the Bible says π = 3:

1 Kings, 7:23 says:
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.

:-D

(I haven't done the math, but it later points out the giant bowl is "an hand breadth thick", so if you account for that it apparently ends up being closer to 3.14...)

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That was heavily implied, as evidenced by the rest of my post and by my previous post as well. I was trying to avoid using the word "integer" so much that it loses all meaning.

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Maes: I wouldn't describe methods to calculate pi as digit-by-digit, as that wouldn't be quite possible. They probably use an infinite series to calculate PI to the desired percision. And yes, calculating it past 30 digits is usually useless anyway.

Hellbent: The problem with that is that you're trying to find PI by using PI.

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Wagi said:

Maes: I wouldn't describe methods to calculate pi as digit-by-digit, as that wouldn't be quite possible. They probably use an infinite series to calculate PI to the desired precision. And yes, calculating it past 30 digits is usually useless anyway


"Digit by digit" is a a bit of a misnomer, but not too far off from what happens with all successive approximation methods, whether they're an infinite series, monte-carlo simulations etc.: in practice you keep applying a finite step as many times as it's necessary to stop observing differences at a certain digit (e.g. if I'm trying to compute the sine of 45 degrees (pi/4) with a McLauryn infinite series with at most 10 digits precision, I can safely stop adding terms when the difference between two successive iterations becomes smaller than 1^-10.

Sure, there isn't a SINGLE function or operation that will compute at arbitrary precision at constant time, but similarly, when a team is set off to discover the n-th digit of pi or e or phi or whatever, they first think of a mathematical method, translate this to numerics, and sloooooowly work their way towards computing it with n-digits precision (which is usually marked by seeing less than 10^-n change in computations). Of course, this may take a ridiculous amount of computing power, especially beyond the precision afforded by IEEE floats.

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I prefer to use Base 2 for liquid measurement since the units are all powers of 2. 8 ounces in a cup, 16 ounces in a pint, 32 in a quart, 64 in a half-gallon/pottle, 128 in a gallon.

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Heh this talk reminded me of a pseudoscientific paper named "The tyranny of the circle". It was written by one of those VMSK delusionals, and was very bitter about how standard signal analysis and trigonometry consider the sine wave to be the base of every signal (and e.g. square waves to be composed of infinite sine waves).

They also proposed an alternative system where the square wave was the base, and the sine was made of infinite square waves. Only that in nature, sine waves really do behave as the fundamentals of signals, and anything "square" will have an infinite -or very large-
sine" bandwidth, therefore any alternative approach would be pretty useless practically.

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