Pi: Pythagoras be Tripping

Well, Queen Brangomar, you could at least claim I'm full of crap.

reply to post by LilDudeissocool


U mad?

I did star you. I actually did have some radunculous reply at the tips of my fingers, but thought the better of it. Oh well, so much for impulse control:

We may as well all convert over to base six now, b/c it'll be mandatory under the reign of our saurian overlords (three dactyls). They'll keep us around for our thumbs.



Just remember. You asked for it, dude.

Originally posted by Eidolon23
reply to post by LilDudeissocool

 

U mad?

You mean insane?

Originally posted by Eidolon23
I did star you. I actually did have some radunculous reply at the tips of my fingers, but thought the better of it. Oh well, so much for impulse control:

Just don't brown dwarf me

I was redefining the argument. It's a cat chasing it's own tail routine. The latest hottest thing in science these days.

Originally posted by Eidolon23 We may as well all convert over to base six now, b/c it'll be mandatory under the reign of our saurian overlords (three dactyls). They'll keep us around for our thumbs.

Just remember. You asked for it, dude.

I learn more and more obscure things reading your posts. en.wikipedia.org... My kids were of the age during that time, and were never into them. We were a Rocko's Modern Life and Pinky and the Brain family.

reply to post by LilDudeissocool

In a 14 digit number system would 7 be considered rational?

Yes, indeed it would, and its value would still be seven.

7 would be simply moved to the 5 value in place in the 10 digit number system within a 14 digit number system, but how about a 21 digit number system?

I'm afraid you misunderstand. A base 14 arithmetic (to give it its proper name) does not have fourteen integers between 1 and 10. It has fourteen integers between 1 and 14. It's just that you count up to 14 before 'carrying one' in your calculations.

Thus, you could write the first 13 digits in a base-14 system

1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D

The number we call 14 would be written in base 14 arithmetic as 10. 28 (base 10) = 20 (base 14), 42 (base 10) would be 30 (base 14), and so on.

The number we call 11 would be written B; the number we call 19 would be written 15; the number we call 25 would be written 1B, and the number we call 100 would be written 52.

Incidentally, fans of Douglas Adams will be cheered to discover that, in base 13 arithmetic, 9 x 6 actually does equal 42. Try it and see; if it checks out, you've understood how to do multiple-base arithmetic!

You will also be ready to understand how Noah and Methuselah came to have such impossibly long lifespans attributed to them, but that's another story.

reply to post by LilDudeissocool

The Babylonians had a 12 digit number system, "Where does pi appear on your watch?" Hint convert 12 O clock to 10 O clock.

A base 12 arithmetic would write 12 as 10 (see my post above), but π would still appear at a little after 3 o'clock.

The ratio between a circumference of a circle to its diameter can be used to construct a number system where the ratio constant could be deemed rational. Simply square root the ratio where a rational number appears by assigning values accordingly to find a starting point for you invented number system, and work out from there in constructing your number system to complete the entire frame work.

Won't work. π does not produce a rational ratio with any integer. Any attempt to produce one would result in an approximate value, not the exact ratio of circumference to radius of a circle. Your number system would come with built-in inaccuracy.

You could, conceivably, construct an arithmetic in which π was the unit. Everything would be measured as multiples and fractions of π. You could certainly perform all mathematical operations based on such an arithmetic. Unfortunately, you would never be able to measure anything exactly using it – except, of course, the ratios of radii, circumferences, etc. of ideal circles and spheres.

Originally posted by Eidolon23
Here's what I wonder: does basing your entire system of mathematics on a figmentary number some Greek genius pulled out of thin air while loopy on (maybe) amanita strike you as a little odd? Especially when there are ways to calculate real-world geometries without resorting to Pi?

Well, the problem I see here is the confusion of pure mathematics with mathematical physics. Pure mathematics consists in deducing conclusions from axioms and aren't applied like the mathematics we use in mathematical physics and engineering.

To put it even simpler: if you want to construct a large building, you refer to mathematical physics. If you want to transcend the laws of physics to make sense of philosophical questions, like the use of imaginary numbers to explain certain portions of quantum physics, you use pure mathematics (or as Niels Bohr put it, the reference to atoms doesn't mean atoms are things but rather provide the conceptual framework to make mental connections). Pythagoras was interested in pure mathematics, and thus is reflected in the transcending nature of things like Pi.

So, the relevancy of Pi is according to your praxeology.

reply to post by Astyanax


> Astyanax, the point is that you could create a number system where the ratio between circumstance and diameter of a circle would be a rational number.

> I can't express 7 as a fraction I can 14.

> See these prime numbers 2 3 5 7 11 13 17... These are composite numbers 4, 6, 8, 9, 10, 12, 14.

This is truly elementary dear, Watson. You really don't want to over analyze and think what I am pointing out here.

The argument over 3.14 being irrational is moot as 3.14 only pertains to a ten digit number system.

Btw what I am doing in the abstract is simply redefining the argument. The pun being, it's pointless to argue in circles. "What came first the chicken or the egg?" Answer T. rex. www.usatoday.com...

I'm trying to be cute in pointing out arguing whether pi (3.14 unique to our 10 digit number system) is an irrational number or not is a pointless exercise.

reply to post by LilDudeissocool

The point is that you could create a number system where the ratio between circumstance and diameter of a circle would be a rational number.

No, you cannot. You can use π itself as a unit (and count π, 2π, 3π, etc.) but notice we're still using integers to multiply π. And, as I pointed out earlier, you'd never get accurate measurements with that system, or accurate calculations based on measurement, either.

I can't express 7 as a fraction I can 14.

Neither 7 nor 14 are fractions. They are integers. Both can be divided by any rational number you please to produce a rational fraction of themselves.

See these prime numbers 2 3 5 7 11 13 17... These are composite numbers 4, 6, 8, 9, 10, 12, 14.

Sorry, I don't see the relevance of this statement.

The argument over 3.14 being irrational is moot as 3.14 only pertains to a ten digit number system.

No. This is your basic error. You think the numerical value of π is irrational. It is not – any numerical value of π is an approximation, but it could be a rational approximation. The ratio of the circumference of a circle to its radius is a number that cannot be expressed numerically, no matter what numbering system you use. That is what is meant by an irrational number. π would be irrational no matter what base you do your arithmetic in.

Arguing whether pi (3.14 unique to our 10 digit number system) is an irrational number or not is a pointless exercise.

π isn't 3.14, it's π – an inexpressible number of which 3.14 is a poor approximation.

However, π would be still approximate to 3.14 in any arithmetical system whose natural base was larger than its actual value, so your statement is incorrect.

Originally posted by Astyanax
reply to post by LilDudeissocool

 

The point is that you could create a number system where the ratio between circumstance and diameter of a circle would be a rational number.

No, you cannot. You can use π itself as a unit (and count π, 2π, 3π, etc.) but notice we're still using integers to multiply π. And, as I pointed out earlier, you'd never get accurate measurements with that system, or accurate calculations based on measurement, either.

Let me respond by segments in individual posts. I''ll respond to a few now, and the rest later during my next work break. I think this is more to do with misunderstanding each other than anything else, so I want to have time to ponder my later responses before posting them.

First off, sorry "circumstance" should've been in quotes, it's a pun spin on the term "circumference." I poorly communicated it, my bad. The pi value as being expressed within a 10 digit number system being the circumstance. The ratio is the constant, but not how it is expressed being my point.

Again what I am merely pointing out is that the constant ratio between the circumference and its diameter of a circle expressed as a value of 3.14159265358979324 to be exact is merely an assigned value of pi within a 10 digit number system. 3.14159265358979324 over a whole. Big deal. It's only assigned that value within the ten digit number system because of our chosen number system. Please think out side the box here, or the ten digit number system rather? That ratio can be expressed as a whole number in an entire number system constructed around the ratio, a constant. Because it's a constant makes what I am illuminating possible. (π ^ π2) 2 = 6506465676 a whole number. Divide that by π and you get another whole number 2071072349 or approximated at the value of 2072122827 using 3.14. If you take 2071072349 and divide it by 6506465676 you get .318309886 divide that by 100, a whole, and you get π of course, coming back full circle, no pun intended, after adjusting the decimal place swinging it back to the left two places. Because that's all we are doing here finding a number system that fits π into a whole number, or assigning it a whole number value, and then you can construct an entire number system around it.

Originally posted by Astyanax
reply to post by LilDudeissocool

 

I can't express 7 as a fraction I can 14.

Neither 7 nor 14 are fractions. They are integers. Both can be divided by any rational number you please to produce a rational fraction of themselves.

There is no argument that 7 and 14 are whole numbers. I'm just suggesting that one can create a number system where the π ratio becomes a whole number. Thus the argument of 3.14 not being a whole number is a matter of "circumstance" contained within a 10 digit number system rendering this debate moot. That's my point. I am simply redefining the "circumstances" surrounding this debate.

I'll explain further in my next post.

Originally posted by Astyanax
reply to post by LilDudeissocool

 

See these prime numbers 2 3 5 7 11 13 17... These are composite numbers 4, 6, 8, 9, 10, 12, 14.

Sorry, I don't see the relevance of this statement.

22/7 22 is a composite and 7 is a prime. Invert and you get .318, an approximation of .318309886 which is the precise number we are looking for here explained above. All we are doing is searching for a whole number that can convert π expressed in a 10 digit number system into a whole number within our new number system on which all mathematics thereof will be based upon.

I'll get back to you later on the rest when I have some more time to spare. I need to ponder where and how we are missing each other on all this.

Hate to be the bearer of bad news, but the paper you linked from the University of Utah's Math department just says that the traditional value of pi is one half of the true constant. In other words, pi is the constant of a half circle and tau (2*pi) is the constant for a whole circle.

If you really want to say pi is flawed I'd recommend going the route of drawing a circle on a positively/negatively curved surface. Then pi is no longer a constant and is related to the radius of curvature of the surface. But it's still not 'wrong'

A number being transcendental is a statement about the amount of information required to express the number, which has nothing to do with the particular way a number is represented. The trick of expressing pi in 'base pi' is just the tautology that
pi = ... + 0*pi^-2 + 0*pi^-1 + 0*pi^0 + pi + 0 * pi^2 + ...
which depends upon the definition of pi.

And regarding pi vs. tau, it's a stupid debate. You can define constants up to any multiplicative factor you want, they're just chosen for convenience. Pi is chosen like it is to make trig and geometry formulas easier to calculate with. There's no fundamental meaning to it. What's fundamental are the forms of the geometric and algebraic relationships.

reply to post by LilDudeissocool

The following illustrates the error into which you have fallen.

the constant ratio between the circumference and its diameter of a circle... is merely an assigned value of pi within a 10 digit number system. It's only assigned that value within the ten digit number system because of our chosen number system.

I don't think you have quite grasped the concept of arithmetical bases. Changing the base in which you do your calculations can alter the numerals in which numbers are represented, but it can't change the numbers themselves.

Thus the argument of 3.14 not being a whole number is a matter of "circumstance" contained within a 10 digit number system rendering this debate moot.

No, that's not correct. The value of π will always be the same, no matter what numerals you use to express it. This is your fundamental error; you seem to believe that changing the notation changes the number. No chance of that. The fundamental constants retain their values in all transformations.

all we are doing here finding a number system that fits π into a whole number, or assigning it a whole number value, and then you can construct an entire number system around it.

I have already dealt with this. It is impossible, for reasons I have made clear in earlier posts.

reply to post by Moduli


What "clear?" THIS crap> "a circle to its radius is a number that cannot be expressed numerically" WTF is THAT

I give up.

reply to post by LilDudeissocool


PS Some 4th grade math for ya to learn cstl.syr.edu...

Yeah it can be done. Cross multiply and divide the numerical value of 22/7... uh an approximation. 7th grade math!

Originally posted by Moduli
A number being transcendental is a statement about the amount of information required to express the number, which has nothing to do with the particular way a number is represented. The trick of expressing pi in 'base pi' is just the tautology that
pi = ... + 0*pi^-2 + 0*pi^-1 + 0*pi^0 + pi + 0 * pi^2 + ...
which depends upon the definition of pi.

And regarding pi vs. tau, it's a stupid debate. You can define constants up to any multiplicative factor you want, they're just chosen for convenience. Pi is chosen like it is to make trig and geometry formulas easier to calculate with. There's no fundamental meaning to it. What's fundamental are the forms of the geometric and algebraic relationships.

You make a lot more sense than Astyanax.

reply to post by LilDudeissocool

What "clear?" THIS crap> "a circle to its radius is a number that cannot be expressed numerically" WTF is THAT

I believe I wrote 'the ratio of the circumference of a circle to its radius is a number that cannot be expressed numerically'. Meaning, it has a fixed value that cannot be expressed in numerals.

Thanks for the fourth-grade maths lesson. Unfortunately, matters like these cannot be worked out using elementary arithmetic. Bye for now.

Originally posted by Astyanax

What "clear?" THIS crap> "a circle to its radius is a number that cannot be expressed numerically" WTF is THAT

I believe I wrote 'the ratio of the circumference of a circle to its radius is a number that cannot be expressed numerically'. Meaning, it has a fixed value that cannot be expressed in numerals.

That's outrageous.

Originally posted by AstyanaxThanks for the fourth-grade maths lesson. Unfortunately, matters like these cannot be worked out using elementary arithmetic. Come back when you've graduated from high school and we'll talk again. Bye now!

This is about freaking pi not rocket science.

It does not require anything complicated.