My Issue With Pi

[Skwim]

That's only because you're using base 10. If you use base pi, then pi is expressed with a single digit.

Yeah, Shaul mentioned the same thing in post 24, and as Polymath257 responded in post 31

"It isn't *rational* in that base. It merely has a terminating expression in that base. Different things."

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[siti]

Yeah, Shaul mentioned the same thing in post 24, and as Polymath257 responded in post 31

"It isn't *rational* in that base. It merely has a terminating expression in that base. Different things."

In base pi only powers of pi would be (or at least could be) simply expressible - pi ^ 0 = 1, pi ^ 1 = 10 and pi ^ 2 = 100 and so on. All the other numbers would be horridly complicated to express. I would not want to try doing arithmetic in base pi. I'm not sufficiently adept to decide whether I think this is a genuine way to express numbers in a non-natural base anyway - it strikes me as a bit contrived to use the natural numbers to express numbers in a non-natural base but I have absolutely no idea how else to do it.

 

[siti]

Um, no. Pi in base_pi would be the base unit, “1” and quite rational in that base system. (Not to be confused with the “1” of base_10) The rational numbers of base_10 would become irrational in the base_pi system.

No - pi ^ 0 would be 1 - as zeroth power of any number is always 1, pi ^ 1 (aka pi itself) could be expressed as 10 - but that seems contrived (as I mentioned in my previous post) and really pi would just be pi, pi ^ 2 could be 100 but again why not just pi ^ 2 (I don't know how to put the proper symbol for pi-squared in here) or some other way of expressing that does not depend on (effectively redefining) the natural numbers.

 

[Shaul]

Here you go,

https://www.quora.com/How-do-I-make-a-number-system-with-base-pi

Of course the author, while right that expressing “normal” base 10 numbers in a pi-nary based system is problematic, expressing base 10 irrational numbers becomes quite elegant. Pi, e(the natural log base), and others become rational if they can be related rationally to pi. And, since there are infinitely more irrational numbers (base 10 perspective) than rational ones, more numbers would rational in a pi-nary system than in a base 10 system.(!) Just not the ones we are used to and most familiar with.

 

[Shaul]

No - pi ^ 0 would be 1 - as zeroth power of any number is always 1, pi ^ 1 (aka pi itself) could be expressed as 10 - but that seems contrived (as I mentioned in my previous post) and really pi would just be pi, pi ^ 2 could be 100 but again why not just pi ^ 2 (I don't know how to put the proper symbol for pi-squared in here) or some other way of expressing that does not depend on (effectively redefining) the natural numbers.

You are correct. My bad.

 

[siti]

Here you go,

https://www.quora.com/How-do-I-make-a-number-system-with-base-pi

Of course the author, while right that expressing “normal” base 10 numbers in a pi-nary based system is problematic, expressing base 10 irrational numbers becomes quite elegant. Pi, e(the natural log base), and others become rational if they can be related rationally to pi. And, since there are infinitely more irrational numbers (base 10 perspective) than rational ones, more numbers would rational in a pi-nary system than in a base 10 system.(!) Just not the ones we are used to and most familiar with.

Interesting, but the parenthetical claim in the linked article that "1, 2 and 3 remain the same, but integers 4 and higher become irrational in base pi" seems really odd to me because clearly the "1" referred to is the same as in any base - i.e. pi divided by pi or pi to the zeroth power and that is definitely 1. But what does "2" mean? What does "3" mean. In base 4 (the nearest natural base to have the same set of natural numbers as digits, these symbols mean exactly the same as they do in the more familiar base 10 - 2 is the same as two ones - but that cannot be the case in base pi can it? And 3 is just a smidgen below pi - the difference in value between 3 and the next digit pi is (very approximately) 7 times the difference between 2 and 3, or 1 and 2, or most importantly zero and 1 - which calls into question whether either zero or 1 could be sensibly used as counting numbers at all in base pi. I can see how you could use those symbols to count in base pi - but they would surely have to mean something other than what they mean in the base of any natural number mustn't they?

 

[Polymath257]

Interesting, but the parenthetical claim in the linked article that "1, 2 and 3 remain the same, but integers 4 and higher become irrational in base pi" seems really odd to me because clearly the "1" referred to is the same as in any base - i.e. pi divided by pi or pi to the zeroth power and that is definitely 1. But what does "2" mean? What does "3" mean. In base 4 (the nearest natural base to have the same set of natural numbers as digits, these symbols mean exactly the same as they do in the more familiar base 10 - 2 is the same as two ones - but that cannot be the case in base pi can it? And 3 is just a smidgen below pi - the difference in value between 3 and the next digit pi is (very approximately) 7 times the difference between 2 and 3, or 1 and 2, or most importantly zero and 1 - which calls into question whether either zero or 1 could be sensibly used as counting numbers at all in base pi. I can see how you could use those symbols to count in base pi - but they would surely have to mean something other than what they mean in the base of any natural number mustn't they?

Nope. The symbols 0,1,2, and 3 would mean exactly the same thing as they usually do. Perhaps a brief refresher on how to write numbers in a base.

Let's start with base 4 for convenience and express the number that in decimals would be 25.3 =253/10. Start by dividing by the base (4) until you get a number less than the base: 25.3/4 = 6.325, 6.325/4=1.38125. The integer part of this number is 1, so 1 will be the leftmost digit base 4. Remember the number of times you divided because that will tell you where the 'decimal' point will be.

25.3 = 1......

Now, subtract that 1 off and look at what is left.... .38125. Multiply this by the base: .38125*4 = 1.525. The integer part of this is 1, so the next digit base 4 will also be 1.

25.3 = 11...

Now, subtract that 1 and get .525. Multiply by the base again: .525*4=2.1. Peel off the integer part:

25.3 = 112....

We place the decimal point here because we originally divided twice and we have now multiplied twice.

Now, subtract the integer part, mulitply by the base: .1 * 4 =.4. The integer part is 0:

25.3 = 112.0

.4*4 = 1.6...

25.3 = 112.01...

.6*4 = 2.4

25.3 = 112.012

.4 has already been seen, so at this point, things start to cycle:

25.3 = 112.01212121212121212.....

So, we have written 25.3 (base 10) as 112.012121212... (base 4). Notice that the base 10 description terminates, but the base 4 description cycles.

So, let's do the same thing and write 5 (base 10) in base pi.

5/pi = 1.59154931....The integer part is less than the base (pi), so we can start:

5 = 1.....

Subtract the integer part and multiply by the base: .59154931*pi=1.858407346...

5 = 11.....

The decimal point goes here. Now subtract the 1 and multiply by pi to get 2.696766...

5 = 11.2....

.696766*pi = 2.188955....

5 = 11.22......

.188955*pi = 0.5936215....

5 = 11.220....

.593621* pi = 1.864917....

5 = 11.2201....

And I'll stop there for a number of reasons. As you can see, the basic procedure is the same no matter what base is used (as long as the base is more than 1).

 

[siti]

And I'll stop there for a number of reasons.

Thanks for the tutorial, but I would like to know what the reasons are for stopping there? And (perhaps related to this question) how would you calculate, say, 2 + 3 in base pi? Also how would you convert the answer 11.2201... (base pi) back to base 10. The problem seems to be - and this is why I wouldn't be able to go any further - because I don't know how to go past 3 or -3 for the exponent without changing what happens when you increase the exponent. Is that why you stopped there?

The digits should represent successively larger or smaller powers of pi, just as the familiar base ten expressions indicate successive powers of ten

The number 25.3 from your example in base ten represents 2 x 10 ^ 1 + 5 x 10 ^ 0 + 3 x 10 ^ -1 - yes?

So the number 11.2201... in base pi represents 1 x pi ^ 1 + 1 x pi ^ 0 + 2 x pi ^ -1 + 2 x pi ^ -2 + 0 x pi ^ -3 + 1 x pi ^ -pi...

So what comes after -pi? And in any case if you calculate that back into base ten even without additional digits, it comes to more than 5 - doesn't it?

Sorry for being a difficult student - I was always a pain in the neck in maths class.

 

[Polymath257]

Thanks for the tutorial, but I would like to know what the reasons are for stopping there? And (perhaps related to this question) how would you calculate, say, 2 + 3 in base pi? Also how would you convert the answer 11.2201... (base pi) back to base 10. The problem seems to be - and this is why I wouldn't be able to go any further - because I don't know how to go past 3 or -3 for the exponent without changing what happens when you increase the exponent. Is that why you stopped there?

I mostly stopped there for accuracy reasons. Every time you multiply by pi, you lose accuracy: I was only working with a few decimal places and going farther would be a problem. Addition and multiplication would be a serious problem because 'carry digits' would not be simple. I doubt there is a simple algorithm for either addition or multiplication base pi.

The digits should represent successively larger or smaller powers of pi, just as the familiar base ten expressions indicate successive powers of ten

The number 25.3 from your example in base ten represents 2 x 10 ^ 1 + 5 x 10 ^ 0 + 3 x 10 ^ -1 - yes?

So the number 11.2201... in base pi represents 1 x pi ^ 1 + 1 x pi ^ 0 + 2 x pi ^ -1 + 2 x pi ^ -2 + 0 x pi ^ -3 + 1 x pi ^ -pi...

No, the next term would have pi^(-4) instead of pi^(-pi), then pi^(-5), pi^(-6), etc... No real difference.

So what comes after -pi? And in any case if you calculate that back into base ten even without additional digits, it comes to more than 5 - doesn't it?

Sorry for being a difficult student - I was always a pain in the neck in maths class.

Hopefully that cleared up a question or two.

 

[siti]

OK - I see where I am going wrong with the exponents but it is definitely a bit contrived (decimal exponents in a fractional or irrational base seems unnatural to me - but there doesn't seem to be any other way). I'm thinking base pi is not terribly useful after all unless you want to express pi itself as simply as possible.

How about (he suggested in the vain hope of regaining at least the tiniest smidgen of mathematical credibility) base pi-squared - in that case pi = 1. Surely can't get much simpler than that. Please, please can I get a tick for that?

PS - it (11.2201...) still adds up to more than 5 if you convert back to decimal - is that just because of rounding pi in each calculation - or is it more fundamental - the error isn't going to get smaller with more digits is it? (Very confusing)

 

[Polymath257]

OK - I see where I am going wrong with the exponents but it is definitely a bit contrived (decimal exponents in a fractional or irrational base seems unnatural to me - but there doesn't seem to be any other way). I'm thinking base pi is not terribly useful after all unless you want to express pi itself as simply as possible.

How about (he suggested in the vain hope of regaining at least the tiniest smidgen of mathematical credibility) base pi-squared - in that case pi = 1. Surely can't get much simpler than that. Please, please can I get a tick for that?

PS - it (11.2201...) still adds up to more than 5 if you convert back to decimal - is that just because of rounding pi in each calculation - or is it more fundamental - the error isn't going to get smaller with more digits is it? (Very confusing)

Nope: 1*pi +1 + 2/pi + 2/pi^2 +0/pi^3 +1/pi^4 is about 4.99110 in decimals. It seems to work!

Remember, use pi=3.1415926535....

 

[siti]

Nope: 1*pi +1 + 2/pi + 2/pi^2 +0/pi^3 +1/pi^4 is about 4.99110 in decimals. It seems to work!

Remember, use pi=3.1415926535...

Drat! He's right - of course (how dare anyone suggest otherwise)! But do I get a tick for pi =1 in base pi-squared though? Please! And I promise I'll try harder on the next lesson.