Integers in Base Pi

Question

Background:

Pi (π) is a transcendental number, and therefore it has a non-terminating decimal representation. Similar, the representation doesn't terminate if written in any other integer base. But what if we wrote it in base π?

Digits in decimal represent powers of 10, so:

π = 3.14… = (3 * 10^0) + (1 * 10^-1) + (4 * 10^-2) + …

So in base π, the digits would represent powers of π:

π = 10 = (1 * π^1) + (0 * π^0)

In this new base, integers now have non-terminating representations. So 10 in decimal now becomes the following:

10 => 100.01022… = (1 * π^2) + (0 * π^1) + (0 * π^0) + (0 * π^-1) + (1 * π^-2) + …

Note that in base π the digits used are 0,1,2,3 because these are the digits less than π.

Challenge:

Given a non-negative integer x, either:

1. Output (without halting) its representation in base π. If the number has a finite representation (0, 1, 2, 3), then the program may halt instead of printing infinite zeros.

2. Take an arbitrarily large integer n, and output the first n digits of x in base π.

Rules:

• Since a number has multiple possible representations, you must output the one that appears the largest (normalized). Just as 1.0 = 0.9999… in decimal, this problem exists in this base, too. In base π, one is still 1.0, but could also be written as 0.3011…, for example. Similarly, ten is 100.01022…, but could also be written as 30.121… or 23.202….
• This is code-golf, so fewest bytes wins. Program or function.
• No built-ins (I'm looking at you, Mathematica)

Results:

0       = 0
1       = 1
2       = 2
3       = 3
4       = 10.220122021121110301000010110010010230011111021101…
5       = 11.220122021121110301000010110010010230011111021101…
6       = 12.220122021121110301000010110010010230011111021101…
7       = 20.202112002100000030020121222100030110023011000212…
8       = 21.202112002100000030020121222100030110023011000212…
9       = 22.202112002100000030020121222100030110023011000212…
10      = 100.01022122221121122001111210201201022120211001112…
42      = 1101.0102020121020101001210220211111200202102010100…
1337    = 1102021.0222210102022212121030030010230102200221212…
9999    = 100120030.02001010222211020202010210021200221221010…

First 10,000 digits of ten in base Pi

Verification:

You can verify any output you want using the Mathematica code here. The first parameter is x, the third is n. If it times out, pick a small n and run it. Then click "Open in Code" to open a new Mathematica worksheet with the program. There's no time limit there.

Convert the resulting output to a number here.

Related:

• Wikipedia: Non-integer representation
• Quora: What is 10 in base pi?
• Math.SE: What would a base π number system look like?
• XKCD forum: Counting in a base pi number system?

Answer 1

Python 3, 471 317 310 bytes

7 bytes thanks to caird coinheringaahing.

Surely there are golfs that I missed. Feel free to point them out in the comments.

def h(Q):
	a=0;C=b=4;c=d=s=1;P=o=3
	while P^C:
		a,b,c,d,s,o,P,A,B=b,s*a+o*b,d,s*c+o*d,s+o,o+2,C,0,1
		for I in Q:B*=c;A=A*a+I*B
		C=A>0
	return P
def f(n,p):
	Q=[-n];R=""
	while h([1]+Q)<1:Q=[0]+Q
	Q+=[0]*p
	for I in range(len(Q)):
		i=3;Q[I]+=3
		while h(Q):Q[I]-=1;i-=1
		R+=str(i)
	return R[:-p]+"."+R[-p:]

Try it online!

Ungolfed version:

def poly_eval_pi_pos(poly,a=0,b=4,c=1,d=1,o=3,s=1,prev=9,curr=6):
	while prev != curr:
		a,b,c,d,s,o=b,s*a+o*b,d,s*c+o*d,s+o,o+2
		prev = curr
		res_n, res_d = 0,1
		for I in poly:
			res_d *= c
			res_n = res_n*a + I * res_d
		curr = res_n > 0
	return prev
def to_base_pi(n,precision):
	poly = [-n]
	res = ""
	while not poly_eval_pi_pos([1]+poly):
		poly = [0]+poly
	poly += [0]*precision
	for index in range(len(poly)):
		i = 3
		poly[index] += 3
		while poly_eval_pi_pos(poly):
			poly[index] -= 1
			i -= 1
		res += str(i)
	return res[:-precision]+"."+res[-precision:]

Try it online!

Answer 2

Ruby -rbigdecimal/math, 111 103 97 bytes

->x,n{q=BigMath::PI n;r=q**m=Math.log(x,q).to_i;n.times{$><<"#{?.if-2==m-=1}%i"%d=x/r;x%=r;r/=q}}

Try it online!

Takes input number as x and desired precision as n. Outputs by printing. Makes use of the built-in BigDecimal library for arbitrary pecision PI value.

Answer 3

Julia 0.6, 81 bytes

f(x,i=log(π,x)÷1)=(y=big(π)^i;d::Int=x÷y;print(i==0?"$d.":"$d");f(x-d*y,i-1))

Prints out digits (and the . which cost me 14 bytes) until it Stack overflows at about 22k digits on TIO. If I'm allowed to pass the input as a BigFloat I can cut 5 bytes. Makes use of the built in arbitrary precision constant π. But its a bit cooler than that, its actually an adaptive precision constant, π*1.0 is a 64 bit floating point number, π*big(1.0) (aka multiplied by an higher precision number) gives π at whatever your precision is set to.

Try it online!

Answer 4

Python 3 + sympy, 144 bytes

from sympy import*
def f(n,p):
	R="";O=n and int(log(n,pi));r=pi**O
	for _ in range(O+p):R+=str(int(n/r));n%=r;r/=pi
	return R[:O+1]+"."+R[O+1:]

Try it online!

Quite slow, actually.