Irrational Base Number System (2000)

Better articles (IMO) at Wikipedia:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Non-integer_base_of_numeration" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Non-integer_base_of_numeration</a><p>Example: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Golden_ratio_base" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Golden_ratio_base</a><p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Non-standard_positional_numeral_systems" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Non-standard_positional_numera...</a> has a long list.<p>An interesting one: base 2i <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quater-imaginary_base" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Quater-imaginary_base</a> -- this was introduced in one of the papers Donald Knuth wrote in college.

The most efficient number system is base e, ie ratio of number of digits that exist to placeholders needed to represent certain numbers.<p>The most efficient integer is actually base 3, which also explains why ternary logic puzzles are so elegant.

The bigger issue seems to be what a digit is. In a &quot;traditional&quot; base system the digits represent a linear spread between base^n and base^{n+1}.

I came up with a slightly different take on irrational radices is to give a finite closed form representation of integers in an irrational base, code and write up here:<p><a href="https:&#x2F;&#x2F;github.com&#x2F;crisdosyago&#x2F;irradix" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;crisdosyago&#x2F;irradix</a>

I wrote something like this also in 2000:<p>&quot;Way Off Base&quot; <a href="https:&#x2F;&#x2F;dwheeler.com&#x2F;essays&#x2F;bases.html" rel="nofollow">https:&#x2F;&#x2F;dwheeler.com&#x2F;essays&#x2F;bases.html</a>

So for instance, in a base pi, pi =1 or pi=10? Does 1 always equal 1? That can’t be right…<p>(Aside, Leibniz invented binary based on inspiration from I Ching. Cool!)

Having multiple representations of the same number isn’t the problem one of the commenters thought. In decimal, for example, 1=0.99999999...

here&#x27;s my first question (I have enough math background to cogitate or research this and find out the answer, but it&#x27;ll just be another rabbithole, so let me just ask):<p>if anything to the zero power is 1, what does that mean for pi? pi to the 0 is 1? or pi to the 0 is &quot;base ten 1 represented in base pi&quot;?<p>asking because that page starts out stating that 0 is always 0 and 1 is always 1.

A bad idea that still merits discussion about the ways in which it is bad...

is the problem essentially not just that if you make irrational numbers the bases, then you make [a significant number of] rational numbers irrational?