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Imagine if we used a base $\pi$ number system, what would it look like? Wouldn't it make certain problems more intuitive (eg: area and volume calculations simpler in some way)?

This may seem like a stupid question but I do not remember this concept ever being explored in my Engineering degree. Surely there is some application to the real world here.

I am interested in answers that demonstrate which problems would become more elegant to represent and compute. I am also interested in any visualizations that leverage the meaning of that scale. Never-mind a logarithmic scale, what would a $\pi$arithmic scale be and what would simple areas on it mean?

From the comments, I realise that the normal representation of numbers is flawed (or difficult to use) for this idea, so maybe it's worth modifying it slightly. eg:

let:

$[1] = 1.\pi^0 = 1$

$[2][1] = 2.\pi^1 + 1.\pi^0 = 2\pi + 1$

$[2.3][1] = 2.3 . \pi^1 + 1.\pi^0 = 2.3\pi + 1$

$[1][2][3] = 1.\pi^2 + 2.\pi^1 + 3.\pi^0 = \pi^2+2\pi+3 $

Peter
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Luke Machowski
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    How do you write the first few integers in this representation? – Simon S Jun 10 '15 at 17:46
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    Should it not be $\pi=10$ and $\pi^2=100$ ? – Peter Jun 10 '15 at 17:51
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    In this representation, no rational number $x$ can be represented exactly because some coefficient in $x=a_n\pi^n+...+a_1\pi+a_0$ must be transcendental. – Peter Jun 10 '15 at 17:53
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    Moreover, the number $\pi^2$ has no intuitive meaning. It is related though to the sum $$\sum_{j=1}^{\infty} \frac{1}{j^2}=\frac{\pi^2}{6}$$ – Peter Jun 10 '15 at 17:58
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    Some related information http://en.wikipedia.org/wiki/Non-integer_representation – pjs36 Jun 10 '15 at 18:01
  • As irrational numbers go, I think algebraic integers (e.g., $\sqrt{2}$) work much better as bases of numeration than transcendental numbers (of which $\pi$ is an example). – Mr. Brooks Jun 10 '15 at 21:01

2 Answers2

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The picture below may answer your queries.

My opinion is that it would help in some problems.Taken from Wiki page as suggested by a comment above.

mbomb007
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Agile_Eagle
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  • I think sqrt 2 cannot be a base with 0 1 digits because the representation is not unique. ($$\sum_{k=0}^\infty\frac{1}{(\sqrt{2})^k}=\frac{\sqrt{2}}{\sqrt{2}-1}\geq\sqrt{2}$$). A base shall have for every number exactly one representation ? – QuantumPotatoïd Dec 12 '23 at 12:08
  • In fact dumbly there is no base of numbers as soon as considering negative powers (e.g. .99(period)=1), hence there is no free part, there is always surrepresentation of numbers. But this is maybe nitpicking. – QuantumPotatoïd Dec 12 '23 at 18:27
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No problem will became easier to solve by choosing a different base, because they depend on the numbers, not on their representation. The solution of some numerical problems might look nicer, for some definition of nice, but that's as far as you'd go.

lhf
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    Spot on. For the OP (who is an engineer), a perfect example would be binary versus octal/hexadecimal. 1 hex digit is 4 binary digits, so it is simply a more compact way of writing the same thing. 1 octal digit is 3 binary digits, but that is becoming less and less common. – FundThmCalculus Jun 10 '15 at 18:24
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    Yes, your comment is valid. I also think that the representation can also make certain problems simpler. Take for example the conversion of signals from time domain to frequency domain. That makes many useful operations much simpler to deal with. I was interested if there was any similar simplification that might arise for certain problems in the pi space. – Luke Machowski Aug 10 '15 at 04:53
  • This answer is not correct. An example of a problem that becomes easier is in computing where the binary system is used. – Hunaphu Oct 24 '22 at 21:52