If a irrational number be the basis of count

[Jhenrique]

In the comum sense, the number 10 is the base of the decimal system and is the more intuitive basis for make counts (certainly because the human being have 10 fingers). But, in the math, the number 10 is an horrible basis when compared with the constant e. You already thought if an irrational number can be the basis of system of count, it make sense?

 

[micromass]

http://en.wikipedia.org/wiki/Non-integer_representation
http://en.wikipedia.org/wiki/Radix_economy

 

[mathman]

In the comum sense, the number 10 is the base of the decimal system and is the more intuitive basis for make counts (certainly because the human being have 10 fingers). But, in the math, the number 10 is an horrible basis when compared with the constant e. You already thought if an irrational number can be the basis of system of count, it make sense?

Why is it horrible?

For arithmetic to work easily, the basis has to be an integer. It is hard to envision what a number would even look like with e or any non-integer.

Since computers became widespread, 8 or 16 might be practical alternatives to 10.

 

[jim hardy]

Have any of you folks ever converted everyday numbers to non-integer radixes?

I tried back in the late seventies, just out of curiosity but my math was not up to the challenge.
I wanted to see what would some of the physical constants, Planck, c, μ₀ , ε₀, look like in bases e pi etc.

Closest i ever came was a Basic program that converted Florida's lotto numbers into 49 bit binary numbers and printed them out as hex, decimal and octal. No visual patterns emerged.

Your links above are quite interesting.

For a non-integer radix β > 1, the value of

x=d _n... d₂d₁ d₀d_-1d_-2...d_-m...
is

x= βⁿd_n + β²d₂ + β¹d₁ + β⁰d₀ β^-1d_-1 +β^md_m

Thanks !

old jim , who is distractable to a fault.

 

[Mark44]

Why is it horrible?

For arithmetic to work easily, the basis has to be an integer. It is hard to envision what a number would even look like with e or any non-integer.

Since computers became widespread, 8 or 16 might be practical alternatives to 10.

Base-8 used to be used a lot, but not as much any more, as far as I can see. Base-2 (binary) and base-16 (hexadecimal) are heavily used in computer programming.

 

[Nugatory]

Many years ago, when one of my daughters was in grade school... Over dinner she asked me what was the most "interesting" base for a number system.

I answered $-2$, because $1+1=110$ and you can get the rest of arithmetic from there.

What I didn't know was that the question was prompted by a school homework assignment: Choose a radix and demonstrate worked addition, subtraction, multiplication, and long division problems in that radix. She pulled it off, although the long division algorithm is not deterministic - when dividing $A$ by $B$, having $nB\le{A}$ and $(n+1)B\gt{A}$ doesn't mean that subtracting $nB$ is the right next step.

 

[Matterwave]

This...this was in grade school? ...

 

[Nugatory]

This...this was in grade school? ...

Somewhere between fifth and eighth grade, don't remember exactly.

 

[jbunniii]

I remember learning about different bases when I was in grade school, maybe 4th or 5th grade. This was in the late '70s, around the tail end of the "new math" era. For some reason base 7 was often used in examples, if I recall correctly. We also learned about the commie metric system before Reagan abolished it.

 

[Jhenrique]

Many years ago, when one of my daughters was in grade school... Over dinner she asked me what was the most "interesting" base for a number system.

I answered $-2$, because $1+1=110$ and you can get the rest of arithmetic from there.

What I didn't know was that the question was prompted by a school homework assignment: Choose a radix and demonstrate worked addition, subtraction, multiplication, and long division problems in that radix. She pulled it off, although the long division algorithm is not deterministic - when dividing $A$ by $B$, having $nB\le{A}$ and $(n+1)B\gt{A}$ doesn't mean that subtracting $nB$ is the right next step.

-2 is interesting, can you give more examples?