Pi -- finite in different bases?

[Baby Astronaut]

I had wondered if looking for Pi in a different base other than "10" would produce finite or repeating decimals.

 

So using the formula circumference/diameter I started with the normal base 10, as a control, putting 23.12/7.36 into the WolframAlpha engine, of course getting 3.14..... (infinitely non-repeating as usual).

 

And then I changed it from base 10, entering "23.12/7.36 in base 5". I also tried it with all other bases from 1-20, also 100, and a few in between. Oddly, every single result had the numbers repeating after only less than fifteen digits.

 

Cap'n Refsmmat advised me just input "pi in base (anything)" to see what happens. And of course doing that returned it back to the familiar infinite non-repeating progression again.

 

However, there's one reasonable possibility I'd like to eliminate before accepting the results. When you enter "Pi in base ___" into the Wolfram engine, does it make the conversion to "base ___" from the already known non-repeating Pi number (located within its database), or does it attempt to calculate Pi from scratch with each different base number you enter?

 

If they calculate from a non-repeating Pi already written in their database, of course it would give non-repeating numbers in any base you calculate for. So might anyone shed expert light on the matter?

[insane_alien]

pi in base pi is 1

[Sisyphus]

I don't know how Wolfram Alpha does it, but pi will definitely be non-repeating in any base. Pi is an irrational number, which means it cannot be expressed as a ratio between two integers. There is nothing peculiar to base 10 about that.

[D H]

pi in base pi is 1

Beat me to it!

[Bignose]

pi in base pi is 1

 

[math]\pi[/math] in base [math]\pi[/math] is 10, not 1.

 

[math] 1\cdot\pi^1 + 0\cdot\pi^0[/math]

[insane_alien]

ahem... whoops.

 

you are of course correct.

[Baby Astronaut]

I don't know how Wolfram Alpha does it, but pi will definitely be non-repeating in any base. Pi is an irrational number, which means it cannot be expressed as a ratio between two integers. There is nothing peculiar to base 10 about that.

Does that mean any irrational number doesn't change type regardless of what base you use?

 

Is there any example where the quality of a number (or its smoothness of use in an equation) changes when switched to a different base?

[the tree]

Does that mean any irrational number doesn't change type regardless of what base you use?

Yes. A number is irrational if it cannot be expressed as a ratio of two integers - nothing to do with the base used.

 

It is a consequence of being irrational that it cannot be expressed as a series of integer powers of any natural base.

 

Is there any example where the quality of a number (or its smoothness of use in an equation) changes when switched to a different base?

There are a handful of well studied properties that are base dependent, but none with much application.

[Zombiemouse]

Sorry for ressurecting a 3 year old forum, but this topic needs to be discussed more.

I would like to know more about base dependant properties and how they differ from decimal.

Especially if this would lead to any semblence of better understanding Pi.

[HalfWit]

Pi is finite in every base. It's between 3 and 4 no matter how you represent it.

 

It can't possibly have a terminating or repeating expansion in any integer or rational base (however you define rational bases, it's tricky) because pi is an irrational number.

[Boson_heavy]

I would like to point out, in regards to pi in base pi being ten, that in fact how would you define pi as a base?

For one, calculated in different bases, the number differs. So which "Pi" would you take as the base? If the number infinitely does not repeat, how could you ever hope to use Pi as a base for establishing Pi as ten?

Perhaps my logic is flawed, would be very pleased to have my error corrected.

[Amaton]

I would like to point out, in regards to pi in base pi being ten, that in fact how would you define pi as a base?

 

For one, calculated in different bases, the number differs.

 

What do you mean by "the number differs"? The way we represent a number in writing changes between different bases, but the actual value of the number does not.

 

You may be asking something else, though, in which case, please correct me.

[Endercreeper01]

Pi in base n*pi is 1/n. In fact, x in base nx is 1/n