# Pi expressed by different bases

Discussion in 'Recreational Math' started by David W. Cantrell, May 3, 2006.

1. ### David W. CantrellGuest

Sure (unless you insist that your bases must be integers).

In a certain base, which is a bit larger than 3, pi = 10 exactly.

Cheers,
David

David W. Cantrell, May 3, 2006

2. ### TarantellaGuest

This is a question from someone with
hardly any math knowledge.

Since the ratio Pi cannot be expressed
wholly in the usual numerical system "base 10",
can it be wholly expressed by choosing a
different base?

Thanks,

Tarantella, May 3, 2006

3. ### Arturo MagidinGuest

Not with an integral basis.

If b>1 is an integer, then the expression

(d_n)(d_{n-1})...(d_1)(d_0).(e_1)(e_2)...(e_m)

where 0<= d_i < b
0<= e_i < b

in "base b" corresponds to the real number

d_n*b^n + d_{n-1}*b^{n-1} + ... + d_1*b + d_0 + e_1/b + e_2/b^2 +... +e_m/b^m.

We can write this as a single fraction by multiplying through by b^m,
so we would obtain

(d_n*b^{n+m} + d_{n-1}*b^{n+m-1} + ... + d_0*b^m + e_1*b^{m-1} + ... + e_m)
---------------------------------------------------------------------------
b^m

and in particular, if the expression terminates then it must represent
a rational number since both numerator and denominators are
integers. It is known that pi is not rational (in fact, it is not even
algebraic), so it cannot be thus expressed.

In general, for b an integer greater than 1, the only numbers that can
be expressed with a terminating "base b" expansion are those that can
be expressed as n = p/q, where p and q are integers, q is not zero,
and the prime divisors of q are all prime divisors of b.

(So, for base 10, only fractions that can be written with a
denominator which can be written as 2^a*5^b, a,b>=0, have terminating
decimal expansion).

However, for any b>0, a number expressed in base b is rational if and
only if its "base b expansion" is eventually periodic (eventually
repeats).

One can define a similar notion of "base b" for nonintegral values of
b, though it is a much more complicated affair; normally the
conditions we put on b to make this sensible also preclude pi from
having a terminating "base b expansion".

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin

Arturo Magidin, May 3, 2006
4. ### Frank J. LhotaGuest

Pi is irrational, so it cannot be wholly expressed in any integer base.

Frank J. Lhota, May 4, 2006
5. ### TarantellaGuest

Thanks for all the replies.

Tarantella, May 5, 2006
6. ### SpazGuest

But a circle has a finite circumference, a finite diameter, and a finite
area. So how can pi be irrational? Must not have been calculated right in
the first place.

Spaz, May 7, 2006
7. ### Dennis MarksGuest

You don't understand what an irrational number is. It means that you can't
express the number as a ratio of 2 integers. Any circle that has an integer
diameter will have an irrational circumference. It has nothing to do with
finite.

--
Dennis

Disclaimer: The above is my opinion. I do not guarantee it. Be sure to back
up any files involved and use at your own risk. Batteries not included. Not
for internal use. Don't run with knives.

Dennis Marks, May 7, 2006
8. ### Arturo MagidinGuest

So? Irrational numbers are still finite. They just cannot be expressed
with a repeating decimal expansion.

What does one thing have to do with the other?

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin

Arturo Magidin, May 8, 2006