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To clarify, that's not a thing you can meaningfully do. It makes sense to define your units such that, e.g. c=1, because c has units (distance per time). You're really adjusting your units, not your numbers. But π is dimensionless. Diameter and circumference are both measures of distance, and they are never equal.

If you say "this number times one equals that number", and the numbers aren't the same, then when you say "one" you don't actually mean "one" the way anyone else speaks about it. (By convention: whenever you can do multiplication, if there's something called "one", then "one" is the multiplicative identity.)

There's nothing to stop you from, e.g. renaming all numbers x -> x/π. But you wouldn't gain anything; you'd just say things like "1/π squared is 1/π", and "1 squared is π", and "π times 1/π is π". It looks weird, but what's happening is perfictly ordinary, just with weird names.

One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system. And yes, mathematicians have studied transcendental bases for numeral systems (Knuth in particular, see the short paragraph on http://mathworld.wolfram.com/Base.html which explicitly mentions pi as basis).

Of course it's not a good idea to base your day-to-day numeral system on pi, but I know that at least for irrational bases there are actual applications.

This is a good point as well, and more in line with what the original commenter was asking.

Using pi as a base is even briefly discussed on wikipedia. I had some interest in using e as the basis for a while, since (as is pointed out on wiki) "The base e is the most economical choice of radix β > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values."

http://en.wikipedia.org/wiki/Non-integer_representation#Exam...

"One could sort of satisfy naturalethic's wish by choosing a numeral system with pi as basis. Then pi would have digit representation '1' in this system."

I'm pretty sure pi comes out to 10 base pi, not 1 base pi.

Numeral systems are not restricted to the set (base N).

"a numeral system with pi as basis" could be (base pi)pi, where pi =-> '1' and pipi -> '10' and unity -> '0.1'.

I'd refer you back to philh's answer. You can do that and it isn't "wrong", but it is surprisingly uninteresting, as compared to a discussion of irrational or negative or imaginary bases.

Ah, right, of course! Still, I think this is the closest possible to the wish the OP had.

Depends on what you mean by setting pi = 1.

You can subtract (pi - 1) from every number, giving you a translation of the real numbers.

You can divide everything by pi, giving you a slight compression that ultimately doesn't mean anything.

You could use it to create a field extension of the rationals, giving you a http://en.wikipedia.org/wiki/Algebraic_number_field where {1, pi} is the basis.

But to answer your question: probably nothing.

Careful, none of these actually "set pi=1".

Also, since pi is transcendental, the field would not be an algebraic number field, but in fact isomorphic to the rational functions over Q, i.e., quotients of polynomials with coefficients in Q.

Yep, you are right. I can't believe I forgot that.

Sounds a little irrational...

Relativistic mathematics.

I think my head just exploded.