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Let's start here:

310^0 + 110^-1 + 410^-2 =
31 + 1*.1 + 4*.01 =
3.14

That's uhh... not pi. The only way to do pi that way is to extend it infinitely.

Also, what you're using is called scientific notation, but it's still in decimal format, i.e. base10

[Edit: just noticed you did say that was decimal notation; my bad).

Any baseX numeral system has X number of integers per digit.

• Base10: {0,1,2,3,4,5,6,7,8,9}
• Base2: {0,1}
• Base3: {0,1,2}
• Base16: {0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f}
• Base60: {[series of 60 sumerian numerals]}

A baseπ numeral system would look like this: {0,1,2,[int(π-3)]}.

But that's not how set theory works. Since integers are by definition whole numbers and their inverse counterparts, it's impossible to have .141592654... of an integer. If you have {0,1,2,3}, that's base4; if you have {0,1,2,n}, that's still base4.

To put it another way, in any baseX system, (if it includes 0), X is the first two-digit number. That means π in baseπ would be written as "10".

• In base2, two is written as "10"
• In base3, three is written as "10"
• In base10, ten is written as "10"
• In base16, sixteen is written as "10"

That means, if you wanted to make a baseπ numeral system, in order to have a consistent interval between integers (without which, integers become meaningless), each numeral would have to represent (π/3).

So in baseπ:

• "0" = base10(0)
• "1" ≈ base10(1.047197551)
• "2" ≈ base10(2.094395102)
• "10" ≈ base10(3.141592654)

[Edit: aaand I just noticed you did say baseπ(10) = base10(π); my bad again. I guess you weren't as wrong as I thought you were. Not bad for being too high for this...]

But that's still technically base3, it's just a wonky base3. And it would have no practical value. Also, the same thing can already be achieved in base10 using radians.

• (0π) rad = 0°
• (π/3) rad = 60°
• (2π/3) rad = 120°
• π rad = 180°

I guess if you really wanted to express radians as whole numbers, you could use baseπ, i.e.:

• baseπ(0) rad = 0°
• baseπ(1) rad = 60°
• baseπ(2) rad = 120°
• baseπ(10) rad = 180°

But again, that's still technically base3, and all it does is confuse people. Plus, if you want to express an angle as a whole number you can choose degrees or mills. The whole point of radians is to express it with reference to pi (as in, the arc corresponding to the length of the radius along the circumference)