How can a transcendental number be a base?

[p1l0t]

I was recently told that base Pi can only be speculation because it irrational. However the Euler formula uses e. e is the base of the natural log and yet it is a transcendental. So is it or is it not possible for an irrational and/or transcendental number to be used as a base?

 

[HallsofIvy]

I think you are confusing two different uses of the word "base". We say that our usual number system is "base 10" because "1232.3" means $$1\times 10^3+ 2\times 10^2+ 3\times 10+ 2\times 10^0+ 3\times 10^{-1}$$. And "binary" is "base 2" because 1232.3 (base 2) means $$1\times 2^3+ 2\times 2^2+ 3\times 2+ 2\times 2^0+ 3\times 2^{-1}$$ which, in base 10, would be 8+ 8+ 6+ 2+ 1/2= 24.5.

But a number being the "base" of an exponential is very different. we can take any (positive) number as a base (I put 'positive' in parentheses because while, for many values of x, a negative number to the x power is perfectly well defined, there are some values of x such that a negative number or 0 to the x power is not defined). For example, for x= 2, $\pi^2$= 9.8696044010893586188344909998762...

And I think you may be misinterpreting "speculation". Of course, because $\pi$ is an irrational number, it cannot be written as a finite number of decimal places and cannot be written as a fraction with integer numerator and denominator so I cannot write it or $\pi^2$ or $\pi$ to any other power as a finite number or decimal places. I don't know what comes after that "09998762" that is indicated by the "...". I could theoretically use a calculator that holds a greater number of decimal places or use a computer program to extend to as many decimal places as I want but I would never get the entire value of $\pi^2$. But whether or not I can write it in a specific way, I know that $\pi^2$ is a specific number.

Similarly, although given a number a, I cannot actually calculate $a_0$, $a_1$, $a_2$, ... so that $a= a_0\pi^0+ a_1\pi^1+ a_2\pi^2+ \cdot\cdot\cdot$ but I know that such number exist so that I can, in fact, write any number in "base $\pi$".

 

[Dale]

I was recently told that base Pi can only be speculation because it irrational. However the Euler formula uses e. e is the base of the natural log and yet it is a transcendental. So is it or is it not possible for an irrational and/or transcendental number to be used as a base?

As Halls said, you should be aware that the same English word often refers to multiple distinct concepts. "Base" is used as a description of different number representations (e.g. binary numbers are base 2, hexadecimal numbers are base 16). "Base" is also used to denote the number which is raised to a power in exponentiation.

The previous discussion (and the first sentence quoted here) referred to the first meaning. In "base N", the N must be a natural number. The Euler formula and so forth use e as the base referring to the second meaning.

 

[jbriggs444]

I was recently told that base Pi can only be speculation because it irrational. However the Euler formula uses e. e is the base of the natural log and yet it is a transcendental. So is it or is it not possible for an irrational and/or transcendental number to be used as a base?

In a standard positional notation system, the base (or "radix") must be a positive integer greater than 1 and all of the digits must be non-negative integers less than the base. However, non-standard notations exist.

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

 

[skiller]

And "binary" is "base 2" because 1232.3 (base 2) means $$1\times 2^3+ 2\times 2^2+ 3\times 2+ 2\times 2^0+ 3\times 2^{-1}$$ which, in base 10, would be 8+ 8+ 6+ 2+ 1/2= 24.5.

Well, that's the first time I've seen a 2 and a 3 in binary!

In any case, your arithmetic is wrong. Please read your posts before posting! :tongue:

 

[p1l0t]

Well, that's the first time I've seen a 2 and a 3 in binary!

In any case, your arithmetic is wrong. Please read your posts before posting! :tongue:

Actually you are right binary would be all 1s and 0s but I knew what he meant. I actually do know the differences between the types of bases too but I did incorrectly assume the wrong type of base. I even thanked Halls for his answer but maybe it does need an edit.

 

[HallsofIvy]

I never was any good at arithmetic! Thanks, skiller, for that correction. It is now too late to edit so I can't pretend I didn't make that foolish mistake.