Register to reply

How can a transcendental number be a base?

by p1l0t
Tags: base, number, transcendental
Share this thread:
p1l0t
#1
Jul10-14, 10:38 AM
P: 58
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?
Phys.Org News Partner Mathematics news on Phys.org
Math journal puts Rauzy fractcal image on the cover
Heat distributions help researchers to understand curved space
Professor quantifies how 'one thing leads to another'
HallsofIvy
#2
Jul10-14, 02:10 PM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
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 [tex]1\times 10^3+ 2\times 10^2+ 3\times 10+ 2\times 10^0+ 3\times 10^{-1}[/tex]. And "binary" is "base 2" because 1232.3 (base 2) means [tex]1\times 2^3+ 2\times 2^2+ 3\times 2+ 2\times 2^0+ 3\times 2^{-1}[/tex] 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, [itex]\pi^2[/itex]= 9.8696044010893586188344909998762...

And I think you may be misinterpreting "speculation". Of course, because [itex]\pi[/itex] 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 [itex]\pi^2[/itex] or [itex]\pi[/itex] 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 [itex]\pi^2[/itex]. But whether or not I can write it in a specific way, I know that [itex]\pi^2[/itex] is a specific number.

Similarly, although given a number a, I cannot actually calculate [itex]a_0[/itex], [itex]a_1[/itex], [itex]a_2[/itex], ... so that [itex]a= a_0\pi^0+ a_1\pi^1+ a_2\pi^2+ \cdot\cdot\cdot[/itex] but I know that such number exist so that I can, in fact, write any number in "base [itex]\pi[/itex]".
DaleSpam
#3
Jul10-14, 05:23 PM
Mentor
P: 17,537
Quote Quote by p1l0t View Post
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
#4
Jul10-14, 07:34 PM
P: 998
How can a transcendental number be a base?

Quote Quote by p1l0t View Post
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
#5
Jul11-14, 03:59 AM
P: 235
Quote Quote by HallsofIvy View Post
And "binary" is "base 2" because 1232.3 (base 2) means [tex]1\times 2^3+ 2\times 2^2+ 3\times 2+ 2\times 2^0+ 3\times 2^{-1}[/tex] 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!
p1l0t
#6
Jul11-14, 08:14 AM
P: 58
Quote Quote by skiller View Post
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!
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
#7
Jul11-14, 08:52 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,682
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.


Register to reply

Related Discussions
Is Pi a rational number in any other base besides base Pi? General Math 19
New irrational number to develop transcendental operators Linear & Abstract Algebra 2
Base changing for transcendental numbers Linear & Abstract Algebra 6
Is this number algebraic or transcendental? Linear & Abstract Algebra 13
Can the probability of an event ever be a transcendental number? Set Theory, Logic, Probability, Statistics 11