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How can a transcendental number be a base?

  1. Jul 10, 2014 #1
    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?
  2. jcsd
  3. Jul 10, 2014 #2


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    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]".
    Last edited by a moderator: Jul 10, 2014
  4. Jul 10, 2014 #3


    Staff: Mentor

    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.
  5. Jul 10, 2014 #4


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    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.

  6. Jul 11, 2014 #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:
  7. Jul 11, 2014 #6
    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.
  8. Jul 11, 2014 #7


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    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.
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