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Thread: irrational powers

  1. #1 irrational powers 
    Moderator Moderator AlexP's Avatar
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    How do you raise a number to a power that is an irrational number? Such as 2<sup>pi</sup>. I understand how to raise something to a power that's a fraction, but you can't really make pi into a fraction, so I don't think it can be done that way.


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  3. #2  
    Forum Professor serpicojr's Avatar
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    You use "continuity" and the fact that the rationals are "dense" in the reals. Instead of defining each of these words, let me put it this way: if you fix a number a and want to raise it to the power x, you find a sequence of rational numbers x<sub>n</sub> that converges to x. Then the sequence a<sup>x<sub>n</sub></sup> converges, and we define a<sup>x</sup> to be the limit of this sequence. You can also think of this as "filling in the holes" of the graph you get when you evaluate y = a<sup>x</sup> for rational values of x.

    I can go on if you wish.


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  4. #3  
    Moderator Moderator AlexP's Avatar
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    ok, I think I have it. If you don't mind, could you give a simple example so I can make sure I have it? Only if you feel like it, I don't want to be a pain.
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  5. #4  
    Forum Professor serpicojr's Avatar
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    So if I wanted to calculate 2<sup>π</sup>, I find a sequence of rationals converging to pie by cutting off its decimal expansion. Then I evaluate 2 to each of these powers. I take the limit, and I get:

    2<sup>3</sup> = 8
    2<sup>3.1</sup> = 8.5741877...
    2<sup>3.14</sup> = 8.81524093...
    2<sup>3.141</sup> = 8.8213533...
    2<sup>3.1415</sup> = 8.82441108...
    2<sup>3.14159</sup> = 8.8249616...
    2<sup>3.141592</sup> = 8.82497383...
    2<sup>3.1415926</sup> = 8.8249775...
    2<sup>3.14159265</sup> = 8.82497781...
    2<sup>3.141592654</sup> = 8.82497783...
    ...
    2<sup>π</sup> = 8.82497783...

    This is valid for a mathematical explanation of what 2<sup>π</sup> is. But we don't actually calculate exponents like this. We (probably) calculate (or, rather, approximate) them using the power series for e<sup>x</sup> and then using that 2<sup>x</sup> = e<sup>x ln 2</sup> (and using power series for ln).
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  6. #5  
    Moderator Moderator AlexP's Avatar
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    So the idea of cutting off the decimal expansion is to make the power you're raising to rational, so you can calculate, for example, <sup>50</sup>rt(2<sup>157</sup>) for 2<sup>3.14</sup>? And then what's the setup for the limit exactly? I won't even attempt the way you said it's really done, I think I have too much lack of background knowledge to tackle that right now.
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  7. #6  
    Forum Professor serpicojr's Avatar
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    Yeah, cutting off a decimal expansion gives a rational approximation. There are "better" approximations, but these are the easiest.

    The limit is more of a mathematical formality. It gives meaning to the expression a<sup>b</sup>, and it tells you how you can approximate this expression, but you can't actually sit down and calculate it. You can approximate a<sup>b</sup> by a<sup>c</sup> if c is rational and really close to b. But then you need some way of calculating a<sup>c</sup>. As I suggested before, this is not as easy as it looks.
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  8. #7  
    Moderator Moderator AlexP's Avatar
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    Got it. Thanks.
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  9. #8  
    Forum Freshman Faldo_Elrith's Avatar
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    If a>0, . That's what I take to be the definition of for any real number x.
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