
#1
Oct2106, 06:03 AM

P: 361

i know what the meaning of [tex]a^p[/tex] is when p is an integer or rational. e.g., [tex]a^3 = a.a.a[/tex] or [tex]a^{\frac{1}{5}}[/tex] is such a number that when multiplied five times gives the number a.
but what is the menaing of [tex]a^p[/tex] when p is an irrational number? 



#2
Oct2106, 06:54 AM

P: 76

same thing just a irrational amount of times instant of a integer/rational times.
like a^3,14 (simple part of pi) its the same as a^3*a^0,14=a^3*a^(7/50)=a^3*(a^(1/50))^7 pi and such can be explained as a infinite amount of sums like this and by that a infintie amount of parts like this 



#3
Oct2106, 08:24 AM

P: 4

The modern way is: 1. Define the Natural logarithm using an integral, viz, [tex]ln(x)[/tex]. This is continuous and continuously differentiable for [tex]x>0[/tex] 2. Its inverse will be [tex]e^x[/tex] for all real [tex]x[/tex]. 3. This means that [tex]a^p = e^(pln(a))[/tex]. This is then taken to be the definition of [tex]a^p[/tex]. 4. This is or was Alevel in England till the year 2000 but you don't need to know that, these days, as things are dummed down. 



#4
Oct2106, 10:33 AM

P: 1,520

meaning of irrational exponent
Well, it depends. For example, the value 2^pi does not mean anything, meaning that all it represent is another irrational number, untill you give pi a rational approximation. However, something like 10^log 2 has a meaning as it is another expression of 2. Here it is important to notice that ln 2 is a limit, as in the more digits you assign to log 2, the closer the expression 10^log 2 is to 2.




#5
Oct2106, 11:34 AM

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You certainly can define, as eds said, 2^pi as e^(pi ln 2) but since pi ln 2 is an irrational number itself, you still haven't answered the original question: 



#6
Oct2206, 11:24 AM

P: 361

so, here r_{i} is a closer and closer approximation to the irrational number p as i becomes larger and larger. am i right? and can we say that [itex]r_{i} \rightarrow p[/itex] as [itex]i \rightarrow \infty[/itex]?




#7
Oct2206, 12:04 PM

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P: 9,398

Werg, what do you mean by 'meaning'. pi, is just as meaningful as 1/2, mathematically, and if you want to discuss the philosophy of it there is another forum entirely devoted to that. Of course, mathematicians bandy about these tongue in cheek statements, but I wouldn't condone doing so here where the opportunity for misapprehension is so large.




#8
Oct2306, 10:35 AM

P: 361





#9
Oct2306, 11:50 AM

P: 1,520





#10
Oct2306, 11:56 AM

P: 361





#11
Oct2306, 11:59 AM

P: 1,520

Sorry I forgot to add the "not". It should be provided that we are not dealing...




#12
Oct2306, 12:09 PM

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#13
Oct2306, 12:14 PM

P: 1,520





#14
Jan807, 08:27 PM

P: 17





#15
Jan807, 08:34 PM

HW Helper
P: 3,353

I prefer the integral definitions. A good way to approximate them by hand if you ever lose your calculator :P Its also interesting when your a is not equal to 0 or 1, you get a nice transcendental number. Look up GelfondSchnieder Theorem




#16
Jan807, 08:38 PM

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