I do not know what your author wants to do; but exponents are definied ONLY for positive real numbers on this level.
Thus, is only defined if is positive. But can only be defined if is positive.
I have questions about one proof.
It concerns numbers with rational exponents.
Having in mind that for all is then author of book wants to show that must be positive real number.
Proof is:
If is rational number then is also a rational number, so:
Similarly, if is rational number then is rational number, so:
which gives us that . Then is
If then is .
Based on above, we get that which means that must be positive number because is positive number so is positive number.
My question is why must be positive number?
If is rational number then is also a rational number, so:
which means that can be also negative number.
Based on that, then can mean that can be also negative number.
In more advanced math we define for all complex numbers (imaginary numbers). For now that is not necessary. Thus, only positive real numbers (non-imaginary) are defined. For example still has meaning but does not. Thus, we only consider postive real numbers. Thus, I do not know what your author is trying to do because as I understand it, it has only meaning for positives numbers thus there is nothing to prove.Originally Posted by DenMac21
Maybe I am wrong and missing something what your author is trying to say.
Neither do I!Originally Posted by ThePerfectHacker
Yes I know that there is no rational solution of but there is rational solution of which author didn't mention. He only took in consideration but not . He said that must be positive real number, but it can be also negative real number (under right condition of course).