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Math 8 lecture notes

• We can extend the definition of logarithm to nonzero complex numbers z if we accept
that it has to be a multivalued function. We called this multivalued function

• Next we defined ab for any a, b ∈C, with a ≠ 0:

This is a set, meaning it could potentially have more than one value!

• This seems a little crazy at first, but it’s not so bad. For example we saw that this
definition of ab as a set agrees with our intuition when b is a rational number: if b = p/q
in lowest terms, then ab will consist of q different values (e.g., 361/2 = {−6, +6}).

• But this definition also has interesting consequences when b is not a rational number: if
b is irrational, then ab will have infinitely many values! For instance, if = 1.4142...
is the positive square root of 2, then

and these numbers are all different.

• I left it as an exercise to show that

which is a little unexpected: it says that all values of are positive real numbers!