Let's first divert to the question ''what are rational numbers?''
Simplistically, we say they are numbers that are the ratio of two whole numbers m/n where n is not zero. But actually if we are going to define them properly, we have to regard them as a pair of whole numbers (m,n) where n is not 0. But then we have to define how to add and multiply two of them.
Multiplying (m,n) and (p,q) is easy - just giving (no,mq), but adding them is a bit tricky.
and we also need to consider when two of them are equal.
Now back to the main question - what are complex numbers? This time it is a pair of real numbers (a,b) where a and b can be any real numbers you like, including zero.
Adding them is easy because
We ''think'' of (a,b) as
So when we multiply
So the rule is
Now that all seems very tricky, but if you work through a few examples such as
Then finally you can go on and check that all the usual rules that apply to real numbers also apply to complex numbers, such as
It turns out that the real numbers and the complex numbers behave very similarly and we call both of them fields which have both addition and multiplication following similar rules.
I assume you mean to ask what the complex logarithm of 1-i is.
Let’s look at the general case.
Given a complex number in cartesian or polar form
We want to know the complex logarithm, which is multi-valued:
In your case:
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