# What are complex numbers? Ln(1-i)

What are complex numbers? $\mathrm{ln}\left(1-i\right)$
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star233

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.
$\left(m,n\right)+\left(p,q\right)=\left(mq+np,nq\right)$ or h pi w we would normally write it $\frac{m}{n}+\frac{p}{q}=\frac{mq+np}{nq}$
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 $\left(a,b\right)+\left(c,d\right)$ is simply $\left(a+c,b+d\right)$ but multiplying two of them is rather special.
We ''think'' of (a,b) as $a+ib$ where i is the imaginary square root of minus 1.
So when we multiply $\left(a+ib\right)\cdot \left(c+id\right)$ we collect the real parts and imaginary parts together giving us $ac+ib\cdot id$ which is $ac-bd!$
and $aid+ibc=i\cdot \left(ad+bc\right)$
So the rule is $\left(a,b\right)\cdot \left(c,d\right)=\left(ac-bd,ad+bc\right)$
Now that all seems very tricky, but if you work through a few examples such as  it will start to feel more natural.
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 $\left(x+y\right)z=xz+yz$ where x,y and z are complex numbers.
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.

###### Not exactly what you’re looking for?
nick1337

Step 1
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
$z=a+bi=R{e}^{\psi i+2\pi ki}$
where $R=|z|=\sqrt{{a}^{2}+{b}^{2}}$ is the called the modulus and is called the argument of z. Note that the argument is multi-valued - if $\psi$ is an argument, so is $\psi +2\pi k$ for any $k\in Z$.
We want to know the complex logarithm, which is multi-valued:
$w=\mathrm{ln}z=\mathrm{ln}\left(R{e}^{\psi i+2\pi ki}\right)=\mathrm{ln}R+\psi i+2\pi ki$
$z=1-i=\sqrt{2}{e}^{-\frac{\pi }{4}i+2\pi ki}\phantom{\rule{0ex}{0ex}}w=\mathrm{ln}z=\mathrm{ln}\left(\sqrt{2}\right)-\frac{\pi }{4}i+2\pi ki$

###### Not exactly what you’re looking for?
alenahelenash

$|1-i|=\sqrt{2}$
Arg $\left(1-i\right)=-{\mathrm{tan}}^{-1}\left(1\right)=-\frac{\pi }{4}$
$Ln\left(1-i\right)=\sqrt{2}-i\frac{\pi }{4}$