2022-01-17
Answered

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

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star233

Answered 2022-01-17
Author has **137** answers

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

and

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.

nick1337

Answered 2022-01-17
Author has **510** answers

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

where

We want to know the complex logarithm, which is multi-valued:

In your case:

alenahelenash

Answered 2022-01-24
Author has **343** answers

Arg

asked 2022-04-29

I need helping with simplifying expression:

$\frac{{(1+i)}^{33}}{{(1-i)}^{33}}+{(1-i)}^{10}+(2+3i)(2-3i)-{i}^{-7}$

What I got is:

$\frac{{(1+i)}^{33}}{{(1-i)}^{33}}+{(1-i)}^{10}+(2+3i)(2-3i)-{i}^{-7}=\frac{{(1+i)}^{33}}{{(1-i)}^{33}}+{(1-i)}^{10}+13-{i}^{-7}$

My problems begin with

$\frac{{(1+i)}^{33}}{{(1-i)}^{33}}$

is it possible to simplify it without de Moivre or Euler's formulas?

What I got is:

My problems begin with

is it possible to simplify it without de Moivre or Euler's formulas?

asked 2021-08-17

Multiplicative Inverse of a Complex Number
The multiplicative inverse of a complex number z is
a complex number zm such that $z\times zm=1$ . Find the
multiplicative inverse of complex number.

$z=-2+8i$

asked 2022-05-05

What is 3+2i minus 3+4i?

asked 2022-03-23

Let $a}_{0},{a}_{1},\dots ,{a}_{n-1},{a}_{n$ complex numbers with ${a}_{n}\ne 0$ . If

$f\left(z\right)=|{a}_{n}+\frac{{a}_{n-1}}{z}+\cdots +\frac{{a}_{0}}{{z}^{n}}|$

Exist$\underset{\left|z\right|\to \mathrm{\infty}}{lim}f\left(z\right)$ ?

Exist

asked 2022-04-04

(6+4i)-(7+2i)=?

asked 2022-04-07

Find the product of the complex numbers

${z}_{1}=4\left({\mathrm{cos}50}^{\circ}+i{\mathrm{sin}50}^{\circ}\right)$

and${z}_{2}=7\left({\mathrm{cos}100}^{\circ}+i{\mathrm{sin}100}^{\circ}\right)$

Leave the answer in polar form.

and

Leave the answer in polar form.

asked 2022-05-20

Calculate $$(\frac{14}{11}+6i)+(6+\frac{4i}{9})$$.