2022-01-17

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

star233

Skilled2022-01-17Added 403 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

Expert2022-01-17Added 777 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

Expert2022-01-24Added 556 answers

Arg

$\frac{20b}{{\left(4{b}^{3}\right)}^{3}}$

Which operation could we perform in order to find the number of milliseconds in a year??

$60\cdot 60\cdot 24\cdot 7\cdot 365$ $1000\cdot 60\cdot 60\cdot 24\cdot 365$ $24\cdot 60\cdot 100\cdot 7\cdot 52$ $1000\cdot 60\cdot 24\cdot 7\cdot 52?$ Tell about the meaning of Sxx and Sxy in simple linear regression,, especially the meaning of those formulas

Is the number 7356 divisible by 12? Also find the remainder.

A) No

B) 0

C) Yes

D) 6What is a positive integer?

Determine the value of k if the remainder is 3 given $({x}^{3}+k{x}^{2}+x+5)\xf7(x+2)$

Is $41$ a prime number?

What is the square root of $98$?

Is the sum of two prime numbers is always even?

149600000000 is equal to

A)$1.496\times {10}^{11}$

B)$1.496\times {10}^{10}$

C)$1.496\times {10}^{12}$

D)$1.496\times {10}^{8}$Find the value of$\mathrm{log}1$ to the base $3$ ?

What is the square root of 3 divided by 2 .

write $\sqrt[5]{{\left(7x\right)}^{4}}$ as an equivalent expression using a fractional exponent.

simplify $\sqrt{125n}$

What is the square root of $\frac{144}{169}$