So Im in the 5th semester of CS and I can calculate all kinds of crazy shit but whenever Im using imaginary numbers and quaternions I dont know what the fuck im doing and why it works.
Why is sqrt(-1) = i ? Ive watched like 100 videos but no one explains how the fucking point with the coordinates (0,1) on the y Axis becomes i.
It becomes worse when they say "the real part is a and the imaginary part is b*i", why do I need b*i at all here, what happend to the fucking Y axis, y = b*i obv but why? Why isnt y = b + or - i or b^i?
Ive seen explanations using rotations but what has a rotation to do with factoring a fucking number, whats the process here?
I NEED DETAILS ANON GIVE ME BOOKS SHOW ME THE WAY
>>7944803
>So Im in the 5th semester of CS
Yeah
>Why is sqrt(-1) = i ?
There it is
Nothing to see here folks, just another retarded CS major flopping around in their own stupidity
>>7944811
Im so sorry for asking pleb questions...
SORRY I FAILED YOU MATH OVERLORD
To understand where the multiplication rules come from, you may think of the set of complex numbers as the set of polynomials in x, where each occurence of x^2 is set to -1.
[math]p(x) := 3-4x+2x^2+7x^3 = 3-4x+2(-1)+7x(-1) = 1-11x[/math]
[math]q(x) := 1-x-3x^2 = 1-x-3(-1)= 4-x[/math]
[math]p(x)·q(x) = (1-11x)·(4-x) = 4 - 45 x + 11 x^2 = 4 - 45 x + 11 (-1) = -7 - 45 x[/math]
And indeed
[math](1-11i)·(4-i)=4-11+(-1-44)i[/math]
The x with the x^2 -> (-1) property is a symbolic tool here.
You may also think of a complex number a+ib as a 2 by 2 matrix ((A11,A12),(A21,A22)).
Then matrix multiplication also leads to i^2=-1. Replace a with ((a,0),(0,a)) and ib with ((0,-b),(b,0)).
Check it.
Although if you watched 100 videos, you will know this already.
>>7944811
>implying he won't make more than you'll ever make
>inb4 "I make 150k with full benefits"
A complex number is composed of z := a + bi
Where a is the real part and b*i is the imaginary part. Look up Gauß (Complex) Plane. Srsly not that hard.
You learn that in the first CS semester..
>>7944803
>imaginary numbers are difficult for me to comprehend
dude you learn this when you're 15 wtf
>>7944803
>Why is sqrt(-1) = i ?
Because someone said so.
Hey OP, I dunno if you're actually serious about learning anything or not. I also don't know if CS even has enough math to really understand the idea (basic fucking trig).
But if you want a good, intuitive, understanding of the complex plane, you should read pic related
>>7944803
It's that way because someone defined it that way.
>>7944803
im a retard but i figured i=sqrt(-1) because you're just assigning it that value.
isnt that what you do in CS? just assign values to variables
>>7945057
i is not a variable
>>7944869
>he doesn't know about 300k starting any job you want
Why is nobody explaining to OP?
OP, the answer to your questions requires you to take a different approach to what a number is.
Start with set theory. You know we have sets and then assume ZFC axioms and use them to define natural numbers. Now we have the natural numbers, which can be extended by defining fractional equivalence classes a/b = c/d if and only if ad = bc. This class is called the rational numbers. Then you can use the dedekind cuts to construct the reals from these.
Now you have to study rings. Having done so, you may observe that the real numbers are actually just the only ordered field with the least upper bound property. This is the definition of the reals, so any physical preconceptions you may have about them can be completely forgotten at this point, as that is all they are. That concept. Now as a field, we can extend it over a vector space to R^2, but only if we define some kind of multiplication on R^2. This multiplication is the use of i, which allows us to define (a+bi)(c+di) = ((ac -bd) + (ad + bc)i and uniquely, by this definition of multiplication in R^2 only, define R^2 as a field.
That is why we use complex numbers.
[eqn](a+bi)(c+di) = ac + adi + cbi + bdi^2[/eqn] Refined it to fit CS math. :^)
>>7944803
multiplication is a rotation
+1 is 0 or 360 rotation
sqrt(+1) = half of 0 or 360 rotation
half of 0 = 0 that is +1
half of 360 = 180 that is -1
So here we have sqrt(+1)=+/- 1
Now, if you squirt the -1, you get halves of 180 or 540 rotation
We get 90 and 270(that is basically -90 on plane)
Now we assign 90 degree rotation to be something called "+i", and 270 as "-i"
Basically, numbers are points in multidimentional. For real numbers we take one dimension, for complex - twp, for quaternions - four, and so on
>>7945062
It's a parameter.
>write once read whenever
>>7944803
Actually, what are you asking about> Do you want to find an intuition for complex number? If so, my fist question is, what is your interpretation of real numbers? What is a real number, what is the meaning of the "+", "-", "*", ":" operations in that interpretation ?
>>7944803
Jesus Christ, imaginary numbers? Is this a joke? You learn that shit in 7th grade.
>>7945065
>300k starting any job you want
take your pedophile cartoons back to>>>/a/
>>7945819
Europeans dont learn about imaginary numbers until grad school
>>7944849
You can actually show how the multiplication of complex numbers using relations.
>>7945087
>>7945356
Assuming you've covered relations and whatever, you're going to want to consider the set of polynomials with equivalence relation
f(x)~g(x) if f(x)-g(x) is divisible by [math]x^{2}+1[/math].
So what we're getting at is that any polynomial is equivalent to some degree 1 polynomial, or
[math][f(x)]=[ax+b][/math]
Now we want to just define some operations [math]\odot, \oplus[/math] by
[math][f]\oplus[g]=[f+g][/math]
[math][f]\odot[g]=[f\cdot g][/math].
You can see now that
[math][ax+b]\oplus[cx+d]=[(a+c)x+b+d][/math]
[math][ax+b]\odot[cx+d]=[acx^{2}+adx+bcx+bd]=[acx^{2}]\oplus[(ad+bc)x+bd][/math].
Now just replace your x's with your i's and you're mofuggin good to go senpai.
