little disclaimer here: i don't know english well, so don't

>for what these imaginary units are made
Many mathematical problems become a whole lot easier in the complex plane.

Quaternions are useless poser shit though.

>>7850546
can you present me these problems?

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>>7850544
Read a book

>>7850554
You can solve some vectorial differential equations by saying x is the real and y the imaginary part of a number.

You can solve complicated real integrals with the residue theorem like it ain't no biz

You can display oscillations with euler's function

Just some from the top of my head.

>>7850544
Normal numbers (1, 6.5, -200, pi, etc.) are all one dimensional numbers. They can be put onto a number line. A line is one dimensional. It only has length.

Complex numbers are two dimensional. They have two dimensions: the real and imaginary parts. In other words, they have a length and a width. They can be expressed as a+bi.

Quaternions are three dimensional numbers. They have a length, a width, and a height. For reasons unknown, quaternions require 4 variables and can be expressed as: a+bi+cj+dk. Just as i^2=-1, the fundamental idea of quaternions is that (i+j+k)^3=1 or something like that. I don't know much about these, though, sorry.

>>7850544
Hello Friend! You need a tachyons.

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>for what these imaginary units are made?

Complex numbers can be seen as a 2D R-vector space with a multiplicative operation (x,y) * (x',y') = (xx' - yy', xy' + x'y). This multiplicative rule makes (C\{0}, *) a commutative group, which makes C a (commutative) field, which is really nice.

Quaternions are the same with a 4D R-vector space, except this time it is not a commutative field.

(You can go further, but you wuold then loose associativity, which fucks everything up).

As for how to represent 3D rotations with quaternions, you just need quaternions of norm 1, so this is really a 3D space, not 4D. I'm not sure how usefull this is though. You can just stick with regular rotation definitions as 3d matrixes of determinant 1. (O+(R3))

>>7850546
>Quaternions are useless poser shit though.
Fucking pleb. Quaternions arise naturally when dealing with the spin Lie algebra.

>>7850702
a ты нe плoх))))))))))))))

>>7850714

He's kind of right though. Most of the time I hear about using unit quaternions, it's for an application where SU(2) would be better. Imagine trying to do complex analysis where
you insisted on keeping x's and y's everywhere rather than using z.

>>7850746
>a ты нe плoх))))))))))))))
To жe caмoe мнe cкaзaлa твoя мaть, кoгдa я дpaл eё кaк пocлeднюю блядь.

>>7850751
нeт ты вpёшь)))))
AЗAЗAЗAЗAЗЗAЗAЗAAЗ))))))))

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>>7850746
>>7850751
>>7850766

>>7850714
>dealing with the spin Lie algebra
got any links, ive never seen then treated with quaternions before.

Извинитe, a чтo здecь пpoиcхoдит?

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>>7852043
here i just present that: imaginary unit is a bull shit
it ruined most needed fundamental rule in mathematics.
imagine that:multiplication of two numbers with similar positive/negative sign equals minus one.
convulsions of insane.

>>7852269
multiplication is actually a rotation
*1 = 360
*(-1) = 180
*i = 90 (CCW)
*(-i) = 270 (CCW)
It's better to imagine numbers as places at whatever-dimensional space, from where we can take as many as possible:
real - one
complex - two
quaternion - four

I like the explanation that negative numbers are the 'proper', everyday numbers, reflected, but complex numbers are rotated. Because negative numbers are imaginary in the sense that you can't 'have' minus 2 apples, but we easily accept that minus 2 is a useful number.

>>7852416
plot twist: negative numbers ARE rotated, but but 180 degrees, while imaginary are rotated by 90

>>7850544
> complex numbers are just vectors?
Not *just* vectors. You can use them to represent 2D vectors. But unlike 2D vectors, they're actual numbers, i.e. practically any mathematical operation which is defined on the real numbers is automatically defined on the complex numbers. Given any expression involving complex variables, just replace e.g. z with a+b*i, then using i^2=-1 to simplify the resulting expression.

> and quaternions... so difficult) why they have 4-dimensional unit to explain 3d rotation. very want to uderstand this. very,very)
When it comes to using unit quaternions for modelling rotations, you don't actually need to understand them. Just learn how to implement the basic operations (multiplication, SLERP, conversion to matrix) and use them.

>>7852700
>you don't actually need to understand them
not good

>>7852364
>shilling DeMoivre's crackpottery this hard

>>7852364
yes, i finally understand this
if multiplicate any number and -1, we just rotate point at number line in 180° and go to -78 (for 78), but if we multiplicate √-1 and any number we rotate point in 90°.
brilliant!
√-1 = 90°
-1 = 180°
√-1 * √-1 = -1

>>7850563
I proved Riemann with the residue theorem just now, thanks anon!