>complex numbers are just like 2d vectors!!! >i dot

>2d vectors are just like complex numbers!!!
ftfy
>i dot i = -1 instead of 1
if you think the dot product corresponds to scalar multiplication, then either you don't understand vectors, or you don't understand complex numbers.

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>>8875086
The real mind fuck is that complex numbers are both vectors (2-d linear objects) and scalars (elements of a field).

>>8875086
>i dot i = -1 instead of 1

Nope.

i dot i is actually 1.

Don't get the complex dot product confused with complex multiplication.

Given a+bi and c+di:

The complex dot product is: ac+bd (and is always a real number).

Complex multiplication is: (ac-bd) + (ad+bc)i

>>8876367
pls report back when you get to complex numbers with dual number coefficients.

A hermitian iner product conjugates the second argument you faglord.

>>8875086
[math]<v,w>=\bar{v}'{w}[/math]

Did you fail linear algebra?

Serious question: Why do we need separate Real numbers when the Complex ones contain the same real numbers as well?

[math]xi^2 = -x[/math]
[math]xi^4 = x[/math]

While you cannot create complex numbers out of Real numbers. So why the division? Why still use simple real numbers when there's a superior version?

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>>8876774
>real numbers
>>>/x/

>>8876774
Because shit that works with reals doesn't always work the same way with complex numbers.

>>8876784
kek

I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.
Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

[math] \displaystyle
\\ z_1 = x_1+y_1i \; \; \; \; \; z_2 = x_2+y_2i
\\ z_1^* = x_1-y_1i \; \; \; \; \; z_2^* = x_2-y_2i
\\| z_1 | = \sqrt{x_1^2+y_1^2} \; \; \; \; \; | z_2 | = \sqrt{x_2^2+y_2^2}
\\ z_1+z_2 = x_1+x_2 + (y_1+y_2)i
\\ \left | z_1+z_2 \right |^2 = \left ( \sqrt{(x_1+x_2)^2+(y_1+y_2)^2} \right ) ^2 = (x_1+x_2)^2 +(y_1+y_2)^2
\\ z_1z_2^* = x_1x_2 -x_1y_2i +y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_2y_1-x_1y_2)i
\\ z_1^*z_2 = x_1x_2 +x_1y_2i -y_1ix_2 -y_1iy_2i = x_1x_2+y_1y_2 +(x_1y_2-x_2y_1)i
\\ z_1z_2^* + z_1^*z_2 = 2(x_1x_2+y_1y_2) = \text{2Re}(z_1z_2^*) = \text{2Re}(z_1^*z_2)
\\ |z_1|^2+|z_2|^2 + z_1z_2^* + z_1^*z_2 = x_1^2+y_1^2 + x_2^2 + y_2^2 + 2(x_1x_2+y_1y_2)
\\ = (x_1^2 + 2x_1x_2 + x_2^2) + (y_1^2 + 2y_1y_2 + y_2^2) = (x_1+x_2)^2 +(y_1+y_2)^2

[/math]

>>8877258
>equation can only be solved by inventing numbers that can't exis

Cubic polynomials are too much for you?
Bless your heart.

https://youtu.be/_qvp9a1x2UM?t=3m10s

>>8875086
>>8876367
>>8876671
Can a 2-component vector that consists of two complex numbers be interpreted as a quaternion?

>>8875086

Complex numbers are actually 2x2 matrices