Can you explain, Veeky Forums?

>complex numbers aren't applicable in the real world
That's equivalent to saying trigonometric functions aren't applicable in the real world.

[math]\displaystyle \frac{d^2\theta}{dt^2}+\omega^2\theta=0[/math]

A differential equation like this (although trivial) can be solved easily with complex numbers. it can also be solved with trigonometric functions, but they are equivalent, and often the exponential form is easier to interpret.

Not to mention the use of complex numbers in quantum mechanics, where many professionals seriously consider complex-valued functions to be physically real.

nice pasta

If a complex number is essentially imaginary, then shouldn't complex numbers be able to solve all problems?

Also, isn't mathematics at the quantum mechanical level just approximations

real numbers are a subset of complex numbers

Well, not [math]\textit{all}[/math] problems, but complex numbers are used to solve a great deal of problems that engineers and scientists come across every day. In fact, the quaternions (an extension of the complex numbers, just as the complex numbers are an extension of the reals) are used by game developers to rotate objects in 3D space. The computations are less intensive than using trigonometric methods to rotate the objects. Quaternions are also crucial to the theory of particle spin in physics, and to the standard model.

There are a lot of approximations, but this is because of perturbation theory. There are some exactly solvable problems, and you don't have to make any approximations to get there. Hydrogen is a good example. When a physicist used approximation methods for quantum mechanics, that really means considering the exact influence of a disturbance (perturbation) of a system is not easily calculated by hand or with a slow computer.

The core of quantum theory relies on the use of complex numbers, and in fact cannot be properly formulated without them.

thats because sqrt(-1) isnt a real number, it is literally called an imaginary number because it does not exist in the real set

...

>ancient math runes
lol

Nice pasta, I had not seen this one in a while.

Sqrt (-1) isn't negative, that's why.

I think your missing the point with complex numbers bud.

Well for one, instead of a number line you have a number plane. Why limit yourself to just a line you plebian.

>Essentially imaginary
>takes the word literally
I bet you think the God particle literally has something to do with God

give a clear definition of complex numbers to prove your understanding then bud

The algebraic closure of the reals.

the algebraic closure of Q
the unique field structure on R^2
R[x] / ~= R[i]

the algebraic closure of Q is not the complex field.

Normie number rules don't apply to imaginary numbers.

You're full of shit, complex numbers are just a shortcut for people too lazy to apply their trigonometry.

>what are imaginary numbers

are you literally in middle school?

In QM though, they're the real stuff, not just a shortcut for trig.

youtu.be/AufmV0P6mA0?t=11m5s

You're using the assumption that [math]i^2 = -1[/math] to prove that [math]i^2 = -1[/math].

Good job.

just showing the mechanics

[math]i^2=-1[/math] doesn't have to be proved,
it's an axiom

sqrt(-1) * sqrt(-1) = x

can be re-written as:

[sqrt(-1)]^2 = x

x is the square of the square root of -1

x = -1 QED

>sqrt(-1)
Doesn't even exist, and shouldn't be written You autistic retards.

Write it sqrt(i^2).

I used to believe in imaginary numbers. For years I did not understand why people would stop and stare at me in the street. Children would run up to me, laughing, and throw dung in my face. I carried on, oblivious to the reason for my public humilation. Then one day a close friend told me why. I didnt understand, imaginary numbers were just so useful, they solved so many problems, especially in engineering. I, foolishly, ignored his advice and continued to use a mathmatical concept cooked up out of nothing to account for what should have been a blindingly obvious flaw at the core of our mathematics. I soldiered on, solving wonderously complex equations with the use of imaginary numbers. All the time I had this vague , sort of itching, discomfort in my anus. I ignored it, but as I continued to work the pain became worse, You could imagine my surprise when one day, in frustration at my increasing rectal irritation, I hurriedly pulled down my pants, pulled up a mirror behind me, and bending down so my head was firmly between my knees, I stared straight into the gaping hole that was my anus. What I saw horrified me. Besides all the wingnuts I caught a glimpse of something so terrifying that my mind almost refused to believe it. But somehow I managed to retain control of my senses and forced myself to observe the crawling horror that was so inflamming the delicate tissues of my sphincter and the passages within. It was an imaginary number dildo. How could I have missed it? There it was! Working its way rthymatically up and down my arse! Oh sweet Jesus the trauma was almost too great for me to bear. But I held on to my sanity. Drastic action was obviously needed. Quickly I read up on Clifford algebra. Within minutes the pain had gone! The dildo of imaginary numbers had vanished! I was saved!

Try becoming an electrical engineer without them

I'm aware of that, but I didn't want to confuse OP by introducing him to the type of advanced procedures that can only be comprehended after, like, *at least* two years of high shool.

why call √(-1) anything other than √(-1) if its only purpose is to solve equations, if the equations can be solved, and the result doesn't include √(-1) then who even thought about naming it i and pretending it exists.

because square root doesn't work on negative numbers, period.

nobody knows wtf i IS,
i^2 is -1 by definition.

the only ones talking about √(-1) are brainlets

I'll give you an analogy to explain what I'm on about.

Say your goal is to reach an indoor swimming pool. the only way to reach it is by walking through a fire. you decide to do it, because the water will extinguish the fire.

this is solving an equation without stopping because square root doesn't work on negative numbers.

Now some genius has another plan. he doesn't actually see the fire. he decides it's actually a sandbox, and starts hanging out and building things in it. he eventually reaches the pool, but keeps coming back to his sandbox despite everyone telling him how crazy he is.

Good god, you suck at analogies.

could it be that you have no imagination?

But I do have an imagination. I'm an art student, i.e. human, unlike you STEMtards.

>I'm an art student
No wonder you didn't understand my analogy, what are you even doing here

Stop being so fucking pedantic. No math professor is going to look down on you for putting a "√(-1)" in one of your equations. If we're talking about complex numbers then, yes, it does probably need to be represents by "i" just so your math doesn't come out looking retarded.

i=√(-1) is a valid equation; if a valid equation contains a term, that term is also valid. √(-1) is only written as "i" for the sake of expedience.

try to solve everything with natural numbers then because negative numbers an fractions are also just a concept not applicable to the (((real))) world.

But I did, and it was mediocre at best.

>But I did
Good.
>it was mediocre at best
formulate a better one please then

Math isn't any more flawed than human perception.

s-sorry senpai

jesus christ you suck at analogies

No. With [math]i^2 = -1[/math], [math]i[/math] is [math]\pm \sqrt{-1}[/math].

>>stop being pedantic
>proceeds to be even more pedantic

[math] \displaystyle
\sqrt {x^2} \ne \pm x, \quad \sqrt {x^2} = \left | x \right |
\\
|x| =
\begin{cases}
\;\;\; x & ,x \geq 0 \\
-x & ,x < 0
\end{cases}
[/math]

brainlets usually do

sqrt(x) * sqrt(x) = x
Always, no matter what x is.

en.wikipedia.org/wiki/Imaginary_unit#Proper_use

for all you brainlets:
betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

i^2 = -1

sqrt(-1) = i

sqrt(-1) is no meaningful statement.
i^2 = -1 though is. 1 = sqrt(1) = sqrt(-1 * -1 ) = sqrt(-1)*sqrt(-1) = -1

Every math operation was created by humans to solve a problem. It wasn't meant to solve every problem. Why was multiplication invented? Because addition is too slow when you have a bunch of identical numbers you want to add. So we invented recursive addition. Whole numbers are great and all, but we can't represent a fraction of a number. So we made fractions. Well, hey guys it looks like fractions can't represent some numbers, so we made decimals. Well, shit guys, what's the square root of a negative number? Should we throw away all our inventions and start from scratch? No, let's slap a new axiom and call it a day.

...

I don't know who's memeing and who's genuinely retarded

+1 senpai

You can model the imaginary numbers as matrices. The matrix
((0 -1)
(1 0))
squares to minus the identity and together with the identity spans a two dimensional real algebra that can be shown to be a field. This field can be taken as an explicit depiction of the complex numbers.

Definition:
let [math]i[/math] be the number such that [math]i^2=-1[/math]