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/sci/ - Science & Math

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12296411 No.12296411 [Reply] [Original]

Brainlet here.
Can someone explain complex numbers to me? It feels like the imaginary unit is just a cheap magic trick without any real purpose.

>> No.12296415

>>12296411
No, it's well defined. It's when you multiply R with R.

>> No.12296453

>>12296411
Imagine the number line, now imagine we were like but what if we had TwO number lines and positioned them perpandicularly. We name the second line with a little i next to the numbers so we dont get confused. Then using trig identities we create relationships between the numberlines. It just so happens that this set of complex and non complex numbers allow us to explain and solve problems in easier ways. Thats it

>> No.12296460

show me a square with complex area.
it's schizo bullshit.

>> No.12296465

>>12296460
Thats like using a hammer to try and write a letter, use a pen to write and a hammer to hit a nail.

>> No.12296484

>>12296411
C is one of the greatest objects in mathematics. There's nothing mystical about it, the complex numbers are as real as the real numbers.
There are many ways to actually construct C from the reals R. One is as follows:
Let C be the set of pairs (a,b) with a,b real numbers and with operations
(a,b)+(c,d)=(a+b, c+d)
(a,b)*(c,d)=(ac-bd, bc+ad).
We usually write a+bi for (a,b) or just a when b=0 or just bi when a=0.
This definition lets you do arithmetic on the complex numbers. It turns out that i*i=-1.
What's special about C is that actually it's a field that contains R. It means that any two elements of C can be divided (if the second one is nonzero) and that the operations + and * satisfy all the usual axioms of arithmetic such as
ab=ba
a(b+c)=ab + ac
etc.
The reason why C is so cool and the reason why it pops up literally everywhere is because of the fundamental theorem of algebra (misnomer, it's actually a theorem in analysis), which says that any nonconstant polynomial p(x) with coefficients in C (possibly R) has a root in C, that means there is a complex number z such that p(z)=0.
From this it follows that you can write any polynomial as a product of linear factors.
In particular, whenever you have a real polynomial you can explain its behavior very well by extending the domain to the complex numbers.
Aside from this fundamental theorem, another reason for considering the complex numbers is that most of the elementary functions like cos, sin, tan, log, e^x, sqrt, and so on are actually much better understood when viewed as functions on the complex numbers. Their analytic properties and integrals become much more tractable. You often see real improper integrals of elementary functions being calculated with complex analysis by complex integration.

>> No.12296495

>>12296411
geometric meaning of multiplication by unit complex numbers is rotation. as such, complex numbers are extremely useful for describing oscillatory phenomena. that's all there is to it.

>> No.12296506

>>12296495
Idiot.

>> No.12296512
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12296512

>>12296484

>> No.12296517

>>12296506
nice argument

>> No.12296521

>>12296517
Just because engineers use complex exponentials to simplify notation that doesn't mean that's all there is to it. You are clearly a moron with no clue about why complex numbers are actually useful.

>> No.12296538

>>12296411
Imaginary or complex numbers is just common term.
Don't get hung up too much on semantics.
My best advice would be to do some quick internet research into the history of why the complex numbers system was formalized and accepted in the mathematics community.
The contemporary debates at the time concerning the mathematical validity of complex numbers will probably be the best place to begin.

>> No.12296543

>>12296521
>You are clearly a moron with no clue about why complex numbers are actually useful
because they describe oscillatory phenomena

>> No.12296550

>>12296521
It's also not the only use for complex numbers for engineers. That is like saying that all of physics is simply looking at the relation between potential and kinetic energy.

>> No.12296555

>>12296521
I don't think it's clear at all that the anon you're replying to is an idiot, without making far-reaching assumptions.
However, it's clear that you're an arrogant asshole, irregardless.

>> No.12296576

>>12296543
Idiot.
>>12296555
>irregardless

>> No.12296597

>>12296576
>>irregardless
My sweet summer child, stop posting completely if you fall so easily for such low-effort bait.

>> No.12296601
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12296601

>>12296597
Fuck off, retard.

>> No.12296614

>>12296484
Ok, cool, but, what about quaternions and octonions?

>> No.12296621

>>12296614
They are also cool and useful for some stuff but they're not fields which makes dealing with them harder. In particular, quaternions are not commutative and octonions are not associative.

>> No.12296627

>>12296411
Some guy i forgot the Name of wanted to get the root of negative numbers
That is impossible so he just said fuck you it is now

>> No.12296708

>>12296576
nice argument

>> No.12296710

>>12296411
>It feels like the imaginary unit is just a cheap magic trick without any real purpose.
Do you actually have a problem with the idea/definition of complex numbers, or do you want convincing that they have applications? Often in pure maths people will just invent shit and see what it does, and C has lots of interesting pure maths reasons for studying it. The fact that it's algebraically closed means its algebraic geometry is very well behaved, and the fact that it's also geometrically a plane makes its geometry even more structured. Likewise doing calculus with it is very different to over R. The basic example is the fact that over C differentiable is the same thing as analytic (i.e having a derivative is the same as having a power series expansion), then also all the stuff about contour integrals, poles and zeroes, residues, it's all very well behaved and powerful.

But what about the real world? Well quantum mechanics absolutely requires complex numbers. The wavefunction is a complex valued function, and you don't get the right cancellations without it being complex. There are loads of other applications, but that's the most fundamental one.

>> No.12296712

>>12296708
Thanks. I tried to make it coherent to you.

>> No.12296730

>>12296411
https://www.youtube.com/watch?v=T647CGsuOVU
watch this to get a flavor. full series:
https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF

>> No.12296782
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12296782

The main purpose is that the fundamental theorem of algebra isn't true if you don't have [math]\sqrt{-1}[/math] as a number.

>> No.12296789

>>12296782
>pic
Imaginary numbers have a well-defined construction.
Infinity can be act like a number in certain aspects if you properly define it (for example, on the Riemann sphere).

>> No.12296797

>>12296782
oh man its tooker

>> No.12297003

>>12296460
show me a square with -1 area. Does that mean negative numbers are schizo bullshit?

>> No.12297081

>>12297003
Yes. Yes it does.

>> No.12297592

>>12296411
Stop thinking in terms of quantities and more in terms of structures. In that pov, complex numbers are incredibly important structures.

>> No.12297609

>>12296411
From watching some jewtube videos it seems like complex numbers are more like vectors, the real component is like the x-axis and imaginary component is like the y-axis

>> No.12297653

>>12296411
Positive and negative numbers can only be reflected. Complex numbers can be rotated.
One real purpose is dealing with numbers at (right) angles to each other, like in electromagnetism.

>> No.12297694

>>12296411
If the square root of a number is negative, then just take the negative sign out, compute it, then say its negative afterwards. That which exists is from all that exists, the claim that there are negative numbers is false. You have to assume a frame of reference. You cannot assume a frame of reference because it can only exist. I posit that all axioms for reality can only be represented by functions that are the thing. two things cannot be equal if they are a component which composes the real. Therefore mathematics as we know it is a lie. so yeah complex numbers are bullshit.

>> No.12297715

>>12297081
Based retard

>> No.12297720

>>12296411 at alternate current there is complex power, complex resistance and so on and the complex power can't be used, but the power plant has to give this additional power too. Actually companies have to pay for that also. So it has actually a very real purpose

>> No.12297723

>>12296411
[math] \displaystyle
\boxed{ \mathbb{T} \;
\boxed{ \mathbb{S} \;
\boxed{ \mathbb{O} \;
\boxed{ \mathbb{H} \;
\boxed{ \mathbb{C} \;
\boxed{ \mathbb{R} \;
\boxed{ \mathbb{Q} \;
\boxed{ \mathbb{Z} \;
\boxed{ \mathbb{N}}}}}}}}}}
[/math]

>> No.12297727

>>12297723
Is Tsohcrqzn a friend of Tzeentch?

>> No.12297733

>>12297723
What is the T

>> No.12297744

>>12297733
Tooternions.

>> No.12297746

>>12297733
32 in latin or greek

>> No.12297753

>>12297723
>tsoh

no idea

>> No.12297779

>>12297753
>schizo has arrived

>> No.12297792

>>12297723
>>12297727
>>12297733
>>12297744
>>12297753
>>12297746

Hypercomplex Numbers
H is the set of quaternions, O is the set of octonians, S is the set of sedenians, and T is the set of trigintaduonions

>> No.12297870

>>12296411
They are crutches to write functions in their simplest, forced terms.

>> No.12297871

>>12296411
it's literally just a macro for √-1.

>> No.12297876

>>12297792
>seden
>triginta
what is the etymology of this

>> No.12297898

>>12296730
would normally never recommend popmath youtube shit but these videos are actually excellent.
>>12297792
16, 32 obviously

>> No.12297977

>>12296614
quaternions and octonions are pretty much useless in their own regard; the quaternions were created so that you could work with 3D "numbers", but vectors were created soon after so no one uses them anymore

>> No.12298063

>>12297733
Torah. All those number systems are from the kabbalah, but people forgot how to interpret the ancient wisdom thus threads like this one.

>> No.12298144

>>12297723
>Believing in numerical embedding

>> No.12298223

>>12296484
That doesn't really explain anything. Seems like you just related C to R but didn't explain how C is a true set, how it can exist.

>> No.12298231

>>12296460
Show me a square with area 1? Even this is abstract nonsense. So why not accept it all?

>> No.12298236

>>12297977
idiot, it's used in 3d graphics all the time to avoid gimbal lock
https://youtu.be/zc8b2Jo7mno?t=2m30s

>> No.12298238

>>12296453
>now imagine we were like but what if we had TwO number lines and positioned them perpandicularly
But that's just a cartesian coordinate system.

>> No.12298708

>>12298238
Indeed

Pretty cool how if you decide to define a number which squares to -1 a 2D coordinate system falls out. It's fun how in mathematics there are often a number of different ways to get to the same thing.

>> No.12298731

>>12296411
The way I see it is as a placeholder. In physics, a complex result to an equation means nothing. But an equation with sqrt -1 can still be used, if there are an even number of it, because they cancel out into a real. So it's like a "hidden dimension" that only works if it cancels out. Kind of an epic hidden dimension too, because so much more stuff happens there since it's 2d.

>> No.12299437

>>12297715
Show me a square with area -1, or x or y. Algebra is useless.

>> No.12299910

>>12296484
High IQ answer here

>> No.12299951

>>12296411
Instead of points of line (real numbers), you consider points on the plane.
On those points/vectors you define two operations:
"addition" of two vectors is achieved by simply concatenating them, i.e. you put on to start at the end of the other and where you end is the new point
"multiplication" is performed by multiplying the lengths of the vectors and adding their angles

If you restrict your view on just the horizontal line, the new multiplication and addition are the same as usual. That's why we use the same names and why that line is identified with the reals.

>> No.12300052

>>12296411
Numbers have phases now

>> No.12300103
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12300103

>>12296415
brainlet

>> No.12300124

>>12297723
everything above C is schitzo bullshit

>> No.12300150

>>12296411
>Brainlet here.
>Can someone explain complex numbers to me? It feels like the imaginary unit is just a cheap magic trick without any real purpose.
just like vectors should not be explained with a geometrical analogy (it completely messes up people's ability to comprehend more abstract concept), complex numbers should not be forcefully explained with any analogy to 'normal' numbers. they are more like 2D vectors, which have a special relation between their two components.

>> No.12300152

>>12296411
>explain complex numbers to me?

they were created to solve some operations not allowed in R.
It was then discovered they can bypass trigonometric crap with simple multiplication and their extension is useful (somewhat) in describing rotations in 3 dimensions without trigonometry.

>> No.12300158

>>12296460
>>12299437
Weak bait. Get out of our math and science board.

>> No.12300165

>>12296797
Thing is, he totally right.

>> No.12300166

>>12300124
Anything about Q is schizo shit.

>> No.12300175

>>12299437
>Show me a square with area -1
Pseudo-Riemannian geometry has negative areas.

>> No.12300184

>>12299437
>Show me a square with area -1, or x or y.

show me a square with an area which is not the result of whatever arbitrary convention.

that still leaves you with the problem of quantifying stuff.

>> No.12300685

>>12300184
>units aren't real
The state of this board.

>> No.12300693

>>12296411
Real and has many practical uses which you personally will never need to worry about at your McJob.

>> No.12300709

>>12298231
imagine a square with an arbitrary width 1 and height 1, thus it is a square that can fit one square with arbitrary width 1 and height 1.
Area is the number of unitary squares that can fit within something.
using the reversal of logic we'd imagine a square with negative area as a square with a side length of i.
now show me on a ruler the approximate length of i.

>> No.12300717

>>12300166
Holy based