How come there are tons of mathematical constants that are real but no constants that are complex numbers?
pic unrelated
>no constants that are complex numbers
That's a strong statement, you have a way to prove that?
>>7984294
i is as famous as pi or e, nigga.
>>7984294
2*pi*i comes up a lot
>>7984294
That's true, maybe nobody felt the need of one. /sci/ should create one.
>>7984294
[math]\mathbb R \subset \mathbb C[/math]
now think before your shitposting kills someone
>>7984332
Are you fucking dense? [math]i \in pi \equiv \pi [/math]
It's literally in the god damn name of the MOST FREQUENTLY USED CONSTANT "pi" you fucking cretin.
>>7984315
thought this was bait at first
fuck off back to school kiddo
take this constant for a test drive /sci/
[math] \frac{ \sqrt{ 2 } }{ 2 } + i \frac{ \sqrt{ 2 } }{ 2 } [/math]
>>7984294
>real constants but not complex
>real numbers aren't complex
wew lad
>>7984440
[math] \sqrt{i} [/math]?
>>7984294
who's that handsome devil on the left?
his costume kinda sucks desu but at least he tried
also, you probably should have specified that you are talking about constants with a complex part, as the autists here will surely point out that you can embed the real numbers in the complex numbers, implying that real numbers are complex numbers
OP asks an interesting question. Any non pedants have ideas?
>>7984529
[math]\mathbb R \subset \mathbb C[/math]
>>7984324
What the fuck are you talking about? Any real number is a complex number with imaginary part equal to zero.
>>7984543
> complex numbers that are not include in the real and the imaginary domain
oh boy, so what part of [math]\mathbb C[\math] would be left after your restriction?
>>7984332
That's not actually true. The real numbers are isomorphic to a subfield of the complex numbers.
>>7984547
Surreal numbers?
>>7984549
the real numbers are isomorphic to all of the complex plane, as well as any nondegenerate subset
>>7984294
Afrobro browses /sci/?
>>7984559
So you're telling me you can build a bijection from R to RXR?
>>7984547
A fucking a+bi where a=/=0 and b=/=0.
A fucking number with a beautifully fucked up polar angle.
Fucking happy now?
>>7984579
yes, this is literally babby tier analysis
>>7984581
thx m8
>>7984543
It's not being pedantic. We only have a few "constants deemed important enough to have their own symbol" to begin with. Any "complex number (potentially) deemed important enough to have its own symbol" would just be formed by just affixing an i as necessary. For instance, say [math]\sqrt{5} + 2\pi i[/math] were as important as something like [math]\varphi[/math] (golden ratio) in equations, theorems, whatever. Would we assign it its own symbol, say [math]\Xi[/math]? It's highly doubtful, since we can express it as a combination of pre-existing real numbers and the imaginary unit.
In other words, it's mainly about practicality. [math]e[/math] and [math]\pi[/math] are "constants deemed important" not just because they're so frequently used, but also because they're transcendental, and we can't express it conveniently via numbers. [math]\sqrt{2}[/math] is incredibly ubiquitous as well, so why don't we designate it its own symbol? Because we don't need to. And that's really all it is, and that's why [math]\mathbf{R} \subset \mathbf{C}[/math] is a completely valid explanation. But if you don't understand why we designate certain numbers are "constants" to begin with, then you have a fundamental misunderstanding about "constants" to begin with, let alone "complex constants."
>>7984584
Okay, show me it
>>7984559
>[math] \mathbb{R} is isomorphic to \mathbb{C} [/math]
inb4 HURR I MEAN ONLY AS SETS :^)
>>7984340
Best chuckle since sin(x)/n = 6
>>7984294
Because constants are mostly related to physical phenomena, which are kind of obliged to be real
>>7984549
let R be the real numbers, and C be some construction of the complex field
and assume that R is not a subset of C
first, let A be the set of all (x,0) in C, for x in R, with C being the complex numbers, and R being the reals
even if you've defined C in such a way that it is formed of pairs of objects, but not necessarily real numbers, there is a bijection (call it ф) such that every complex number is of the form (ф(a),ф(b)) for real a and b
now, form the set B by taking the complement C\A
then, form set D by taking B union R
because we assume R is not a subset of C, B and R are disjoint, so an element lies either in B, or in R, but not both
this allows us to define well-defined addition and multiplication on D
define addition on D like such:
a+b =
if a and b are both in R, then a+b on D is the same thing as a+b on R, so define a+b for real a,b to be the a+b in R
if a is a member of B, and b is in R, then a can be expressed as a pair (x,y), with x,y real, or at least bijective to two reals
take the element of R which x is bijective with, (call it ж) and return (ф^-1(ж+b),y)
if ф(y) = real 0, then return the ж+b in R
you can define multiplication in a similar manner; constructing it carefully piece-by-piece to comply with the new structure
if you've done it properly, you should have a field of complex numbers which contains R as a subset
i happen to use this field rather than some lazy definition of C which contains only equivalence classes
disclaimer: the purpose of this post is nothing but autism
>>7984626
you would have loved the "solve w = x" by substitution of "w = 2u" post made a few days ago
>>7984653
Fascinating
>>7984654
What else shall we discuss on the math and science board? I guess everyone prefers to bash eachothers majors instead...
>>7984678
>/sci/
>IQ threads
>"pls do my derivative calculus hmwk"
>WHICH BOOK IS BEST TO LEARN ANALYSIS??? AND DONT SAY RUDIN XDDDDDDDDDDDDD
> debug my code pls
> engineering ranking thread
> math undergrads pretending to have phds in topics that had learned the class before
> more IQ threads
> nootropic threads
> IQ threads
welcome to /sci/
>>7984695
this post gave me a prolapsed rectum
>>7984691
Yet we still come back here. This place is like a bad relationship
probably because they reduce to (more convenient) real pairs, although multiples of roots of unity [math]\zeta_n = e^{iπ/2n} [/math] seem to be pretty common.
Just a notational thing, but I found [math](-1)^{p/q} = e^{iπp/q} [/math] to be neat when working with number theory stuffs. Which incidentally has nontrivial real and imaginary values for coprime [math]p,q[/math]. I'd imagine such progressions might lead to interesting constants, but I couldn't tell you any.
>>7984701
Good. I hope you saved it
>>7984695
Similarly, 6 = (sin x)/(nx)
>>7985152
6 = (sin x^2)/(nx)
You stand corrected :^)
>>7984581
OP here, this guy has got the right idea. I understand there are tons of real constants that are technically complex but I want to know a well known constant that is complex yet not real or imaginary.
>>7984602
Is it weird that all known complex constants can be represented using existing real constants mixed with i? Like when we found pi or e, we only denoted it that way because we couldn't represent it using the numbers we already had. So why does the complex plane not have any special properties separate from the real numbers that leads to it's own constants?
>>7984440
This is probably the closest we will come to an answer in this thread, but for hard mode, are there any constants that are complex, not real, not imaginary, and not on the unit circle?
>>7984469
He's just a cosplayer but the dude on the right is the creator of giantdad.
>>7984653
BTFO
