are imaginary numbers just things we forced into existence so we could sqrt -1?

[Anonymous No.15567861]

seems kinda like a brutal way to make it work for us

[Anonymous No.15567868]

>>15567861
This has been hashed out before but imaginary numbers are a convenient way to define periodicity (e.g., rotational motion) using algebraic notation. It doesn't need to be more complex than that. Also, the actual definition is i^2 = -1

[Anonymous No.15567870]

>>15567868
sorry could you elaborate?

[Anonymous No.15567896]

>>15567870
Multiplication by -1 is 180 degree rotation so the square root of -1 is 90 degree rotation

[Anonymous No.15567908]

>>15567861
Yes. We plucked them out of the void, their existence is pure agony, they know only pain and anguish, for their very being is a torturous logical conundrum.

[Anonymous No.15567919]

>>15567896
>Multiplication by -1 is 180 degree rotation
No its not

[Anonymous No.15567929]

>>15567919
>*turns -1*-1 degress and walks away*

[Anonymous No.15567937]

>>15567861
>forced square root

[Anonymous No.15567940]

if a graph is 2d then imaginary numbers exist on a 3rd plane

[Anonymous No.15567944]

>>15567940
if you cross your eyes hard enough you will unlock your third eye and be able to see into the complex dimension

[Anonymous No.15567946]

>>15567929
Go play xbugs 360 Michael

[Anonymous No.15567968]

>>15567946
nooooooooooooooo

[Anonymous No.15567979]

>>15567940
> -1 does not exist on a 2d number line

[Anonymous No.15567981]

>>15567979
who are you quoting?

[Anonymous No.15567986]

>>15567981
>who are you quoting

[Anonymous No.15567987]

[math] \sqrt{-2} = ?????????[/math]

[Anonymous No.15567989]

>>15567987
20E

[Anonymous No.15567990]

>>15567987
i*sqrt(-1)

[Anonymous No.15568019]

>>15567990

[Anonymous No.15568026]

>>15568019
2 3D 4 me

[Anonymous No.15568039]

>>15567861
So what is stoping them from doing 1+i+j?

[Anonymous No.15568073]

>>15567896
The square root of 180 is not 90, though, why would it be that?
What exactly are you rotating into, some imaginary dimension, are you saying there are actually 6 dimension of space and 2 of time since each dimension has to have the ability to rotate into its accompanying imaginary dimension?

[Anonymous No.15568082]

>>15568073
yea it is
its exactly like that

[Anonymous No.15568089]

>>15568082
No, the sqrt of 180 is 13.4164079 rather than 90.
To demonstrate the exactness, can you show an empty spatial graph for those 6 dimensions like pic related, but 6 orthogonal axis rather than the 3 traditional ones?

[Anonymous No.15568092]

>>15568089
sqrt(180 degress) = i * sqrt(degress)
its basic science, einstein

[Anonymous No.15568096]

>>15568092
>sqrt(degress)

[Anonymous No.15568838]

>>15567861
This question gets asked every week, but considering the way you framed it, it seems like you're asking in good faith.
You have to realize that many number systems are just extensions of previous, simpler ones. These extensions allow you to do do more stuff, but often sacrifice certain properties and intuitive analogs. Consider that the natural numbers, by definition, nicely describe discrete stuff. By contrast, the positive numbers are needed to describe continuous quantities of real things. You can cut 3 yards of fabric into two sections of 1.5 yards. But if you tried to evenly divide 3 people in half, you'd get two people and a corpse.

You can extend the positive numbers to the negative numbers and get the "real" number line. This is very useful for describing one dimensional relations. After all, it's perfectly analogous to a line. The key here is it can describe *relations,* like spatial relations and debt. The negative sign means 'below' or 'against' some standard zero point, but trying to interpret it as an absolute amount of a real thing yields nonsense. You can't have a -3 apples, or walk a negative distance. You can owe a guy 3 apples, or walk backwards.

There's this famous movie line, "A negateev times a negateev ees a positeev." https://www.youtube.com/watch?v=2k_jS1zVLWw
Why? It's because of the relational structure. Say you start with an establish forward direction. Then, you turn around 180° (-1), and then walk backwards (another negative value) relative to your new bearing. You will progress in the direction originally specified as positive: -1*(-D) = D

The complex numbers extend this concept beyond only two forward and backward directions (a line). This implies a whole plane (a full 360° of possible directions). Consider: root(-1) can't possibly be positive OR negative, and thus can't exist on the number line. But if it exists off the number line, remember that a line and a disjoint point literally define a plane.

[Anonymous No.15568857]

>>15567861
>>15568838
Continued:

Indeed, it turns out that the complex numbers perfectly fill a plane in the same way the "reals" perfectly fill a line. But unlike a Cartesian plane where you make two real number lines perpendicular, the complex numbers have the added benefit that rotations (necessarily, transformations *between* the x and y terms) are encoded in the pure algebra. Multiplying by 1 is a 0° rotation (multiplying by the identity changes nothing). Doing it with -1 is 180°, as discussed before. Necessarily, because i*i = -1, multiplying i must correspond to exactly half a 180° rotation, ie: 90°. It's standard to go counterclockwise.
Thus, the expression i*i = -1 is logically equivalent to saying "two left turns face you backwards". In the same way -1 can mean "backwards," i just means "left."

If the idea of "left" apples strikes you as odd, it should. But no more than "negative" apples. But I want to stress the point that none of this is "forced." It's very natural and actually quite simple once you get the hang of it. It's one of those constructions that's so elegant it looks more like a discovery than an invention.

[Anonymous No.15569127]

>>15567987
[math]\sqrt{-2}=\sqrt{(-1)(2 )}=\sqrt{2}\sqrt{-1}=k\sqrt{-1}[/math]
k as a constant.

[Anonymous No.15569131]

>>15569127
Fck me, wysiwig this editor my ass.

[Anonymous No.15569237]

[math]sqrt{-1}=i[/math]
source?

[Anonymous No.15569239]

>>15567861
[math]\sqrt{-1}=i[/math]
OP, where is the source of this claim?

[Anonymous No.15569264]

>>15568039
only the fact that you also need a +k

[Anonymous No.15569445]

>>15568039

[Anonymous No.15569448]

>>15569445
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]

[Anonymous No.15569454]

>>15569445
[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]

[Anonymous No.15569483]

>>15567861
That is just a derogatory term that remained in use just because people are fucking stupid. Those are side/lateral numbers, they are as imaginary as any other number.

[Anonymous No.15569711]

>>15567861
i4 = 1
root factors
i4 - 1 = (i-1)(i+1)(i-x)(i+x)
= (i2-1)(i2-x2)= i4 - i2x2 - i2 + x2 = i4 + x2(-i2+1) -i2
x2 = (-1 + i2)/(1 - i2) = -1

otherwise there is no root factors equation. What is forceful about this? If Reals are defined then there is a negation of the Reals set. Reals is a bad name I guess, I guess it makes you think anything else is Not Real lmao

[Anonymous No.15569900]

it's an interesting algebraic and geometric structure. that's all there is to it, doesn't matter what you call it.

[Anonymous No.15569902]

>>15569454
[math]\displaystyle \boxed{ \mathbb{D}_s \; \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]

[Anonymous No.15569906]

>>15569902
What does it mean and why should i care

[Anonymous No.15569948]

>>15569906
Number systems which contain other number systems. Look into Cayley-Dickson construction

[Anonymous No.15569963]

all numbers are "forced into existence" in some sense
it's up to you what you decide to work with and complex numbers are convenient and interesting so they we use them
there are also other extensions of the reals such as the 16-dimensional sedenions, but they are not particularly interesting so we dont really worry about them

[Anonymous No.15569974]

>>15569906
Each set contains the previous plus the previous' negation. Positives("Naturals") + negatives = Integers. Integers + Non-integers(fractions)= Rationals. Rationals + irrationals = Reals. Reals + Non-reals = Complex. And it goes on forever, since for any set there is its negation and a superset containing the two.

[Anonymous No.15570058]

>>15569902
What's D_s, T, S, O, H?

[Anonymous No.15570112]

>>15570058
I can tell you H is the "Quaternions" which is a four dimensional extension of the two dimensional complex numbers. It adds imaginary numbers "j" and "k" which, coupled only with the real numbers, work identically to "i". What's interesting/useful abut them is that, taken altogether, they perfectly fill a four dimensional space. And like the complex numbers, algebraic operations with them perfectly correspond to transformations (rotations, translations, etc.) in the space. In the 2-space of the complex numbers, there's only one possible axis of rotation. This is not true in higher dimensions where you have distinguish between pitch, yaw, roll, etc. The distinction between the additional "imaginary" numbers j and k provides that difference.
However, quaternions must also have this weird property that multiplication is no longer necessarily commutative. In fact, the more generalized these number system get, the more such properties you lose. Overall, they can be neatly summarized with the following rules: i2 = j2 = k2 = ijk = -1 , ij = k , jk = i , ki = j , BUT: ji = -k , kj = -i , ik = -j

Honestly don't know much about the Octonians (O), other than they're an 8 dimensional extension of the quaternions and that multiplication under them can't even be associative.

[Anonymous No.15570554]

>>15569974
This is beyond retarded, well done

[Anonymous No.15570624]

>>15570058
>What's D_s
[math]\mathbb{D}_s[/math] nuts

[Anonymous No.15570688]

>>15570058
Sedenion & trigintaduonion
https://en.wikipedia.org/wiki/Sedenion

Ds is a joke, see >>15569454

[Anonymous No.15570695]

https://youtu.be/GBTUVg91bao?t=10s