IJ=JK=KI=IJK=I^2=J^2=K^2=-1

IJ=JK=KI=IJK=I^2=J^2=K^2=-1

What could I, J and K be? If you don't answer it means all of maths since the 1840s is a fallacy and anyone who goes past arithmetic and calculus has been fooled.

they're the basis of the 4d clifford algebra

>>2
4d clifford algebra must be bullshit then.

i don't answer.

>>1
I've seen a lot of silly grudges before, but a grudge against quaternions? Yikes.
Also, they're 2x2 complex matrices. I don't remember off the top of my head which each one is, but you can find that at mathworld if you want. Oh, and that -1 actually means -1*U where U is the 2x2 identity matrix.

>>5
Welcome to 4chan.

>>3
Ah, the rancid smell of defeat...

>>4
it means all of maths since the 1840s is a fallacy and anyone who goes past arithmetic and calculus has been fooled

>>8
I must admit you have me confused. Do you think the complex numbers are also fraud? Or just the quaternions?

complex numbers are clifford(1) and quarternions are clifford(2) SO IT"S ALL THE SAME MANG~

>>5
No, they aren't 2x2 complex matrices. They're just arbitrary group elements. You can literally define a numbering system as {1, i, j, k} (and their inverses), and those are your numbers; they don't have to "be" anything else.

You can however represent the group using 2x2 complex matrices in a variety of ways; this representation is called clifford(2).

gb2/grouptheory/

>>11
Any group {1, -1, i, -i, j, -j, k, -k} whose group operation table is what we expect for the quaternions is isomorphic to the group {U, -U, I, -I, J, -J, K, -K} with U,I,J,K being complex 2x2 matrices. Objecting to my 'defining' the quaternions in terms of those complex matrices is similar to objecting to one of the various constructions of the real numbers by saying that the reals are an arbitrary complete ordered archimedean field. You've added nothing to the conversation but a half-hearted attempt to show off how totally awesome at math you are.

>>12
"with U,I,J,K being complex 2x2 matrices" should read "with U,I,J,K being complex 2x2 matrices which also satisfy the group operation table we expect for the quaternions."

No. The group operation is just good old matrix multiplication.

Sorry, but no. Quaternions are not a part of mathematics and do not model anything in the real world.

Logic is simple

A = the property of something being white
1-A = the property of something not being white

A^2 = A
A-A = 1-A
A^2 + A = A

etc etc.. It makes perfect sense in the context that we are using it, however if we were to declare these equations were arithmetic and then declare that A = 5, the equations would have to be false.

However since quaternions are in the context of normal arithmetic and no numbers fit into the equation, then quaternions are bullshit and everything, tensors, harmonic resonance etc etc.. is bullshit and a huge fraud.

>>15
Do you believe irrational numbers exist? If so, why?

>>16
They exist if they are logically applicable. Otherwise I could just come up with any old shit and say "oh look im super wonderful suck my dick".

i = -1^0.5 = 2 = 3673467346 = A small pineapple.

>>17
What do you mean by "logically applicable"? Quaternions have "logical applications" in computer graphics because they are an excellent way of representing rotations around arbitrary vectors in 3 dimensions. Similarly, complex numbers are useful for simplifying notation of electrical engineering.

On the other hand, if you continue to insist that complex numbers and quaternions are not "logically applicable," what makes irrational numbers "logically applicable"? No measured quantity in the universe can be measured infinitely precisely, so saying that irrational numbers exist is a statement of faith.  If we say that "the solution of x^2 = -1" does not exist, why should sqrt(2) exist just because it is "the solution of x^2 = 2"? Or Pi, why should that exist solely because it is the ratio of the circumference of a circle to its diameter when a perfect circle exists only as an abstract idea?

IJK=I^2=J^2=K^2=-1

IJ=K=-JI
JK=I=-KJ
KI=J=-IK

fixed

>>19
whatevs

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This is... interesting. Interesting like Candlejack...
...
IJ=JK=KI=IJK=

Stand in front of a mirror in the middle of the dark and say IJ=JK=KI=IJK=I^2=J^2=K^2=-1 10 times and candle jack will stab you in the back.

ZA WARUDO