Debates - Potential Faulty Science

The issue is that you’re treating i,j,ki, j, ki,j,k as if they are all the same number −1\sqrt{-1}−1. That’s exactly where things break.
 
So quaternions are flawed, since we need axioms that says i^2=k^2 but i not = k which is unmathematical.
 
talanum1 said:
But their squares are equal. You are saying we cannot always draw the square root on both sides of any equation.
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The claim comes from a misuse of algebra rather than a flaw in quaternions.

Even though i2=j2=k2=−1i^2 = j^2 = k^2 = -1i2=j2=k2=−1, it does not follow that i=j=ki = j = ki=j=k. Equality of squares does not imply equality of the values (e.g., 22=(−2)22^2 = (-2)^222=(−2)2, but 2≠−22 \ne -22=−2).

Square roots are also not single-valued. In Complex numbers, −1\sqrt{-1}−1 can be iii or −i-i−i, so “taking the square root” is not a reversible or unique operation.

More importantly, in Quaternion algebra, i,j,ki, j, ki,j,k are not defined as square roots of −1-1−1. They are distinct basis elements with specific multiplication rules, and the system is non-commutative.

Treating them as equal collapses the structure and introduces contradictions that do not exist within the actual rules of quaternions.
 
talanum1 said:
So quaternions are flawed, since we need axioms that says i^2=k^2 but i not = k which is unmathematical.
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  • Having i2=k2i^2 = k^2i2=k2 but i≠ki \ne ki=k is not unmathematical.
    • Example: 22=(−2)22^2 = (-2)^222=(−2)2 but 2≠−22 \ne -22=−2
  • Mathematics does not require uniqueness from squaring.
    • The function x↦x2x \mapsto x^2x↦x2 is not injective in most systems.
  • In Quaternion algebra:
    • i,j,ki, j, ki,j,k are defined axiomatically as distinct
    • Their shared property x2=−1x^2 = -1x2=−1 does not imply equality
    • The structure is internally consistent and widely used (e.g., 3D rotations)
  • The objection assumes an unstated rule:
    • “If a2=b2a^2 = b^2a2=b2, then a=ba = ba=b”
    • This rule is false in general algebra
Bottom line
  • No inconsistency exists.
  • The claim depends on an invalid assumption, not a flaw in quaternions.
 
Prawnapple said:
Example: 22=(−2)22^2 = (-2)^222=(−2)2 but 2≠−22 \ne -22=−2
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I can't read that: "22 not = (-2)22^2" or what do you mean?

So you can draw the negative square root also, this just leads to: -sqrt(-1)*sqrt(-1)*sqrt(-1)= sqrt(-1).
 
I think x=+-sqrt(n) comes from the intuition and is not derived since:

x^2 = n

now draw the negative square root on both sides to get:

-x = -sqrt(n)

and cancell the sign both sides to get:

x=sqrt(n).
 
It also implied you can't draw the square root both sides. This conflicts with the rules of mathematics.

I don't need to disbelieve: I can just derive.
 
You should stick to one thread, come up with a universal title. Others have to....

You can always quote yourself from earlier pages when you later need to.

It's looking for trouble not to.
 
talanum1 said:
It also implied you can't draw the square root both sides. This conflicts with the rules of mathematics.

I don't need to disbelieve: I can just derive.
Click to expand...
That's the thing. One can't trust AI for important stuff like this. The Universe will disrupt into chaos and before we know it a second big bang hit us.
 
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