The discussion revolves around whether quaternions should be considered an extension of the real numbers, similar to how complex numbers are viewed as extensions of the real numbers. Participants explore the nature of quaternions, their components, and their relationship to complex numbers and vectors.
Participants express differing views on whether quaternions are extensions of real numbers or complex numbers, indicating that multiple competing views remain without a clear consensus.
Some statements rely on historical context regarding Hamilton's motivations and the definitions of quaternions, which may not be universally agreed upon or fully resolved in the discussion.
Wouldn't they be considered generalizations of 4-vectors? A quaternion has four components.Topolfractal said:Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?
The "quatern" part comes from Latin, meaning "four times."Topolfractal said:Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.
Topolfractal said:Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.
Thank you, and you for the longest time having not researched quaternions in depth, I always thought they were a 3-d version of the real numbers with three components. After reading the Wikipedia article on quaternion I understand Hamilton's motivation behind the four parts now.DrewD said:You are not alone. My recollection is that Hamilton was trying to add one more dimension to complex number to make a three part number. He wasn't able to find a way to make add just one dimension and have it be an extension of the complex numbers. Quaternions, however, are an extension to the complex numbers.
Yes, undoubtedly!Topolfractal said:Quaternions are generalizations of 3- vectors, in the same as complex numbers are generalizations of 2- vectors. Should quaternions be considered an extension of the real numbers as the complex numbers were?