Quaternions: Extension of Real Numbers?

[Topolfractal]

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?

 

[Mark44]

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?

Wouldn't they be considered generalizations of 4-vectors? A quaternion has four components.

 

[Topolfractal]

Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.

 

[Mark44]

Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.

The "quatern" part comes from Latin, meaning "four times."

A quaternion has the form q = ai + bj + ck + d, where i, j, and k are such that i² = j² = k² = -1.

 

[DrewD]

Oh I thought quaternions were just adding 1 more part to a complex number. I must be way wrong.

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.

 

[Topolfractal]

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.

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.

 

[blueberry]

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?

Yes, undoubtedly!