I am not a mathematician. I studied electrical engineering. I encountered quaternions while trying to understand motion of mobile robots and how rotations are achieved. This question occurred to me when I got to know of the inverse of a quaternion.
2026-04-01 22:10:50.1775081450
Why is '1' the multiplicative identity of complex numbers and quaternions?
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The symbol 1 is often used as the identity element in notation for mathematical structures known as fields. The real numbers are field. The complex numbers and the hypercomplex set of quaternions is not quitec a field (fields are required to be commutative), but they have a lot of the same properties (quaternions are something called a skew field). The rules for a field dictate that the multiplicative identity element is unique. The complex numbers and the quaternions are both viewed as supersets of the reals, so if we consider complex numbers of the form $x+0i$, and expect them to behave the same as real numbers, we had better use the same identity element as already built into the reals. The same goes for the quaternions.