imaginary number

A complex number $z$ is **imaginary** if it is not real; it is **purely imaginary** if its real part $\Re{z}$ is zero. For purposes of constructive mathematics, we only accept $z$ as imaginary if its imaginary part $\Im{z}$ is apart from zero, or equivalently if $z$ is apart from $\Re{z}$. This all generalizes to other kinds of hypercomplex numbers.

People often get these two notions mixed up. (For example, $2 + 3\mathrm{i}$ is imaginary but not purely imaginary; while $0$ is the unique purely imaginary number that is not imaginary.) This may be because the imaginary numbers, as is typical for things defined by an inequality, do not form an interesting collection as a whole (for example, they are not even closed under addition). Compare irrational number.

The purely imaginary numbers, on the other hand, form the Lie algebra $\mathfrak{u}(1)$. Often people substitute $\mathbb{R}$ (the algebra of real numbers), which is simpler, when they only care about this Lie algebra up to isomorphism. However, using $\mathrm{i}\mathbb{R}$ (the algebra of purely imaginary numbers) makes $\mathfrak{u}(1)$ fit with the matrix formulas used in higher dimensions.

Revised on January 6, 2018 14:25:27
by Urs Schreiber
(84.63.72.123)