nLab imaginary number

Definitions

A complex number $z$ is purely imaginary if its real part $\Re{z}$ is zero, hence if under complex conjugation we have $\overline{z} = - z$. The complex number $z$ is imaginary if it is not real, in other words if $z \ne \Re{z}$, or equivalently if its imaginary part $\Im{z}$ is nonzero, or if $z \ne \overline{z}$. (In constructive mathematics, we mean apartness here.)

The same terminology applies to quaternions, octonions, and other $*$-algebras of hypercomplex numbers. (In yet other $*$-algebras, such as in matrix algebras, one tends to say “skew-hermitian” instead of “purely imaginary”. There is no clear analogue of “imaginary” here since $z \notin \mathbb{R}$ and $z \ne \overline{z}$ diverge.)

Beware that often one says just “imaginary” for “purely imaginary”. (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 complex numbers, on the other hand, form the Lie algebra $\mathfrak{u}(1)$. Often one substitutes $\mathbb{R}$ (the algebra of real numbers), which is simpler, when one only cares 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.

Related concepts

• imaginary unit

• imaginary part, real part