nLab
imaginary number

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Definition

A complex number zz is called imaginary if it is not real; it is purely imaginary if its real part z\Re{z} is zero, hence if under complex conjugation we have z¯=z\overline{z} = - z.

The same terminology applies to quaternions and octonions. (In more general star-algebras, such as in matrix algebras, one tends to say “skew-hermitian” instead of “purely imaginary”.)

Beware that often one says just “imaginary” for “purely imaginary”.

People often get these two notions mixed up. (For example, 2+3i2 + 3\mathrm{i} is imaginary but not purely imaginary; while 00 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 𝔲(1)\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 i\mathrm{i}\mathbb{R} (the algebra of purely imaginary numbers) makes 𝔲(1)\mathfrak{u}(1) fit with the matrix formulas used in higher dimensions.

In constructive mathematics

For purposes of constructive mathematics, we only accept zz as imaginary if its imaginary part z\Im{z} is apart from zero, or equivalently if zz is apart from z\Re{z}. This all generalizes to other kinds of hypercomplex numbers.

Last revised on April 27, 2020 at 10:09:26. See the history of this page for a list of all contributions to it.