imaginary number (changes) in nLab

Showing changes from revision #6 to #7: Added | Removed | Changed

Contents

Definition Definitions

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

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

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

People The often purely get imaginary these complex two numbers, notions on mixed the up. other (For hand, example, form the2+3i2 + 3\mathrm{i}Lie algebra is imaginary but not purely imaginary; while0𝔲(1) 0 \mathfrak{u}(1) . is Often the one unique substitutes purely imaginary number that is not imaginary.) This may be because the imaginary numbers, as is typical for things defined by aninequalityℝ\mathbb{R} , do (the not algebra form of an real interesting numbers), collection which as is a simpler, whole when (for one example, only they cares are about not this even Lie closed algebra under up addition). to Compare irrational isomorphism number . However, usingiℝ\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.

The purely imaginary complex numbers, on the other hand, form the Lie algebra 𝔲(1)\mathfrak{u}(1). Often people substitute ℝ\mathbb{R} (the algebra of real numbers), which is simpler, when they only care about this Lie algebra up to isomorphism. However, using iℝ\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 ℑz\Im{z} is apart from zero, or equivalently if zz is apart from ℜz\Re{z}. This all generalizes to other kinds of hypercomplex numbers.

• imaginary unit

• imaginary part, real part