I formats as :

The typeset form  can be entered as ii (for "imaginary i"):

Generate from square roots of negative real numbers:

Use I in exact and approximate calculations:

Built-in mathematical functions work with complex numbers:

Extract imaginary parts:

Use jj to enter the engineering notation  for I:

Use as a direction in infinite quantities:

Use as a direction in Limit:

Use as a generator of extension fields:

Factor integers over the Gaussians:

Use as an expansion point for series:

Convert a complex number from polar to rectangular form:

Flow around a cylinder as the real part of a complex‐valued function:

I is represented as a complex number with vanishing real part:

I is an exact number:

Use ComplexExpand to extract real and imaginary parts:

Use ExpToTrig to convert exponentials containing I into trigonometric form:

Simplify expressions containing I:

I is an algebraic number:

Trigonometric functions with purely imaginary arguments evaluate to simpler forms:

Obtain I in solutions of polynomial equations:

Roots of quadratic polynomials can evaluate to complex numbers:

Use Chop to remove small imaginary parts:

Use I as limits of integration:

Sort numbers by increasing imaginary parts:

Evaluated complex numbers are atomic objects and do not explicitly contain I:

Patterns of the form Complex[x_,y_] can be used to match the whole complex number:

If I is inside of a held expression, it will not become an expression with head Complex:

Compare with the evaluated form:

In particular, an unevaluated I is a symbol rather than a number:

Machine‐precision evaluation of I yields an approximate zero real part:

Arbitrary‐precision evaluation yields an exact zero real part:

Disguised purely real quantities that contain I cannot be used in numerical comparisons:

Use FullSimplify or ComplexExpand to convert to manifestly real expressions first:

Finite imaginary quantities are absorbed by infinite real or complex quantities:

I cannot be used in intervals:

Nested powers of I:

Find the limit in closed form:

Generate all possible nestings of powers of I:

Plot the points in the complex plane: