# I represents the imaginary unit .

# Details • Numbers containing I are converted to the type Complex.
• I can be entered in StandardForm and InputForm as , ii or \[ImaginaryI].
• , jj and \[ImaginaryJ] can also be used.
• In StandardForm and TraditionalForm, I is output as .

# Examples

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## Basic Examples(3)

I can be entered as ii (for "imaginary i"):

Generate from square roots of negative real numbers:

Use I in exact and approximate calculations:

## Scope(2)

Built-in mathematical functions work with complex numbers:

Extract imaginary parts:

## Generalizations & Extensions(6)

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:

## Applications(2)

Convert a complex number from polar to rectangular form:

Flow around a cylinder as the real part of a complexvalued function:

## Properties & Relations(12)

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:

## Possible Issues(9)

Machineprecision evaluation of I yields an approximate zero real part:

Arbitraryprecision evaluation yields an exact zero real part:

Real and imaginary parts of complex numbers can have different precisions:

Arithmetic operations will typically mix them:

The overall precision of a complex number depends on both real and imaginary parts:

Complex numbers are atomic objects and do not explicitly contain I:

Disguised purely real quantities that contain I cannot be used in numerical comparisons: Real roots of irreducible cubics still contain I in their algebraic forms:

Machine-precision numerical evaluation gives a spurious imaginary part:

Arbitrary-precision evaluation still leaves an imaginary part:

Use Reduce with an option to get explicitly real roots:

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

I cannot be used in intervals:

The symbol I needs to be evaluated to become a complex number:

## Neat Examples(2)

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: