represents the imaginary unit .



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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:

Wolfram Research (1988), I, Wolfram Language function, https://reference.wolfram.com/language/ref/I.html (updated 2002).


Wolfram Research (1988), I, Wolfram Language function, https://reference.wolfram.com/language/ref/I.html (updated 2002).


Wolfram Language. 1988. "I." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/I.html.


Wolfram Language. (1988). I. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/I.html


@misc{reference.wolfram_2023_i, author="Wolfram Research", title="{I}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/I.html}", note=[Accessed: 07-December-2023 ]}


@online{reference.wolfram_2023_i, organization={Wolfram Research}, title={I}, year={2002}, url={https://reference.wolfram.com/language/ref/I.html}, note=[Accessed: 07-December-2023 ]}