I 
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
open allclose allBasic Examples (3)Summary of the most common use cases
I formats as :

https://wolfram.com/xid/0p2m-j38lbg

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

https://wolfram.com/xid/0p2m-fau3ev

Generate from square roots of negative real numbers:

https://wolfram.com/xid/0p2m-qoti6

Use I in exact and approximate calculations:

https://wolfram.com/xid/0p2m-fou7ea


https://wolfram.com/xid/0p2m-cdn33q

Scope (2)Survey of the scope of standard use cases
Generalizations & Extensions (6)Generalized and extended use cases
Use jj
to enter the engineering notation for I:

https://wolfram.com/xid/0p2m-bvaaf2

Use as a direction in infinite quantities:

https://wolfram.com/xid/0p2m-bv51gx


https://wolfram.com/xid/0p2m-ecf5dv

Use as a direction in Limit:

https://wolfram.com/xid/0p2m-b9u4dk

Use as a generator of extension fields:

https://wolfram.com/xid/0p2m-gmtpc4


https://wolfram.com/xid/0p2m-f1ueqr

Factor integers over the Gaussians:

https://wolfram.com/xid/0p2m-css4z

Use as an expansion point for series:

https://wolfram.com/xid/0p2m-ej9f1g

Applications (2)Sample problems that can be solved with this function
Properties & Relations (12)Properties of the function, and connections to other functions
I is represented as a complex number with vanishing real part:

https://wolfram.com/xid/0p2m-gq3ox5

I is an exact number:

https://wolfram.com/xid/0p2m-dp7bzc

Use ComplexExpand to extract real and imaginary parts:

https://wolfram.com/xid/0p2m-g3cvu6

Use ExpToTrig to convert exponentials containing I into trigonometric form:

https://wolfram.com/xid/0p2m-gncjc5

Simplify expressions containing I:

https://wolfram.com/xid/0p2m-fvcoau


https://wolfram.com/xid/0p2m-dhixjr

I is an algebraic number:

https://wolfram.com/xid/0p2m-oba3l

Trigonometric functions with purely imaginary arguments evaluate to simpler forms:

https://wolfram.com/xid/0p2m-0f18d


https://wolfram.com/xid/0p2m-hb4nvp

Obtain I in solutions of polynomial equations:

https://wolfram.com/xid/0p2m-qngszv

Roots of quadratic polynomials can evaluate to complex numbers:

https://wolfram.com/xid/0p2m-il5cym

Use Chop to remove small imaginary parts:

https://wolfram.com/xid/0p2m-fj091z


https://wolfram.com/xid/0p2m-bfzwxp

Use I as limits of integration:

https://wolfram.com/xid/0p2m-hbgyu4


https://wolfram.com/xid/0p2m-bgb3y5

Sort numbers by increasing imaginary parts:

https://wolfram.com/xid/0p2m-lgd927

Possible Issues (6)Common pitfalls and unexpected behavior
Evaluated complex numbers are atomic objects and do not explicitly contain I:

https://wolfram.com/xid/0p2m-cyo6ip


https://wolfram.com/xid/0p2m-e572tr


https://wolfram.com/xid/0p2m-b0hdu

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

https://wolfram.com/xid/0p2m-573i47


https://wolfram.com/xid/0p2m-2eitph

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

https://wolfram.com/xid/0p2m-1a08ur

Compare with the evaluated form:

https://wolfram.com/xid/0p2m-teprh4

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

https://wolfram.com/xid/0p2m-dwjgkg


https://wolfram.com/xid/0p2m-jkz9mr

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

https://wolfram.com/xid/0p2m-fsros9

Arbitrary‐precision evaluation yields an exact zero real part:

https://wolfram.com/xid/0p2m-e3pew1

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

https://wolfram.com/xid/0p2m-lnets6



https://wolfram.com/xid/0p2m-h6gkcf

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

https://wolfram.com/xid/0p2m-983kq


https://wolfram.com/xid/0p2m-lcutla

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

https://wolfram.com/xid/0p2m-kt0r7j


https://wolfram.com/xid/0p2m-bzay1e

I cannot be used in intervals:

https://wolfram.com/xid/0p2m-mf5n3k

Neat Examples (2)Surprising or curious use cases
Nested powers of I:

https://wolfram.com/xid/0p2m-gpao8l

Find the limit in closed form:

https://wolfram.com/xid/0p2m-edhwkt



https://wolfram.com/xid/0p2m-fj98iw

Generate all possible nestings of powers of I:

https://wolfram.com/xid/0p2m-ii9glx

Plot the points in the complex plane:

https://wolfram.com/xid/0p2m-ca2h6e

Wolfram Research (1988), I, Wolfram Language function, https://reference.wolfram.com/language/ref/I.html (updated 2002).
Text
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).
CMS
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. Wolfram Research. Last Modified 2002. https://reference.wolfram.com/language/ref/I.html.
APA
Wolfram Language. (1988). I. Wolfram Language & System Documentation Center. Retrieved from 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
BibTeX
@misc{reference.wolfram_2025_i, author="Wolfram Research", title="{I}", year="2002", howpublished="\url{https://reference.wolfram.com/language/ref/I.html}", note=[Accessed: 20-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_i, organization={Wolfram Research}, title={I}, year={2002}, url={https://reference.wolfram.com/language/ref/I.html}, note=[Accessed: 20-June-2025
]}