WOLFRAM

I

represents the imaginary unit .

Details

Examples

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Basic Examples  (3)Summary of the most common use cases

I formats as :

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

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Generate from square roots of negative real numbers:

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Use I in exact and approximate calculations:

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Scope  (2)Survey of the scope of standard use cases

Built-in mathematical functions work with complex numbers:

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Extract imaginary parts:

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Generalizations & Extensions  (6)Generalized and extended use cases

Use jj to enter the engineering notation for I:

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Use as a direction in infinite quantities:

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Use as a direction in Limit:

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Use as a generator of extension fields:

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Factor integers over the Gaussians:

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Use as an expansion point for series:

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Applications  (2)Sample problems that can be solved with this function

Convert a complex number from polar to rectangular form:

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

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Properties & Relations  (12)Properties of the function, and connections to other functions

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

I is an exact number:

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Use ComplexExpand to extract real and imaginary parts:

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Use ExpToTrig to convert exponentials containing I into trigonometric form:

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Simplify expressions containing I:

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I is an algebraic number:

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Trigonometric functions with purely imaginary arguments evaluate to simpler forms:

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Obtain I in solutions of polynomial equations:

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Roots of quadratic polynomials can evaluate to complex numbers:

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Use Chop to remove small imaginary parts:

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Use I as limits of integration:

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Sort numbers by increasing imaginary parts:

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Possible Issues  (6)Common pitfalls and unexpected behavior

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

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

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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:

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Machineprecision evaluation of I yields an approximate zero real part:

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Arbitraryprecision evaluation yields an exact zero real part:

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

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

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

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I cannot be used in intervals:

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Neat Examples  (2)Surprising or curious use cases

Nested powers of I:

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Find the limit in closed form:

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Generate all possible nestings of powers of I:

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Plot the points in the complex plane:

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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).

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 ]}

@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 ]}

@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 ]}