i

E157389

i is the imaginary unit, a complex number satisfying i² = −1 and serving as a fundamental building block in complex number systems such as the Gaussian integers.

All labels observed (1)

Label Occurrences
i canonical 1

How this entity was disambiguated

Statements (47)

Predicate Object
instanceOf complex number ⓘ
imaginary unit ⓘ
absoluteValue 1 ⓘ
additiveInverse -i ⓘ
alternativeNotation j (in electrical engineering) ⓘ
appearsInIdentity e^{iπ} + 1 = 0 ⓘ
argument π/2 ⓘ
basisElementOf C as a 2-dimensional R-vector space ⓘ
belongsTo Gaussian integers ⓘ
complex numbers ⓘ
cartesianForm 0 + 1i ⓘ
conjugate -i ⓘ
EulerFormulaSpecialCase e^{iπ/2} = i ⓘ
fieldExtensionOf real numbers ⓘ
generatesField R(i) = C ⓘ
hasPrincipalLogarithm i(π/2 + 2kπ) for k=0 ⓘ
imaginaryPart 1 ⓘ
isAlgebraicNumber true ⓘ
isNotReal true ⓘ
isPurelyImaginary true ⓘ
isRootOf x^2 + 1 = 0 ⓘ
isSolutionOf z^4 = 1 ⓘ
isUnitIn Gaussian integers ⓘ
iToThePower1 i ⓘ
iToThePower2 -1 ⓘ
iToThePower3 -i ⓘ
iToThePower4 1 ⓘ
liesOn imaginary axis ⓘ
liesOnUnitCircle true ⓘ
magnitude 1 ⓘ
minimalPolynomialOverR x^2 + 1 ⓘ
multiplicativeInverse -i ⓘ
normSquared 1 ⓘ
notationIntroducedBy Leonhard Euler ⓘ
orderUnderMultiplicationModuloPowers 4 ⓘ
orthogonalTo 1 in the standard inner product on C ⓘ
polarForm (1, π/2) ⓘ
e^{iπ/2} ⓘ
powerCycle [i, -1, -i, 1] ⓘ
realPart 0 ⓘ
satisfiesEquation i^2 = -1 ⓘ
square -1 ⓘ
unitGroupOrderInGaussianIntegers 4 ⓘ
usedIn complex analysis ⓘ
electrical engineering ⓘ
quantum mechanics ⓘ
signal processing ⓘ

How these facts were elicited

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.