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.