Field theory and polynomials Proof
Proof that i squared is minus one
Builds on Imaginary unitComplex numbers
Proof
- The imaginary unit is the complex number , with real part and imaginary part .
- The product of complex numbers is .
- Take and . Then .
- The complex number is the real number , since sits in as the numbers . So .
Idea
The rule for multiplying complex numbers is built so that the two imaginary parts multiply to a negative real number. For this gives exactly .
Reading path
To understand this page from the ground up, read these 3 pages in order. The foundations come first.
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- Complex numbers object
- Imaginary unit object
- Proof that i squared is minus one proof
Where this leads
159 pages build on this one.
- exp z = 1 iff z is an integer multiple of 2πi
- Residue at a simple pole
- The complex numbers cannot be ordered
- Arguments add under multiplication
- Characteristic function of a sum of independent random variables
- The complex exponential is 2πi-periodic
- Residue theorem
- de Moivre's formula
- Euler's formula
- The nth roots of unity
- Ptolemy's theorem
- De Moivre–Laplace theorem
- Existence of Laurent expansions on an annulus
- i squared is minus one
- Euler's identity
- Power series of the complex logarithm
- Solutions for complex conjugate roots
- Weak law of large numbers
- Polar form
- exp(2 pi i/n) is a primitive root of unity
- Impossibility of doubling the cube
- Rouché's theorem
- Strong law of large numbers
- Number of nth roots of unity
- Central limit theorem
- Impossibility of trisecting the angle
- Lévy's continuity theorem
- Transcendence of pi
- Inscribed angle theorem
- Angles in the same segment are equal
- Residue
- Underdamped
- Quadratic formula
- Characteristic function
- Principal part
- Characteristic function of the normal distribution
- Sum of independent normal random variables
- Classification of isometries of the plane
- Argument principle
- The constructible numbers form a field closed under square roots
- Jordan's lemma
- Damped harmonic oscillator
- Wedderburn's little theorem
- Pendulum equation
- Irreducibility of the cyclotomic polynomials over the rationals
- Hurwitz's theorem
- The degree of a constructible number is a power of two
- The characteristic function determines the distribution
- Forced oscillation and resonance
- Impossibility of squaring the circle
- The principal logarithm is holomorphic off the branch cut
- Gauss–Wantzel theorem
- Harmonic oscillator
- Existence of nth roots of complex numbers
- Proof of the argument principle by the residue theorem
- Proof of Wedderburn's little theorem by the class equation and cyclotomic polynomials
- Sample average approximation
- Proof of the impossibility of doubling the cube
- Simple harmonic motion
- Proof of de Moivre's formula from Euler's formula
- Ergodic decomposition
- Formally real field
- Proof of the impossibility of squaring the circle
- Proof of the transcendence of pi from the Lindemann–Weierstrass theorem
- Koksma–Hlawka inequality
- Circle criterion
- Cramér–Wold theorem
- Proof of the solutions for complex conjugate roots
- Anharmonic oscillator
- Consistency of the maximum likelihood estimator
- Perron's irreducibility criterion
- Asymptotic equipartition property
- Phonon dispersion in lattices
- Proof of Euler's identity
- Proof of the description of the nth roots of unity
- Finite Desarguesian planes are Pappian
- Splitting circle method
- Source–channel separation theorem
- Proof that the constructible numbers form the smallest subfield of the complex numbers closed under square roots
- Proof that exp(2 pi i/n) is a primitive root of unity
- Kingman's subadditive ergodic theorem
- Asymptotic normality of the maximum likelihood estimator
- Typical set
- Brunt–Väisälä frequency
- Damping
- Edgeworth expansion
- Proof that the complex numbers cannot be ordered
- Gain and phase margin
- Lorentz oscillator model
- Proof of Jordan's lemma by Jordan's inequality
- Cyclotomic field
- Proof of the residue formula at a simple pole
- Erdős–Kac theorem
- Proof of the residue theorem by subtracting principal parts
- Proof that there are exactly n nth roots of unity
- Euler's elastica
- Proof of the de Moivre–Laplace theorem from the central limit theorem
- Inverted pendulum
- Proof that the complex exponential is 2πi-periodic
- Liénard equation
- Monte Carlo integration
- Delta method
- Proof that the degree of a constructible number is a power of two
- RLC circuit
- Proof that the principal logarithm is holomorphic off the branch cut
- Nyquist stability criterion
- Shannon–Hartley theorem
- Proof that exp z = 1 iff z is an integer multiple of 2πi
- Matsubara frequency
- Proof of the sum of independent normal random variables by characteristic functions
- Proof of the irreducibility of the cyclotomic polynomials by reduction modulo primes
- Proof of the central limit theorem by characteristic functions
- Berry–Esseen theorem
- Parametric oscillator
- Furuta pendulum
- Proof of the existence of nth roots of complex numbers
- Proof of Rouché's theorem by the argument principle and continuity
- Birkhoff's ergodic theorem
- Proof of the power series of the complex logarithm
- Q factor
- Random vibration
- Ideal chain
- Random vibration
- Proof of the formula for the characteristic function of the normal distribution
- Oseledets theorem
- Veblen–Young theorem
- Proof of Ptolemy's theorem by complex numbers
- Parametric oscillator
- Nörlund–Rice integral
- Noisy-channel coding theorem
- Proof that angles in the same segment are equal
- Anharmonic oscillator
- Proof of the impossibility of trisecting the angle
- Flory–Huggins solution theory
- Elastic pendulum
- Liouville's theorems on elliptic functions
- Proof of the polar form of a complex number
- Proof of the existence and uniqueness of Laurent expansions on an annulus
- Kapitza's pendulum
- Kelly criterion
- Bertrand's theorem
- Proof of the Gauss–Wantzel theorem
- Spherical pendulum
- Jury stability criterion
- Pesin theory
- Reptation
- Uniqueness theorem for Fourier–Stieltjes transforms
- Impossibility of the classical construction problems
- Van der Pol oscillator
- Proof of the weak law of large numbers by characteristic functions
- Proof of Euler's formula by power series
- Monte Carlo methods in finance
- Proof of the product formula for the characteristic function of a sum of independent random variables
- Artin–Schreier theorem on formally real fields
- Proof of the classification of isometries of the plane
- Worm-like chain
- Proof of the inscribed angle theorem by complex numbers
- Proof of Hurwitz's theorem by Rouché's theorem
- Residue calculus for Laplace inversion
Further reading
- Web https://en.wikipedia.org/wiki/Imaginary_unit, Definition
About this page
| Tag | EVARG9RYTF |
|---|---|
| URI | m2://EVARG9RYTF |
| Kind | proof |
| Writing | W0 stub |
| MSC | 12D99 |
| Proves | i squared is minus one |
| Uses | Imaginary unit, Complex numbers |
| Changed | 4 November 2026 |
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