Complex numbers

• Intuitively (and historically) we start with real numbers R,
  introduce an imaginary number i (sometimes denoted j)
  and require that i ² = -1.

  History:

  • The number i first appeared in print in Ars Magna by Cardano (1545), which contained Tartaglia's and Ferrari's formulas for the solutions of polynomial equations of degree 4.
  • The algebra of C was worked out in L'Algebra by Bombelli (1560, publ. 1572), who used complex arithmetic to find real roots of cubic equations.
  • Stereographic projection (introduced by Ptolemy c. 150) is discovered to be conformal (angle preserving) by Harriot (1590)
  • Albert Girard claims that every polynomial equation of degree n has n roots (L'invention en algébre, 1629).
  • Near miss at a geometrical interpretation by Wallis (1673).
  • Partial fraction decomposition of 1/(1+z²) by Johann Bernoulli (1702)
  • An n angle formula for tan by Johann Bernoulli via complex logs (1712)
  • ln(cosx+i sinx)=ix (Harmonia Mensurarum), (a, b) represented by a+ib - Cotes (1714)
  • Geometric factorization of 1-xⁿ, roots of unity - Cotes (1722)
  • exp(ix)=cosx+i sinx - Euler (1740, pub. 1748)
  • The first semi-rigorous proofs of the Fundamental Theorem of Algebra due to d'Alembert (1746) and Gauss (1799).
  • Cauchy-Riemann equations for the components of 2-d steady irrotational flow discovered by d'Alembert (1752).
  • Geometrical interpretation of complex numbers by Wessel (1799), Argand (1806), and Gauss, who gave complex numbers their name and their popularity.
  • Complex integrals, path deformation principle - Gauss (1811)
  • Argand publishes a simple proof of the Fundamental Theorem of Algebra (1814)
  • Development of complex calculus by Cauchy, Riemann, et al. (1814-1851).
  • Branch points, covering surfaces - Riemann's doctoral thesis (1851)
  • Genus, classification of closed surfaces by Möbius (1863)

  Definition: Complex numbers are those obtained by the usual algebraic operations of addition and multiplication from real numbers and i, subject to the distributive and associative laws and the extra condition i ² = -1.

  We see that any complex number can be written in the form a+ib, where a, b R.

• Algebraic definition: C = R [x]/(x ²+1)

  In other words, we take the polynomial ring in one variable over the R and mod out the ideal generated by x ²+1.

  In this formulation i is the equivalence class of x.

  Letting z = a+ib and w = c+id we obtain the following algebraic rules:

  • z+w = (a+c)+i(b+d)
  • z·w = (a·b-b·d)+i(b·c+a·d)

  Letting = a-ib (the conjugate of - the term coined by Cauchy in 1821) we obtain

  • a = (z+)/ 2
  • b = i·(-z)/ 2
  • z· = a ²+b ²

  z· is real, so if z is nonzero, it has a multiplicative inverse /(z·).
  Thus, C is a field.

• Geometric interpretation: C = R²

  A complex number z = a+ib represents the point p = (a, b) in the plane (using cartesian coordinates).

  Notation: a = Re z (real part of z), b = Im z (imaginary part of z).

  

  Relationship to polar coordinates: denote the Euclidean norm of the vector by r and the angle by . Then

  • r ² = a ²+b ² = z·
  • z = r(cos + i sin )

  Notation: r = |z| (modulus of z), = arg z (argument of z).

  If z is nonzero, the multivalued function arg z can be computed, e.g. as follows:
  

  • If a > 0, then = arctan (b/a)+2n
  • If a < 0, then = arctan (b/a)+(1+2n)
  • If a = 0 and b > 0, then = (1/2+2n)
  • If a = 0 and b < 0, then = (-1/2+2n)
  where n Z.

• Interpretation through linear algebra:

  C can be identified with the set of all 2 x 2 orthogonal matrices with real coefficients of the form:
  

  z = a+ib <-> A = [ a -b ]
                   [ b  a ]
  

  Complex multiplication is now interpreted as matrix multiplication:

  [ a -b ] . [ c -d ] = [ ac-bd  -(bc+ad)]
  [ b  a ]   [ d  c ]   [ bc+ad    ac-bd ]
  

  Of course, this is equivalent to the matrix-vector product:

  [ a -b ] . [ c ] = [ ac-bd ]
  [ b  a ]   [ d ]   [ bc+ad ]
  

  so we may think of complex multiplication as acting on R² or as composition of such actions.

  Many results about complex numbers follow immediately from standard results from linear algebra.

  • |z|² = det A. Thus, modulus is multiplicative.
  • = A^t. Thus, conjugation is multiplicative.
  • The fact that 1/z = / |z|² is a manifestation of Cramer's rule.

  Polar coordinate representation z = r(cos + i sin ) can be written as a matrix product

  [ r 0 ] . [ cos   -sin  ]
  [ 0 r ]   [ sin    cos  ]
  

  The first matrix is an isotropic dilation by r and the second is a rotation by .

  Needham calls this combination an amplitwist.

  Note that the first matrix is in the center of the matrix ring, so commutes with all other matrices.

Geometric interpretation of operations

• Conjugation = reflection with respect to the x-axis.
• Addition is the usual parallelogram law for vector addition and satisfies the triangle and Bernoulli inequalities:
  • |z₁+z₂| |z₁|+|z₂|
  • |z₁-z₂| |z₁|-|z₂|
  
• Matrix interpretation of multiplication (or a simple calculation) shows that the norms of the vectors are multiplied and the angles are added, i.e.
  • |z₁·z₂| = |z₁|·|z₂|
  • arg(z₁·z₂) = arg z₁+arg z₂
  
  

• Inversion = circle inversion followed by conjugation (1/z = / |z|²)
  

Check out real time Java operations on complex numbers.