Definition 81 (Landau 65). For x = [Ξ₁, Ξ₂], the number x* = [Ξ₁, −Ξ₂] is the conjugate of x.
Conjugates
Definition 81 · Theorems 389–395
Theorem 389 (Landau 257). (x*)* = x.
[Ξ₁, −(−Ξ₂)] = [Ξ₁, Ξ₂].
Theorem 390 (Landau 258). x* = o if and only if x = o.
Ξ₁ = 0 and −Ξ₂ = 0 is the same as Ξ₁ = 0 and Ξ₂ = 0.
Theorem 391 (Landau 259). x* = x if and only if x is of the form [Ξ, 0].
−Ξ₂ = Ξ₂ if and only if Ξ₂ = 0, by Theorem 310.
Theorem 392 (Landau 260). (x + y)* = x* + y*.
[Ξ₁ + Θ₁, −(Ξ₂ + Θ₂)] = [Ξ₁ + Θ₁, −Ξ₂ + (−Θ₂)].
Theorem 393 (Landau 261). (x • y)* = x* • y*.
[Ξ₁Θ₁ − Ξ₂Θ₂, −(Ξ₁Θ₂ + Ξ₂Θ₁)] = [Ξ₁Θ₁ − (−Ξ₂)(−Θ₂), Ξ₁(−Θ₂) + (−Ξ₂)Θ₁] = [Ξ₁, −Ξ₂] • [Θ₁, −Θ₂].
Theorem 394 (Landau 262). (x − y)* = x* − y*.
x = (x − y) + y, so x* = (x − y)* + y* by Theorem 392.
Theorem 395 (Landau 263). If y ≠ o, then (x/y)* = x*/y*.
y* ≠ o by Theorem 390. x = (x/y) • y, so x* = (x/y)* • y* by Theorem 393.