Definitions 79–80 · Theorems 352–361
Definition 79 (Landau 63). x • y = [Ξ₁ • Θ₁ − Ξ₂ • Θ₂, Ξ₁ • Θ₂ + Ξ₂ • Θ₁]. It is the product of x and y.
The sense: x as multiplier, applied to y. By the remark to Definition 73 it does Ξ₁-fold of (Θ₁, Θ₂) together with Ξ₂-fold of the exchanged (−Θ₂, Θ₁). That is the pair above.
In the proofs below, sums and products of several real numbers are written without brackets where Theorems 320 and 333 allow any bracketing, and their terms are reordered by Theorems 309 and 328.
Theorem 352 (Landau 220). x • y = y • x. (Commutative Law of Multiplication)
[Ξ₁ • Θ₁ − Ξ₂ • Θ₂, Ξ₁ • Θ₂ + Ξ₂ • Θ₁] = [Θ₁ • Ξ₁ − Θ₂ • Ξ₂, Θ₁ • Ξ₂ + Θ₂ • Ξ₁], by Theorems 328 and 309.
Theorem 353 (Landau 221). x • y = o if and only if at least one of x, y is o.
Proof: (1) If x = o, then x • y = [0 • Θ₁ − 0 • Θ₂, 0 • Θ₂ + 0 • Θ₁] = o. If y = o, then x • y = y • x = o.
(2) Let x • y = o and y ≠ o. Then Θ = Θ₁ • Θ₁ + Θ₂ • Θ₂ > 0, since each term is ≧ 0 and not both are 0 (Definition 71). By hypothesis
Ξ₁ • Θ₁ − Ξ₂ • Θ₂ = 0 and Ξ₁ • Θ₂ + Ξ₂ • Θ₁ = 0.
Multiply the first by Θ₁, the second by Θ₂, and add. The terms Ξ₂ • Θ₂ • Θ₁ cancel, leaving Ξ₁ • Θ = 0, so Ξ₁ = 0 by Theorem 326. Then Ξ₂ • Θ₂ = 0 and Ξ₂ • Θ₁ = 0, and since Θ₁, Θ₂ are not both 0, Ξ₂ = 0.
Theorem 354 (Landau 222). x • e = x.
[Ξ₁ • 1 − Ξ₂ • 0, Ξ₁ • 0 + Ξ₂ • 1] = [Ξ₁, Ξ₂].
Theorem 355 (Landau 223). x • (−e) = −x.
[Ξ₁ • (−1) − Ξ₂ • 0, Ξ₁ • 0 + Ξ₂ • (−1)] = [−Ξ₁, −Ξ₂].
Theorem 356 (Landau 224). (−x) • y = x • (−y) = −(x • y).
Proof: (−x) • y = [(−Ξ₁) • Θ₁ − (−Ξ₂) • Θ₂, (−Ξ₁) • Θ₂ + (−Ξ₂) • Θ₁] = [−(Ξ₁ • Θ₁ − Ξ₂ • Θ₂), −(Ξ₁ • Θ₂ + Ξ₂ • Θ₁)] = −(x • y), by Theorems 331, 314 and 315. And x • (−y) = (−y) • x = −(y • x) = − (x • y).
Theorem 357 (Landau 225). (−x) • (−y) = x • y.
(−x) • (−y) = x • (−(−y)) = x • y, by Theorems 356 and 347.
Theorem 358 (Landau 226). (x • y) • z = x • (y • z). (Associative Law of Multiplication)
Proof: By Definition 79 and Theorems 335 and 336,
(x • y) • z = [(Ξ₁Θ₁ − Ξ₂Θ₂) • Λ₁ − (Ξ₁Θ₂ + Ξ₂Θ₁) • Λ₂, (Ξ₁Θ₁ − Ξ₂Θ₂) • Λ₂ + (Ξ₁Θ₂ + Ξ₂Θ₁) • Λ₁]
= [Ξ₁Θ₁Λ₁ − (Ξ₂Θ₂Λ₁ + Ξ₁Θ₂Λ₂ + Ξ₂Θ₁Λ₂), (Ξ₁Θ₁Λ₂ + Ξ₁Θ₂Λ₁ + Ξ₂Θ₁Λ₁) − Ξ₂Θ₂Λ₂].
Again by Definition 79 and Theorems 335 and 336,
x • (y • z) = [Ξ₁ • (Θ₁Λ₁ − Θ₂Λ₂) − Ξ₂ • (Θ₁Λ₂ + Θ₂Λ₁), Ξ₁ • (Θ₁Λ₂ + Θ₂Λ₁) + Ξ₂ • (Θ₁Λ₁ − Θ₂Λ₂)]
= [Ξ₁Θ₁Λ₁ − (Ξ₁Θ₂Λ₂ + Ξ₂Θ₁Λ₂ + Ξ₂Θ₂Λ₁), (Ξ₁Θ₁Λ₂ + Ξ₁Θ₂Λ₁ + Ξ₂Θ₁Λ₁) − Ξ₂Θ₂Λ₂].
The two pairs have the same terms in a different order. So they are the same, by Theorems 309 and 320.
Theorem 359 (Landau 227). x • (y + z) = x • y + x • z. (Distributive Law)
Proof: x • (y + z) = [Ξ₁ • (Θ₁ + Λ₁) − Ξ₂ • (Θ₂ + Λ₂), Ξ₁ • (Θ₂ + Λ₂) + Ξ₂ • (Θ₁ + Λ₁)]
= [(Ξ₁Θ₁ − Ξ₂Θ₂) + (Ξ₁Λ₁ − Ξ₂Λ₂), (Ξ₁Θ₂ + Ξ₂Θ₁) + (Ξ₁Λ₂ + Ξ₂Λ₁)] = x • y + x • z,
by Theorems 335 and 314.
Theorem 360 (Landau 228). x • (y − z) = x • y − x • z.
x • (y + (−z)) = x • y + x • (−z) = x • y + (−(x • z)).
Theorem 361 (Landau 229). If y ≠ o, the equation y • u = x has exactly one solution.
Proof: (1) At most one. If y • u₁ = y • u₂, then y • (u₁ − u₂) = o by Theorem 360, so u₁ − u₂ = o by Theorem 353, so u₁ = u₂ by Theorem 345.
(2) There is one. Θ = Θ₁ • Θ₁ + Θ₂ • Θ₂ > 0. Let u = [Θ₁/Θ, −(Θ₂/Θ)] • x. Then
y • u = ([Θ₁, Θ₂] • [Θ₁/Θ, −(Θ₂/Θ)]) • x = [(Θ₁Θ₁ + Θ₂Θ₂)/Θ, (−(Θ₁Θ₂) + Θ₂Θ₁)/Θ] • x = [1, 0] • x = x.
Definition 80 (Landau 64). The u of Theorem 361 is the quotient of x by y, written x/y.