Theorems 430–434 · Definitions 89–90
Theorem 430 (Landau 298). [Ξ + Θ, 0] = [Ξ, 0] + [Θ, 0]. [Ξ − Θ, 0] = [Ξ, 0] − [Θ, 0]. [Ξ • Θ, 0] = [Ξ, 0] • [Θ, 0]. [Ξ/Θ, 0] = [Ξ, 0]/[Θ, 0] if Θ ≠ 0. [−Ξ, 0] = −[Ξ, 0]. |[Ξ, 0]| = |Ξ|.
Proof: (1) [Ξ, 0] + [Θ, 0] = [Ξ + Θ, 0 + 0]. (2) [Ξ, 0] − [Θ, 0] = [Ξ − Θ, 0 − 0]. (3) [Ξ, 0] • [Θ, 0] = [Ξ • Θ − 0 • 0, Ξ • 0 + 0 • Θ]. (4) By (3), [Θ, 0] • [Ξ/Θ, 0] = [Θ • (Ξ/Θ), 0] = [Ξ, 0]. (5) Theorem 346. (6) |[Ξ, 0]| = √(Ξ • Ξ) = |Ξ|, since |Ξ| ≧ 0 and |Ξ| • |Ξ| = Ξ • Ξ.
Theorem 431 (Landau 299). What holds of the positive whole numbers p under the adding of 1 holds of the numbers [p, 0] under the adding of e: [1, 0] is first among them; [p, 0] + e = [p + 1, 0] is one of them; it is never [1, 0]; different p give different [p, 0]; and whatever holds of [1, 0], and holds of [p + 1, 0] whenever it holds of [p, 0], holds of each [p, 0].
By Theorem 430 and Definition 73, each of these is the same statement about p. Landau states this as: the [x, 0] satisfy the five axioms of the natural numbers. The present system does not rest on those axioms, so the theorem is stated as what it comes to.
Definition 89 (Landau 72). When a real number Ξ is applied to a twofold operand, or is combined with complex numbers, it is taken as [Ξ, 0]. So 0 is written for o and 1 for e. [Ξ, 0] is called real, rational, irrational, whole, positive or negative as Ξ is.
Landau says here that the symbol Ξ is "freed of its previous meaning" and that the old real numbers are replaced. Nothing is replaced. [Ξ, 0] as addend adds Ξ to the first and nothing to the second; as multiplier it takes each of the two Ξ-fold. That is Ξ itself applied to a twofold operand. Ξ and [Ξ, 0] are the same where both apply, and Theorem 430 is the proof that they agree in every operation.
Definition 90 (Landau 73). i = [0, 1].
Theorem 432 (Landau 300). i • i = −1.
[0, 1] • [0, 1] = [0 • 0 − 1 • 1, 0 • 1 + 1 • 0] = [−1, 0].
No real number does this, since Ξ • Ξ ≧ 0 for every real Ξ (Definition 71). That is why i needs a twofold operand. Elsewhere i is named only as the root of the binomial x • x + 1. Here it has an operand and a definition, and that it is such a root is proved.
Theorem 433 (Landau 301). For real u₁ and u₂, u₁ + u₂ • i = [u₁, u₂]. So for each complex number x there is exactly one pair of real numbers u₁, u₂ with x = u₁ + u₂ • i.
Proof: [u₁, 0] + [u₂, 0] • [0, 1] = [u₁, 0] + [u₂ • 0 − 0 • 1, u₂ • 1 + 0 • 0] = [u₁, 0] + [0, u₂] = [u₁, u₂]. The rest is Definition 73.
So the bracket is no longer needed. The complex numbers are the numbers u₁ + u₂ • i with u₁ and u₂ real, and they are added, subtracted and multiplied by the laws already proved, with i • i = −1.
This ends Landau's book.
Theorem 434. For each complex number x there is a w with w • w = x.
Proof: Let x = [Ξ₁, Ξ₂].
(1) Let Ξ₂ = 0. If Ξ₁ ≧ 0, let w = [√Ξ₁, 0]. If Ξ₁ < 0, let w = [0, √|Ξ₁|]; then w • w = [0 − |Ξ₁|, 0] = [Ξ₁, 0].
(2) Let Ξ₂ ≠ 0. Then |x| • |x| = Ξ₁Ξ₁ + Ξ₂Ξ₂ > Ξ₁Ξ₁, so |x| > |Ξ₁| (Lemma to Theorem 397), and both |x| + Ξ₁ and |x| − Ξ₁ are positive. Let P = √((|x| + Ξ₁)/2) and Q = √((|x| − Ξ₁)/2). Then P • P − Q • Q = Ξ₁. And (P • Q) • (P • Q) = (|x| • |x| − Ξ₁Ξ₁)/4 = (Ξ₂Ξ₂)/4, so P • Q = |Ξ₂|/2 (Theorem 255). Let w = [P, Q] if Ξ₂ > 0, and w = [P, −Q] if Ξ₂ < 0. Then w • w = [P • P − Q • Q, Ξ₂] = [Ξ₁, Ξ₂].
So with i the taking of a side is always possible. In particular i itself has a side.