Definition 60 (Landau 44). |Ξ| is ξ if Ξ is +ξ or −ξ, and is 0 if Ξ is 0. It is the absolute value of Ξ: the bound that defines the operation.
Ordering
Definitions 60–67 · Theorems 300–308
Theorem 300 (Landau 166). |Ξ| is positive for positive and for negative Ξ. Definition 60.
Definition 61 (Landau 45). If Ξ and Θ are not both positive, then Ξ > Θ when: Ξ and Θ are negative and |Ξ| < |Θ|; or Ξ = 0 and Θ is negative; or Ξ is positive and Θ is negative; or Ξ is positive and Θ = 0.
The sense is that Ξ, applied to any bound great enough for both to apply, yields more than Θ does.
Definition 62 (Landau 46). Ξ < Θ means Θ > Ξ.
For positive Ξ and Θ both definitions agree with Definitions 45 and 46.
Theorem 301 (Landau 167). For any Ξ and Θ, exactly one of Ξ = Θ, Ξ > Θ, Ξ < Θ is the case. For positive Ξ and Θ this is Theorem 217. If Ξ is positive and Θ is 0 or negative, or Ξ is 0 and Θ is negative: Ξ ≠ Θ, Ξ > Θ, and not Ξ < Θ, by Definitions 61 and 62. If Ξ is 0 and Θ is positive, or Ξ is negative and Θ is 0 or positive: Ξ ≠ Θ, not Ξ > Θ, and Ξ < Θ. If both are 0 they are equal and neither is greater. If both are negative, then by Theorem 217 exactly one of |Ξ| < |Θ|, |Ξ| = |Θ|, |Ξ| > |Θ| holds, giving Ξ > Θ, Ξ = Θ, Ξ < Θ respectively.
Definition 63 (Landau 47). Ξ ≧ Θ means Ξ > Θ or Ξ = Θ.
Definition 64 (Landau 48). Ξ ≦ Θ means Ξ < Θ or Ξ = Θ.
Theorem 302 (Landau 168). If Ξ ≧ Θ then Θ ≦ Ξ, and conversely. Definition 62.
Theorem 303 (Landau 169). The positive numbers are those greater than 0, and the negative numbers are those less than 0.
By Definition 61, +ξ > 0, and if Ξ > 0 then Ξ is some +ξ. By Definition 62, −ξ < 0, and if Ξ < 0 then Ξ is some −ξ.
Theorem 304 (Landau 170). |Ξ| ≧ 0. Definition 60 and Theorems 300 and 303.
Theorem 305 (Landau 171). If Ξ < Θ and Θ < Λ, then Ξ < Λ. (Transitivity of Ordering)
Proof: If Λ > 0: when Ξ > 0, then Θ > 0 and this is Theorem 220; when Ξ ≦ 0, certainly Ξ < Λ. If Λ = 0: then Θ < 0, so Ξ < 0 and Ξ < Λ. If Λ < 0: then Θ < 0 and Ξ < 0, with |Ξ| > |Θ| and |Θ| > |Λ|, so |Ξ| > |Λ| and Ξ < Λ.
Theorem 306 (Landau 172). If Ξ ≦ Θ and Θ < Λ, or Ξ < Θ and Θ ≦ Λ, then Ξ < Λ.
Obvious if an equality holds; otherwise Theorem 305.
Theorem 307 (Landau 173). If Ξ ≦ Θ and Θ ≦ Λ, then Ξ ≦ Λ.
Obvious if two equalities hold; otherwise Theorem 306.
Definition 65 (Landau 49). A number Ξ ≦ 0 is rational if it is 0, or if it is negative and |Ξ| is a rational bound.
Definition 66 (Landau 50). A negative number is irrational if it is not rational.
Definition 67 (Landau 51). A number Ξ ≦ 0 is whole if it is 0, or if it is negative and |Ξ| is a whole bound.
So there are positive whole numbers, 0, and negative whole numbers. These are the whole derived numbers +n, 0 and −n with which the order of derivation began.
Theorem 308 (Landau 174). Every whole number is rational.
Known for positive numbers. For 0 and negative numbers, by Definitions 65 and 67.