Definition 71 (Landau 55). For positive Ξ and Θ the product Ξ • Θ is that of Definition 51. Otherwise:
If one is positive and the other negative, Ξ • Θ = −(|Ξ| • |Θ|).
If both are negative, Ξ • Θ = |Ξ| • |Θ|.
If either is 0, Ξ • Θ = 0.
The sense: to multiply the operation Θ by +ξ is to take Θ ξ-fold. To multiply it by −ξ is to take the contrary of Θ ξ-fold. So −ξ times −η is the contrary of removing η, taken ξ-fold, which is adding ξ • η. The rule of signs follows from what the operations are. It is not a convention.
Theorem 326 (Landau 192). Ξ • Θ = 0 if and only if at least one of Ξ, Θ is 0.
Definition 71, since |Ξ| • |Θ| is a bound and so not 0.
Theorem 327 (Landau 193). |Ξ • Θ| = |Ξ| • |Θ|. Definition 71.
Theorem 328 (Landau 194). Ξ • Θ = Θ • Ξ. (Commutative Law of Multiplication)
For positive numbers, Theorem 236. Otherwise Definition 71 is symmetric in Ξ and Θ, with | Ξ| • |Θ| = |Θ| • |Ξ|.
Theorem 329 (Landau 195). Ξ • 1 = Ξ.
For Ξ > 0, Theorem 245. For Ξ = 0, Definition 71. For Ξ < 0, Ξ • 1 = −(|Ξ| • 1) = −|Ξ| = Ξ.
Theorem 330 (Landau 196). If neither Ξ nor Θ is 0, then Ξ • Θ is |Ξ| • |Θ| when none or both are negative, and −(|Ξ| • |Θ|) when exactly one is negative. Definition 71.
Theorem 331 (Landau 197). (−Ξ) • Θ = Ξ • (−Θ) = −(Ξ • Θ).
Proof: If either is 0, all three are 0. Otherwise all three have absolute value |Ξ| • |Θ|, by Theorems 327 and 312. Among −Ξ, Θ and among Ξ, −Θ exactly one is negative when among Ξ, Θ none or both are, and conversely. Then Theorem 330.
Theorem 332 (Landau 198). (−Ξ) • (−Θ) = Ξ • Θ.
(−Ξ) • (−Θ) = −(Ξ • (−Θ)) = −(−(Ξ • Θ)) = Ξ • Θ, by Theorems 331 and 311.
Theorem 333 (Landau 199). (Ξ • Θ) • Λ = Ξ • (Θ • Λ). (Associative Law of Multiplication)
Proof: If any of the three is 0, both sides are 0. Otherwise both sides have absolute value (| Ξ| • |Θ|) • |Λ| = |Ξ| • (|Θ| • |Λ|), by Theorems 327 and 237. By Theorem 330 applied twice, each side is negative exactly when one or all three of Ξ, Θ, Λ are negative.
Theorem 334 (Landau 200). For positive Θ and Λ: Ξ • (Θ − Λ) = Ξ • Θ − Ξ • Λ.
Proof: (1) Let Ξ > 0. If Θ > Λ: Θ = (Θ − Λ) + Λ, so Ξ • Θ = Ξ • (Θ − Λ) + Ξ • Λ by Theorem 238, and the claim follows by Theorem 321. If Θ = Λ: both sides are 0. If Θ < Λ: by the first case Ξ • (Λ − Θ) = Ξ • Λ − Ξ • Θ, so Ξ • (Θ − Λ) = Ξ • (−(Λ − Θ)) = −(Ξ • Λ − Ξ • Θ) = Ξ • Θ − Ξ • Λ, by Theorems 315 and 331.
(2) Let Ξ = 0. Both sides are 0.
(3) Let Ξ < 0. By (1), (−Ξ) • (Θ − Λ) = (−Ξ) • Θ − (−Ξ) • Λ. Taking the contrary of each side, by Theorems 331 and 315, Ξ • (Θ − Λ) = (−Ξ) • Λ − (−Ξ) • Θ = Ξ • Θ − Ξ • Λ.
Theorem 335 (Landau 201). Ξ • (Θ + Λ) = Ξ • Θ + Ξ • Λ. (Distributive Law)
Proof: First, for positive η and ζ, Ξ • (η + ζ) = Ξ • η + Ξ • ζ: for Ξ > 0 this is Theorem 238; for Ξ = 0 it is obvious; for Ξ < 0, Ξ • (η + ζ) = −(|Ξ| • η + |Ξ| • ζ) = Ξ • η + Ξ • ζ, by Theorem 314.
Now by Theorem 318, Θ = η₁ − η₂ and Λ = ζ₁ − ζ₂ with all four positive. By Theorem 319, Θ + Λ = (η₁ + ζ₁) − (η₂ + ζ₂). So by Theorem 334 and the first step,
Ξ • (Θ + Λ) = (Ξ • η₁ + Ξ • ζ₁) − (Ξ • η₂ + Ξ • ζ₂) = (Ξ • η₁ − Ξ • η₂) + (Ξ • ζ₁ − Ξ • ζ₂) = Ξ • Θ + Ξ • Λ,
the middle step by Theorems 314, 309 and 320, the last by Theorem 334.
Theorem 336 (Landau 202). Ξ • (Θ − Λ) = Ξ • Θ − Ξ • Λ.
Ξ • (Θ + (−Λ)) = Ξ • Θ + Ξ • (−Λ) = Ξ • Θ + (−(Ξ • Λ)), by Theorems 335 and 331.
Theorem 337 (Landau 203). Let Ξ > Θ. If Λ > 0 then Ξ • Λ > Θ • Λ. If Λ = 0 then Ξ • Λ = Θ • Λ. If Λ < 0 then Ξ • Λ < Θ • Λ.
Proof: Ξ − Θ > 0 by Theorem 316. So (Ξ − Θ) • Λ is positive, 0, or negative according as Λ is, by Definition 71. And (Ξ − Θ) • Λ = Ξ • Λ − Θ • Λ by Theorems 336 and 328. Then Theorem 316.
Theorem 338 (Landau 204). If Θ is not 0, the equation Θ • Φ = Ξ has exactly one solution.
Proof: There is at most one. If Θ • Φ₁ = Θ • Φ₂, then Θ • (Φ₁ − Φ₂) = 0 by Theorem 336, so Φ₁ − Φ₂ = 0 by Theorem 326, so Φ₁ = Φ₂ by Theorem 316.
There is one. If Θ > 0, let Φ = (1/Θ) • Ξ, with 1/Θ from Theorem 246. Then Θ • Φ = (Θ • (1/Θ)) • Ξ = 1 • Ξ = Ξ. If Θ < 0, let Φ = −((1/|Θ|) • Ξ). Then Θ • Φ = |Θ| • ((1/|Θ|) • Ξ) = Ξ, by Theorem 332.
Definition 72 (Landau 56). The Φ of Theorem 338 is the quotient of Ξ by Θ, written Ξ/Θ.
With Theorems 321 and 338, every specific operation of adding, removing, multiplying and dividing (except by 0) can be done on any real number and yields a real number.