Theorem 235 (Landau 141). I) Let ξ and η be bounds. Call a ratio lower when it is X•Y for some lower ratio X for ξ and some lower ratio Y for η, and upper otherwise. This gives a bound. II) No lower ratio of it is the product of an upper ratio for ξ and an upper ratio for η.
Proof: (1) There are lower X and Y, so X•Y is lower. For upper X1 and Y1 and any lower X and Y, X•Y < X1•Y1 by Theorem 200, so X1•Y1 is not lower. This proves II) and the first condition. (2) Let Z < X•Y with X and Y lower. Then Z/X = (1/X)•Z < (1/X)•(X•Y) = Y, so Z/X is lower for η, and Z = X•(Z/X) is lower. (3) Given X•Y, take a lower X1 > X for ξ. Then X1•Y > X•Y and is lower.
Definition 51 (Landau 36). The bound of Theorem 235 is ξ•η, the product of ξ and η.
Theorem 236 (Landau 142). ξ•η = η•ξ. (Commutative Law of Multiplication)
Every X•Y is a Y•X, and conversely (Theorem 195).
Theorem 237 (Landau 143). (ξ•η)•ζ = ξ•(η•ζ). (Associative Law of Multiplication)
Every (X•Y)•Z is an X•(Y•Z), and conversely (Theorem 196).
Theorem 238 (Landau 144). ξ•(η + ζ) = ξ•η + ξ•ζ. (Distributive Law)
Proof: I) Every lower ratio for ξ•(η + ζ) is X•(Y + Z) = X•Y + X•Z with X, Y, Z lower for ξ, η, ζ. This is lower for ξ•η + ξ•ζ. II) Every lower ratio for ξ•η + ξ•ζ is X•Y + X1•Z with X and X1 lower for ξ. Let X2 be the greater of X and X1; it is lower for ξ. Then X•Y + X1•Z ≦ X2•Y + X2•Z = X2•(Y + Z), which is lower for ξ•(η + ζ). So X•Y + X1•Z is lower for it, by Theorem 214.
Theorem 239 (Landau 145). If ξ > η, or ξ = η, or ξ < η, then ξ•ζ > η•ζ, or =, or <, respectively.
If ξ > η, then ξ = η + υ by Theorem 234, so ξ•ζ = (η + υ)•ζ = η•ζ + υ•ζ > η•ζ by Theorems 238 and 227. The second case is obvious. The third follows from the first.
Theorem 240 (Landau 146). The converse of Theorem 239.
The three cases are mutually exclusive and exhaust all possibilities.
Theorem 241 (Landau 147). If ξ > η and ζ > υ, then ξ•ζ > η•υ.
By Theorem 239, ξ•ζ > η•ζ, and η•ζ = ζ•η > υ•η = η•υ.
Theorem 242 (Landau 148). If ξ ≧ η and ζ > υ, or ξ > η and ζ ≧ υ, then ξ•ζ > η•υ.
By Theorem 239 if an equality holds; otherwise Theorem 241.
Theorem 243 (Landau 149). If ξ ≧ η and ζ ≧ υ, then ξ•ζ ≧ η•υ.
Obvious if two equalities hold; otherwise Theorem 242.
Theorem 244 (Landau 150). For any ratio R, call a ratio lower when it is less than R and upper otherwise. This gives a bound.
(1) By Theorem 183 some X < R; and R is not less than R. (2) If X < R and X1 ≧ R then X < X1. (3) If X < R, then by Theorem 184 some X1 has X < X1 < R.
Definition 52 (Landau 37). The bound of Theorem 244 is R*.
R is then the least upper ratio of R*. This is the case from which the notion of relative maximum and minimum begins: a given ratio as the maximum of the ratios less than it.
Theorem 245 (Landau 151). ξ•1* = ξ.
Proof: A lower ratio for ξ•1* is X•Y with X lower for ξ and Y < 1. Then X•Y < X, so it is lower for ξ. Conversely let X be lower for ξ. Take a lower X1 > X and let Y = (1/X1)•X. Then Y < 1, and X = X1•Y is lower for ξ•1*.
Theorem 246 (Landau 152). For any ξ, the equation ξ•υ = 1* has a solution υ.
Proof: Call a ratio lower when it is 1/X for some upper ratio X for ξ that is not the least upper ratio. This gives a bound. (1) If X is upper for ξ, so is X + X, and it is not the least; so 1/(X + X) is lower. If X1 is lower for ξ, then every upper X differs from X1, so 1/X differs from 1/X1; so 1/X1 is not lower. (2) Let 1/X be lower and U < 1/X. Then U•X < 1 = U•(1/U), so X < 1/U. So 1/U is upper for ξ and not the least, and U = 1/(1/U) is lower. (3) Let 1/X be lower. Take an upper X1 < X, and by Theorem 184 an X2 with X1 < X2 < X. Then X2 is upper and not the least, and 1/X2 > 1/X. Call this bound υ. It remains to show ξ•υ = 1*. A) A lower ratio for ξ•υ is X•(1/X1) with X lower and X1 upper for ξ. Since X < X1, it is less than 1. B) Let U < 1. Take any lower X for ξ, and by Theorem 226 take X1 lower and X2 upper for ξ with X2 − X1 = (1 − U)•X. Then X2 − X1 < (1 − U)•X2, so U•X2 < X1, and X2 < X1/U. So X1/U is upper for ξ and not the least. Then U = X1•(1/(X1/U)) is the product of a lower ratio for ξ and a lower ratio for υ.
Theorem 247 (Landau 153). For any ξ and η, the equation η•υ = ξ has exactly one solution υ.
At most one, by Theorem 239. If τ solves η•τ = 1* (Theorem 246), then υ = τ•ξ is a solution: η•(τ•ξ) = (η•τ)•ξ = 1*•ξ = ξ, by Theorems 237 and 245.
Definition 53 (Landau 38). The υ of Theorem 247 is ξ/η, the quotient of ξ by η.