Definition 37 (Landau 16). Two fractions have the same ratio when they are equivalent. A ratio is what equivalent fractions have in common. Capital letters X, Y, Z, U stand for ratios, and X is called the ratio of any fraction that has it.
Landau here defines a rational number as the set of all fractions equivalent to a given one. No such set is taken here. By Theorems 130 to 132, having the same ratio is reflexive, symmetric and transitive, which is all that is needed to speak of the ratio of a fraction.
Definition 38 (Landau 17). X = Y when the fractions of X and of Y are equivalent. Otherwise X ≠ Y.
Definition 39 (Landau 18 to 24). X > Y, X < Y, X ≧ Y, X ≦ Y, X + Y, X − Y and X•Y are each said of ratios when the like is true of a fraction of X and a fraction of Y. By Theorems 137, 138, 149, 160 and 161 it does not matter which fractions are taken.
Each of the following is the theorem on fractions named beside it, said of ratios.
Theorem 171 (Landau 78). X = X.
This is Theorem 130, said of ratios.
Theorem 172 (Landau 79). If X = Y then Y = X.
This is Theorem 131, said of ratios.
Theorem 173 (Landau 80). If X = Y and Y = Z then X = Z.
This is Theorem 132, said of ratios.
Theorem 174 (Landau 81). Exactly one of X = Y, X > Y, X < Y.
This is Theorem 134, said of ratios.
Theorem 175 (Landau 82). If X > Y then Y < X.
This is Theorem 135, said of ratios.
Theorem 176 (Landau 83). If X < Y then Y > X.
This is Theorem 136, said of ratios.
Theorem 177 (Landau 84). If X ≧ Y then Y ≦ X.
This is Theorem 141, said of ratios.
Theorem 178 (Landau 85). If X ≦ Y then Y ≧ X.
This is Theorem 142, said of ratios.
Theorem 179 (Landau 86). If X < Y and Y < Z then X < Z.
This is Theorem 143, said of ratios.
Theorem 180 (Landau 87). If X ≦ Y, Y < Z or X < Y, Y ≦ Z then X < Z.
This is Theorem 144, said of ratios.
Theorem 181 (Landau 88). If X ≦ Y and Y ≦ Z then X ≦ Z.
This is Theorem 145, said of ratios.
Theorem 182 (Landau 89). For any X there is a Z > X.
This is Theorem 146, said of ratios.
Theorem 183 (Landau 90). For any X there is a Z < X.
This is Theorem 147, said of ratios.
Theorem 184 (Landau 91). If X < Y there is a Z with X < Z < Y.
This is Theorem 148, said of ratios.
Theorem 185 (Landau 92). X + Y = Y + X.
This is Theorem 151, said of ratios.
Theorem 186 (Landau 93). (X + Y) + Z = X + (Y + Z).
This is Theorem 152, said of ratios.
Theorem 187 (Landau 94). X + Y > X.
This is Theorem 153, said of ratios.
Theorem 188 (Landau 95). If X > Y then X + Z > Y + Z.
This is Theorem 154, said of ratios.
Theorem 189 (Landau 96). If X > Y, or =, or <, then X + Z > Y + Z, or =, or <.
This is Theorem 155, said of ratios.
Theorem 190 (Landau 97). The converse of Theorem 189.
This is Theorem 156, said of ratios.
Theorem 191 (Landau 98). If X > Y and Z > U then X + Z > Y + U.
This is Theorem 157, said of ratios.
Theorem 192 (Landau 99). If X ≧ Y, Z > U or X > Y, Z ≧ U then X + Z > Y + U.
This is Theorem 158, said of ratios.
Theorem 193 (Landau 100). If X ≧ Y and Z ≧ U then X + Z ≧ Y + U.
This is Theorem 159, said of ratios.
Theorem 194 (Landau 101). If X > Y then Y + U = X has exactly one solution U.
This is Theorem 160, said of ratios.
Theorem 195 (Landau 102). X•Y = Y•X.
This is Theorem 162, said of ratios.
Theorem 196 (Landau 103). (X•Y)•Z = X•(Y•Z).
This is Theorem 163, said of ratios.
Theorem 197 (Landau 104). X•(Y + Z) = X•Y + X•Z.
This is Theorem 164, said of ratios.
Theorem 198 (Landau 105). If X > Y, or =, or <, then X•Z > Y•Z, or =, or <.
This is Theorem 165, said of ratios.
Theorem 199 (Landau 106). The converse of Theorem 198.
This is Theorem 166, said of ratios.
Theorem 200 (Landau 107). If X > Y and Z > U then X•Z > Y•U.
This is Theorem 167, said of ratios.
Theorem 201 (Landau 108). If X ≧ Y, Z > U or X > Y, Z ≧ U then X•Z > Y•U.
This is Theorem 168, said of ratios.
Theorem 202 (Landau 109). If X ≧ Y and Z ≧ U then X•Z ≧ Y•U.
This is Theorem 169, said of ratios.
Theorem 203 (Landau 110). Y•U = X has exactly one solution U.
This is Theorem 170, said of ratios.