Addition

Theorem 223 (Landau 129). 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 sum 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. Let X1 and Y1 be upper for ξ and η. For any lower X and Y, X < X1 and Y < Y1, so X + Y < X1 + Y1 by Theorem 191. So X1 + Y1 is not any such X + Y. This proves II) and the first condition. 2) Let Z < X + Y with X and Y lower. Then Z/(X + Y) < 1 by Theorem 199, so X•(Z/(X + Y)) < X and Y•(Z/(X + Y)) < Y by Theorem 198. These are lower for ξ and η by Theorem 214, and their sum is (X + Y)•(Z/(X + Y)) = Z by Theorem 197. So Z is lower. 3) Given X + Y, take a lower X1 > X for ξ. Then X1 + Y > X + Y and is lower.

Theorem 224 (Landau 130). ξ + η = η + ξ. (Commutative Law of Addition)

Every X + Y is a Y + X, and conversely (Theorem 185).

Theorem 225 (Landau 131). (ξ + η) + ζ = ξ + (η + ζ). (Associative Law of Addition)

Every (X + Y) + Z is an X + (Y + Z), and conversely (Theorem 186).

Theorem 226 (Landau 132). For any ratio R and any bound, there are a lower ratio X and an upper ratio U for it with U − X = R.

Proof: Let X1 be lower, and consider the ratios X1 + n•R for numbers n. Not all are lower: if Y is upper then Y > X1, and by Theorem 208 some n has n•R > Y − X1, so X1 + n•R > Y is upper by Theorem 213. By Theorem 22 there is a least such n; call it D. If D = 1, let X = X1 and U = X1 + R. If D > 1, let X = X1 + (D − 1)•R and U = X + R. In each case X is lower, U is upper, and U − X = R.

Remark. This is the exact content of relative similarity. The lower and upper ratios of a bound come as near to one another as any given ratio, however small. Nothing more is needed to make the bound the maximum of the one and the minimum of the other.

Theorem 227 (Landau 133). ξ + η > ξ.

Let Y be lower for η. By Theorem 226 take X lower and U upper for ξ with U − X = Y. Then U = X + Y is upper for ξ and lower for ξ + η.

Theorem 228 (Landau 134). If ξ > η then ξ + ζ > η + ζ.

Proof: Some upper ratio Y for η is lower for ξ. Take a greater lower ratio X > Y for ξ; it is upper for η. By Theorem 226 take Z upper and U lower for ζ with Z − U = X − Y. Then Y + Z = Y + ((X − Y) + U) = X + U. This ratio is lower for ξ + ζ, and by Theorem 223 II) it is upper for η + ζ.

Theorem 229 (Landau 135). If ξ > η, or ξ = η, or ξ < η, then ξ + ζ > η + ζ, or =, or <, respectively.

The first is Theorem 228, the second is obvious, the third follows from the first by Theorems 216 and 215.

Theorem 230 (Landau 136). The converse of Theorem 229.

The three cases are mutually exclusive and exhaust all possibilities (Theorem 217).

Theorem 231 (Landau 137). If ξ > η and ζ > υ, then ξ + ζ > η + υ.

By Theorem 228, ξ + ζ > η + ζ, and η + ζ = ζ + η > υ + η = η + υ.

Theorem 232 (Landau 138). If ξ ≧ η and ζ > υ, or ξ > η and ζ ≧ υ, then ξ + ζ > η + υ.

By Theorem 228 if an equality holds; otherwise Theorem 231.

Theorem 233 (Landau 139). If ξ ≧ η and ζ ≧ υ, then ξ + ζ ≧ η + υ.

Obvious if two equalities hold; otherwise Theorem 232.

Theorem 234 (Landau 140). If ξ > η, then η + υ = ξ has exactly one solution υ.

If ξ ≦ η there is none, by Theorem 227.

Proof: I) At most one, by Theorem 229. II) Call a ratio lower when it is X − Y for some lower ratio X for ξ and some upper ratio Y for η with X > Y. This gives a bound.

1. Such an X − Y exists, by the first step of the proof of Theorem 228. No upper ratio X1 for ξ is of this form, since X − Y < X < X1.
2. If U < X − Y, then U + Y < X, so U + Y = X2 is lower for ξ, and U = X2 − Y.
3. Take a lower X3 > X for ξ. Then X3 − Y > X − Y. Call this bound υ. It remains to show υ + η = ξ. A) Every lower ratio for υ + η is (X − Y) + Y1, with X lower for ξ, Y upper and Y1 lower for η, and X > Y. Since Y > Y1, ((X − Y) + Y1) + (Y − Y1) = X, so (X − Y) + Y1 < X and is lower for ξ. B) a) Let a lower ratio for ξ be upper for η; call it Y. Take a lower X > Y for ξ, and by Theorem 226 take Y1 lower and Y2 upper for η with Y2 − Y1 = X − Y. Then Y > Y1, and Y2 + (Y − Y1) = X, so Y − Y1 = X − Y2 and Y = (X − Y2) + Y1, which is lower for υ + η. b) If a lower ratio for ξ is also lower for η, it is less than those of a), and so is lower for υ + η by Theorem 214.