Definition 28. A fraction, A1/A2, is an ordered relation between two numbers such that when both are multiplied by the same number, n, A2 divides A1 into n equal parts.
Fractions and Their Equivalence
Definitions 28–29 · Theorems 130–133
Definition 29. A1/A2 ~ B1/B2 (“~” to be read “equivalent”) if A1•B2 = B1•A2.
Theorem 130 (Landau 37).
A1/A2 ~ A1/A2.
Proof: A1•A2 = A1•A2.
Theorem 131 (Landau 38).
If A1/A2 ~ B1/B2, then B1/B2 ~ A1/A2.
Proof: A1•B2 = B1•A2; hence, B1•A2 = A1•B2.
Theorem 132 (Landau 39).
If A1/A2 ~ B1/B2 and B1/B2 ~ C1/C2, then A1/A2 ~ C1/C2.
Proof: A1•B2 = B1•A2, B1•C2 = C1•B2, hence, (A1•B2)(B1•C2) = (B1•A2)(C1•B2);
it is always true that (A•B)(C•D) = A(B(C•D)) = A(B•C(D)) = A(D(B•C)) = (A•D)(B•C) = (A•D)(C•B).
Therefore, (A1•B2)(B1•C2) = (A1•C2)(B1•B2), and (B1•A2)(C1•B2) = (B1•B2)(C1•A2) = (C1•A2)(B1•B2).
This gives us (A1•C2)(B1•B2) = (C1•A2)(B1•B2).
Theorem 133 (Landau 40).
A1/A2 ~ (A1•A)/(A2•A).
Proof: A1(A2•A) = A1(A•A2) = (A1•A)A2.