| (121) | a | divᵈ. by | c | or | a / b | = | ad |
|---|---|---|---|---|---|---|---|
| b | d | c / d | bc |
- These results are true even when the letters themselves represent fractions. For example, take the fraction
- a/b
- c/d
whose numerator and denominator are fractional, and multiply its numerator and denominator by the fraction
| e | , which gives | ae / bf |
|---|---|---|
| f | ce / df | |
| which (121) is | aedf | |
| bfce |
which, dividing the numerator and denominator by ef (108), is
- ad
- bc
But the original fraction itself is
- ad
- bc
hence
| a / b | = | a / b | × | e / f |
|---|---|---|---|---|
| c / d | c / d | × | e / f |
which corresponds to the second formula4 in (124). In a similar manner it may be shewn, that the other formulæ of the same article are true when the letters there used either represent fractions, or are removed and fractions introduced in their place. All formulæ established throughout this work are equally true when fractions are substituted for whole numbers. For example (54), (m + n)a = ma + na. Let m, n, and a be respectively the fractions
| p | , | r | , and | b |
|---|---|---|---|---|
| q | s | c |
Then m + n is