Quantum redactiones paginae "Arithmetica" differant

 

:<math>\frac{1}{4} + \frac{1}{3} = ?</math>

:<math>\frac{1 \times 3}{4 \times 3} + \frac{1 \times 4}{3 \times 4} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}</math>

 

Aliud exemplum:

:<math>\frac{3}{7} + \frac{2}{5} = ?</math>

:<math>\frac{3 \times 5}{7 \times 5} + \frac{2 \times 7}{5 \times 7} = \frac{15}{35} + \frac{14}{35} = \frac{29}{35}</math>

 

Aliud exemplum:

:<math>\frac{1}{6} + \frac{1}{9} = ?</math>

:<math>\frac{1 \times 9}{6 \times 9} + \frac{1 \times 6}{9 \times 6} = \frac{9}{54} + \frac{6}{54} = \frac{15}{54}</math>

:et fractio reducta est <math>\frac{5}{18}</math>.

 

 

:<math>\frac{1}{6} + \frac{1}{9} = ?</math>

:Quod 6 = 3 × 2 et 9 = 3 × 3, minimus communis dividuus est 3 × 2 × 3 = 18.

:<math>\frac{1 \times 3}{6 \times 3} + \frac{1 \times 2}{9 \times 2} = \frac{3}{18} + \frac{2}{18} = \frac{5}{18}</math>

 

Exempla:

:<math>\frac{3}{7} \times \frac{2}{5} = ?</math>

:<math>= \frac{3 \times 2}{7 \times 5} = \frac{6}{35}</math>

 

Et:

:<math>\frac{2}{7} \times \frac{3}{4} = ?</math>

:<math>= \frac{2 \times 3}{7 \times 4} = \frac{6}{28} = \frac{3}{14}</math>

 

Exempla:

:<math>\frac{1}{2} \div \frac{3}{5} = ?</math>

:<math>= \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}</math>

 

Aut:

:<math>\frac{2}{3} \div \frac{2}{5} = ?</math>

:<math>= \frac{2}{3} \times \frac{5}{2} = \frac{10}{6} = \frac{5}{3}</math>

 

Quare? Scimus N × 1/N semper est 1 (nisi N sit 0). Pone N = 1/M -- tunc 1/M × 1/(1/M) = 1. Sed M × 1/M = 1 (quod semper verum est, nisi M sit 0); hoc est, 1/M × 1/(1/M) = M × 1/M, vel 1/(1/M) = M. Nunc velimus scire quid sit

<math>\frac{a}{b} \div \frac{c}{d}</math>:

:est <math>\frac{a}{b} \times \frac{1}{\frac{c}{d}}</math>,

:hoc est <math>\frac{a}{b} \times \frac{d}{c}</math>

 

:<math>1000 \times D = abc.abcabcabc....</math>

:<math>D = 0.abcabcabc...</math> Subtrahe:

 

(Multiplicavimus per 1000 = 10³ quod 3 figurae sunt in periodo; si plus vel minus insunt, aliam potestam eligere debemus.)

:<math>1000000 \times D = 142857.142857142857....</math>

:<math>D = 0.142857142857....</math>

 

== De machinis arithmeticis ==