ratio(redirected from A-G ratio)
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The ratio of two numbers is the quotient from the division of the first number, by the second. The ratio of two homogeneous magnitudes is the number obtained by measuring the first magnitude when the second is chosen as the unit of measurement. If two magnitudes are measured in the same unit of measurement, their ratio is equal to that of the numbers that measure them.
The ratio of the lengths of two segments may be expressed by a rational or irrational number. In the former case the segments are said to be commensurable, and in the latter incommensurable. Mathematicians of the ancient world had no knowledge of irrational numbers. For them the concept of the ratio of two segments did not reduce to the concept of number. In their conception the geometrical theory of the ratios of magnitudes was not connected with the concept of number and played an independent role. In a sense, it substituted for a theory of real numbers. Indeed, according to Euclid the four segments, a, b, a’, and b’ form the proportion a: b = a’:b’ if for any natural numbers m and n one of the relations ma = nb, ma > nb, ma < nb is satisfied simultaneously with the corresponding relation ma’ = nb’, ma’> nb’, or ma’ < nb’. It follows that when a and b are incommensurable the subdivision of the rational numbers (x = m/n) into two classes according to whether a > xb or a < xb coincides with the subdivision according to whether a’ > xb’ or a’ < xb’ —this is the idea behind the modern theory of Dedekind cuts.