If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur.

Then the denominator of the fraction, the numerical aperture, must be correspondingly increased, in order to ascertain the real resolving power.

The Romans commonly used fractions with denominator 12; these were described as unciae (ounces), being twelfths of the as (pound).

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Must have a least value, which is moreover positive, since the numerator and denominator are both essentially positive.

The modern system of placing the numerator above the denominator is due to the Hindus; but the dividing line is a later invention.

The frequency ratios in the diatonic scale are all expressible either as fractions, with i, 2, 3 or 5 as numerator and denominator, or as products of such fractions; and it may be shown that for a given note the numerator and denominator are smaller than any other numbers which would give us a note in the immediate neighbourhood.

Fraction in its Lowest Terms.-A fraction is said to be in its lowest terms when its numerator and denominator have no common the more correct method is to write it a: b.

And that thence every symbolic product is equal to a rational function of covariants in the form of a fraction whose denominator is a power of f x.

Can be applied to n, n', the denominator n remaining unaltered.

Hence, so long as the denominator remains unaltered, we can deal with, exactly as if they were numbers, any operations being performed on the numerators.