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The numerator of the third is the sum of the numerators of the first and second, its denominator, the sum of the two denominators. The same is true of the fourth fraction and the two immediately preceding it. Continuing the series, we get the fractions 5/13, 8/21, 13/34.
Learn the rules, and that's all. . . . To divide a fraction by a fraction you must multiply the numerator of the first fraction by the denominator of the second, and that will be the numerator of the quotient. . . . In this case, the numerator of the first fraction. . . ." "I know that without your telling me," Styopa interrupts him, flicking a walnut shell off the table. "Show me the proof."
And again he says: "Natural man has an absolute value; he is a numerical unit, a complete integer and has no relation save to himself and to his fellow man. Civilized man is only a relative unit, the numerator of a fraction whose value depends upon its dominator, its relation to the integral body of society. Good political institutions are those which make a man unnatural."
We will look at it like this. . . . Suppose we want to divide seven eighths not by two fifths but by two, that is, by the numerator only. We divide it, what do we get? "Seven sixteenths." "Right. Bravo! Well, the trick of it is, my boy, that if we . . . so if we have divided it by two then. . . . Wait a bit, I am getting muddled.
Every one knows, that, when an accident happens on a railway, "no one is to blame," which means, that everybody should have so much blame as can be expressed by a fraction whose numerator is unity and whose denominator represents the whole number of employees. Such an infinitesimal dose of censure, contrary to the homeopathic doctrine, always produces infinitesimal results.
The infinitely microscopic chance that each of us had for life cannot be approximated. All the drops of water in the ocean, or all the grains of sand upon the shore, or all the leaves on all the trees, if converted into numbers and used as a denominator, with one for a numerator, could hardly tell the fraction of a chance that gave us life.
The amount is always relative, that is to say, just so much as will maintain the proportion between what he wants and what he gets; for to measure a man's happiness only by what he gets, and not also by what he expects to get, is as futile as to try and express a fraction which shall have a numerator but no denominator.
The phenomena of isomerism, or identity of composition and proportions of constituents with difference of qualities, and of isomorphism, or identity of form in crystals which have one element substituted for another, were equally surprises to science; and although the mechanism by which they are brought about can be to a certain extent explained by a reference to the hypothetical atoms of which the elements are constituted, yet this is only turning the difficulty into a fraction with an infinitesimal denominator and an infinite numerator.
The height of lift, contained in the numerator of the fraction which expresses the efficiency of the pumps, is not to be taken as the difference in level between the surface of the water in the reservoir and the surface of the water whence the pumps draw their supply; but as this difference in level, plus the loss of pressure in the suction pipe, which is usually very short, and plus the loss in the channel to the reservoir, which may be very long.
The description of the hiding-place was written in a most complicated cipher dispatch, the key to which was contained in a stanza of a song known to Kossuth's correspondent in Pesth. Each letter in the dispatch was represented by a fraction, of which the numerator was the number of the letter in one of the lines of the song, and the denominator the number of the line.
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