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(v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (ï¿½ï¿½ 41, 44)ï¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

00In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods.

00There are extensions of the binomial theorem, by means of which approximate calculations can be made of fractions, surds, and powers of fractions and of surds; the main difference being that the number of terms which can be taken into account is unlimited, so that, although we may approach nearer and nearer to the true value, we never attain it exactly.

00On the other hand, this new series is not continuous; for we know that there are some points on the line which represent surds and other irrational numbers, and these numbers are not contained in our series.

00It includes the properties of numbers; extraction of roots of arithmetical and algebraical quantities, solutions of simple and quadratic equations, and a fairly complete account of surds.

00But much more it belongs to his transformation of the epistemological problem, and to the suggestiveness of his philosophy as a whole for an advance in the direction of a speculative construction which should be able to cancel all Kant's surds, and in particular vindicate a " ground of the unity of the supersensible which lies back of nature with that which the concept of freedom implies in the sphere of practice," I which is what Kant finally asserts.

00Roots and Surds 8.2 85.

00Multiplication and Division of Surds 8.4 87.

00We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).

00Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations.

00To add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index.

00(v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theroem (Ã¯¿½Ã¯¿½ 41, 44)Ã¯¿½ (vi.) Surds and Approximate Logarithms. - From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers.

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