The number-system of algebra : treated theoretically and historically . t a complex number may bewritten in the form p(GOsO -j-isinO), where p and 6 havethe meanings just given them. The theorem will be demon-strated, therefore, when it shall have been shown that e»^ = cos 0 -{-i sin 6. If n be any positive integer, we have, by § 36 and thebinomial theorem, nj n 2! n^ , n(n-l)(n-2) (iOy C 3! n^ 1 + 1- i-^Yi-? +^—^T—-(i»y+ Let n be indefinitely increased; the limit of the right sideof this equation will be the same as that of the left. GRAPHICAL REPRESENTATION OF NUMBERS. 49 But the limit of th
The number-system of algebra : treated theoretically and historically . t a complex number may bewritten in the form p(GOsO -j-isinO), where p and 6 havethe meanings just given them. The theorem will be demon-strated, therefore, when it shall have been shown that e»^ = cos 0 -{-i sin 6. If n be any positive integer, we have, by § 36 and thebinomial theorem, nj n 2! n^ , n(n-l)(n-2) (iOy C 3! n^ 1 + 1- i-^Yi-? +^—^T—-(i»y+ Let n be indefinitely increased; the limit of the right sideof this equation will be the same as that of the left. GRAPHICAL REPRESENTATION OF NUMBERS. 49 But the limit of the right side is 2! 3! e i9 * Therefore e^^ is the limit of (1 H— ) as n approaches oo. iOY n To construct the point representing On the axis of real numberslay off OA = 1. Draw AP equal to 6 and par-allel to OB, and divide it into nequal parts. Let AAi be oneof these parts. Then Ai is the •4-1 L*<^ point 1 + — •n Through Ai draw A1A2 atright angles to OAi and con-struct the triangle OA1A2 simi-lar to OAAi. A2 is then the point (1 H— ). +!)?. FiQ. 3. For ^0^2 = 2^0^1; and since 0^42: OA^: : OA^: OA, and OA = 1,the length OA2 = the square of length OA^. (see § 49) In like manner construct ^3 to represent j 1 -^— J, A^ for Let n be indefinitely increased. The broken line will approach as limit an arc of length 6 of the circleof radius OA and, therefore, its extremity, A,„ will approachas limit the point representing cos 0 -[-isinO (§ 47). * This use of the symbol e«^ will be fully justified in § 73, 50 NUMBER-SYSTEM OF ALGEBBA. Therefore the limit of f 1 H— ) as n is indefinitely in- V »v creased is cos 0 -{-1 sin &. But this same limit has already been proved to be e*X Hence e*^ = cos 0 + i sin 0.^ VII. THE PUNDAMEIJfTAL THEOEEM OF ALGEBEA. 52. The General Theorem. If where ri is a positive integer, and cIq, a^, ..., a^ any numbers,real or complex, independent of z, to each value of z corre-sponds a single value of w. We proceed to d
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