The elements of algebra1857 |
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Page 223
... convergent is obtained from the preceding one by merely bringing to account another quotient , it is clear that may be found 1 N D Ꭰ , from N ; by writing q + instead of q , that is , N ( 93+ 1 ) N2 + N , = ̧qs ( q3N2 + N1 ) + N2 _ q1N ...
... convergent is obtained from the preceding one by merely bringing to account another quotient , it is clear that may be found 1 N D Ꭰ , from N ; by writing q + instead of q , that is , N ( 93+ 1 ) N2 + N , = ̧qs ( q3N2 + N1 ) + N2 _ q1N ...
Page 224
... convergent , it also holds for the next ; and as we know it does hold for each of those near the beginning of the series , it follows that the law holds generally . ( NOTE 4. ) 84 Ex . To find a series of converging fractions for 227 84 ...
... convergent , it also holds for the next ; and as we know it does hold for each of those near the beginning of the series , it follows that the law holds generally . ( NOTE 4. ) 84 Ex . To find a series of converging fractions for 227 84 ...
Page 227
... convergent for the true fraction a · b N1 N Let and D1 D2 be any two consecutive convergents , then we know that N , D , -N , D , = 1 , ( Art . 349 ) ; and of N1 N2 and we know that D1 D1 a ō < one , and > the other , that is , lies ...
... convergent for the true fraction a · b N1 N Let and D1 D2 be any two consecutive convergents , then we know that N , D , -N , D , = 1 , ( Art . 349 ) ; and of N1 N2 and we know that D1 D1 a ō < one , and > the other , that is , lies ...
Page 228
... convergent is , and d < D1 . Then since lies between n therefore N1 N1 N2 1 ~ N1 D1 a and lies between N1 and D1 D2 N , 2 , d : . nD1 ~ N1d < D. ' d D < ~ D1 D2 < than b N1 a and D1 b ' 1 < or a whole number < a proper fraction , which ...
... convergent is , and d < D1 . Then since lies between n therefore N1 N1 N2 1 ~ N1 D1 a and lies between N1 and D1 D2 N , 2 , d : . nD1 ~ N1d < D. ' d D < ~ D1 D2 < than b N1 a and D1 b ' 1 < or a whole number < a proper fraction , which ...
Page 235
... convergent immediately preceding then , by Art . 349 , since P and q are contiguous convergents , q a aq − bp = ± 1 , ( + or – according as - q > or < 2 , P , that is , accord- ing as Р occupies an even or an odd place in the series ...
... convergent immediately preceding then , by Art . 349 , since P and q are contiguous convergents , q a aq − bp = ± 1 , ( + or – according as - q > or < 2 , P , that is , accord- ing as Р occupies an even or an odd place in the series ...
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algebraical quantities annuity Arithmetical Progression Binomial Theorem coefficient common denominator Comp contain continued fraction convergent cube root denary difference digits divided divisible divisor Edition equal equation EXERCISES expressed Extract the square factors Find the G.C.M. find the number Find the square find the sum Find the value given greater Greatest Common Measure Hence hour latter Least Common Multiple less lowest terms miles Mult multiplied N₁ negative nth root number of combinations number of terms numerator and denominator positive integers Post 8vo Present Value prime number proportion proposed quantities prove quadratic quadratic equation quotient radix ratio reduced remainder represented rule scale shew shillings square root ST JOHN'S COLLEGE subtracted suppose surd unknown quantities vols vulgar fraction whole number Woodcuts
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