A Treatise on Algebra: By Charles Smith |
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Page 441
... + ... The rth convergent of any continued fraction will be denoted by Pr qr • b d The fractions a , & c . will be called the first , с second , third , & c . elements of the continued fraction . 350. In a continued fraction of the form a +.
... + ... The rth convergent of any continued fraction will be denoted by Pr qr • b d The fractions a , & c . will be called the first , с second , third , & c . elements of the continued fraction . 350. In a continued fraction of the form a +.
Page 443
... element which is taken into account ; and a similar law holds for the denominator . We will now shew by induction that all the convergents after the second are formed according to the above law provided there is no cancelling at any ...
... element which is taken into account ; and a similar law holds for the denominator . We will now shew by induction that all the convergents after the second are formed according to the above law provided there is no cancelling at any ...
Page 444
... elements . m n For let be the given fraction ; then , if m be greater than n , divide m by n and let a be the quotient and p the remainder , so that = a + Now divide n by p m n P. n and let b be the quotient and q the remainder ; then Ρ ...
... elements . m n For let be the given fraction ; then , if m be greater than n , divide m by n and let a be the quotient and p the remainder , so that = a + Now divide n by p m n P. n and let b be the quotient and q the remainder ; then Ρ ...
Page 453
... elements of a continued fraction continually recur in the same order , the fraction is said to be a periodic continued fraction ; and a periodic continued fraction is said to be simple or mixed according as the recurrence begins at the ...
... elements of a continued fraction continually recur in the same order , the fraction is said to be a periodic continued fraction ; and a periodic continued fraction is said to be simple or mixed according as the recurrence begins at the ...
Page 459
... element , is a root of a quadratic equation with rational coefficients whose roots are both of the same sign . Let the fraction be 1 1 1 1 + k + a + B + x = a + b + ... and let 1 y = a + ... B + A ' A Let and B ' B periodic part ; then ...
... element , is a root of a quadratic equation with rational coefficients whose roots are both of the same sign . Let the fraction be 1 1 1 1 + k + a + B + x = a + b + ... and let 1 y = a + ... B + A ' A Let and B ' B periodic part ; then ...
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Common terms and phrases
a+b+c a₁ arithmetic mean arithmetical progression ax² b₁ binomial Binomial Theorem C₁ Cambridge coefficient contain continued fraction Crown 8vo denominator determinant digits divergent divide divisible divisor Edition equal equation ax example expansion Extra fcap Fcap Find the number Find the sum follows geometrical progression given expression Globe 8vo greater Hence the required infinity letters logarithm MACMILLAN'S Multiply negative nth convergent nth root nth term number of terms obtained P₁ periodic continued fraction positive integer powers prime number Professor prove quadratic equation quadratic surd quotient ratio remainder School Shew Solve the equation square root subtract suppose Theorem Trinity College u₁ unknown quantities white ball whole number zero
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