Advanced AlgebraGinn, 1905 - 285 páginas This book is designed for use in secondary schools and in short college courses. It aims to present in concise but clear form the portions of algebra that are required for entrance to the most exacting colleges and technical schools. The chapters in 'Algebra to Quadratics' are intended for a review of the subject. The rest of the text concentrates on subjects that are most vital, which is why topics that demand a knowledge of calculus for complete comprehension have been omitted. |
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Página 258
... convergent , 1+ 1 = 3 3 10 = 1 + = is the second convergent , is the third convergent , etc. When there is no whole number preceding the fractional part of the continued fraction the first convergent is zero . Thus in is called the ...
... convergent , 1+ 1 = 3 3 10 = 1 + = is the second convergent , is the third convergent , etc. When there is no whole number preceding the fractional part of the continued fraction the first convergent is zero . Thus in is called the ...
Página 259
... convergent we have 1 a1 + A 2 + 1 = a1 + az As a 2 + 1 = a3 A3 ( A2α1 + 1 ) + α1 _ P3 , аza 2 + 1 q3 or P3 = а3 ( α2α1 + 1 ) + α1 = а3P2 + P1 , 23 = A3α2 + 1 = α322 + 21 . This indicates that the form of the nth convergent is Pn anPn ...
... convergent we have 1 a1 + A 2 + 1 = a1 + az As a 2 + 1 = a3 A3 ( A2α1 + 1 ) + α1 _ P3 , аza 2 + 1 q3 or P3 = а3 ( α2α1 + 1 ) + α1 = а3P2 + P1 , 23 = A3α2 + 1 = α322 + 21 . This indicates that the form of the nth convergent is Pn anPn ...
Página 260
... convergent is evidently 0 , the second is 1 , and the third is 1 1+ 2 The fourth convergent is = 94 P5 P4 a4P3 + P2 1.2+ = a493 +92 1.3+ a5p4 + P3 - 3 + 2 The fifth convergent is The sixth convergent is The seventh convergent is = = 95 ...
... convergent is evidently 0 , the second is 1 , and the third is 1 1+ 2 The fourth convergent is = 94 P5 P4 a4P3 + P2 1.2+ = a493 +92 1.3+ a5p4 + P3 - 3 + 2 The fifth convergent is The sixth convergent is The seventh convergent is = = 95 ...
Página 261
... convergent The third convergent is b P3 . = ab + 1 93 The fourth convergent , or x , gives us P4 a4 P3 + P2 x = = 94 a 493 + I2 Simplifying , we get = ( c + x ) b + 1 ( c + x ) ( ab + 1 ) + a - ( ab + 1 ) x2 + [ c ( ab + 1 ) + a − b ] ...
... convergent The third convergent is b P3 . = ab + 1 93 The fourth convergent , or x , gives us P4 a4 P3 + P2 x = = 94 a 493 + I2 Simplifying , we get = ( c + x ) b + 1 ( c + x ) ( ab + 1 ) + a - ( ab + 1 ) x2 + [ c ( ab + 1 ) + a − b ] ...
Página 267
... convergent of a con- tinued fraction for the fraction itself . THEOREM . The maximum limit of error in taking the nth 1 convergent for the continued fraction is less than In In + 1 Since by the theorem of the last section the value of ...
... convergent of a con- tinued fraction for the fraction itself . THEOREM . The maximum limit of error in taking the nth 1 convergent for the continued fraction is less than In In + 1 Since by the theorem of the last section the value of ...
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Términos y frases comunes
a₁ a²b ab² algebraic arithmetical means assume axis b₁ binomial called coefficients column complex numbers constant term continued fraction coördinates COROLLARY corresponding curve decimal determinant digits Divide division divisor equa equal equation ax² exponents expression Extract the square factor Find the 7th following equations following RULE geometrical means graph Hence identity integers irrational numbers least common multiple left-hand member letters linear equation logarithm monomial Multiply negative number nth term obtain operations pair of values pairs of equations perfect square permutations Plot polynomial positive integer quadratic equation quotient radical ratic rational number real number remainder represent result rule of signs satisfy the equation Solution square root Substituting subtract symbols synthetic division THEOREM tion Transpose vanish variable x₁ zero
Pasajes populares
Página 237 - The characteristic of the logarithm of a number greater than 1 is a positive integer or zero, and is one less than the number of digits to the left of the decimal point.
Página 53 - Find the square root of the first term, write it as the first term of the root, and subtract its square from the given polynomial. Divide the first term of the remainder by...
Página 87 - Pythagorean theorem, which states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
Página 54 - Separate the given number into periods of two figures each, beginning from the units' place. 2d. find the greatest number whose square is contained in the left-hand period ; this is the first figure of the required root. >Subtract its square from the first period, and to the remainder bring down the second period for a dividend.
Página 13 - Multiply the divisor by the first term of the quotient and subtract the product from the dividend.
Página 157 - The number of permutations of n things taken r at a time is n!
Página 12 - The square of any polynomial equals the sum of the squares of the terms plus twice the product of each term by each term which follows it.
Página 53 - Divide the first term of the remainder by twice the first term of the root, and add the quotient to the part of the root already found, and also to the trial-divisor.
Página 53 - We next multiply the complete divisor by the last term of the root and subtract the product from the last remainder.
Página 27 - The numerator and denominator of a fraction may be multiplied or divided by the same number without changing the value of the fraction.