Higher Algebra

HIGHER ALGEBRA by S. BARNARD. First published in 1936. Contents include: ix CHAPTER EXERCISE XV ( 128). Minors, Expansion in Terms of Second Minors ( 132, 133). Product of Two Iteterminants ( 134). Rectangular Arrays ( 135). Reciprocal Deteyrrtlilnts, Two Methods of Expansion ( 136, 137). Use of Double Suffix, Symmetric and Skew-symmetric Determinants, Pfaffian ( 138-143), EXERCISE XVI ( 143) X. SYSTEMS OF EQUATIONS. Definitions, Equivalent Systems ( 149, 150). Linear Equations in Two Unknowns, Line at Infinity ( 150-152). Linear Equations in Three Unknowns, Equation to a Plane, Plane at Infinity ( 153-157). EXERCISE XVII ( 158). Systems of Equations of any Degree, Methods of Solution for Special Types ( 160-164). EXERCISE XVIII ( 164). XL RECIPROCAL AND BINOMIAL EQUATIONS. Reduction of Reciprocal Equations ( 168-170). The Equation x n - 1= 0, Special Roots ( 170, 171). The Equation x n - A = 0 ( 172). The Equation a 17 - 1 == 0, Regular 17-sided Polygon ( 173-176). EXERCISE XIX ( 177). AND BIQUADRATIC EQUATIONS. The Cubic Equation ( roots a, jS, y), Equation whose Roots are ( - y) 2 , etc., Value of J, Character of Roots ( 179, 180). Cardan's Solution, Trigonometrical Solution, the Functions a - f eo/ - f-> V> a-f a> 2 4-a> y ( 180, 181). Cubic as Sum of Two Cubes, the Hessftfh ( 182, 183). Tschirnhausen's Transformation ( 186). EXERCISE XX ( 184). The Biquadratic Equation ( roots a, , y, 8) ( 186). The Functions A= y + aS, etc., the Functions /, J, J, Reducing Cubic, Character of Roots ( 187-189). Ferrari's Solution and Deductions ( 189-191). Descartes' Solution ( 191). Conditions for Four Real Roots ( 192-ty). Transformation into Reciprocal Form ( 194). Tschirnhausen's Trans formation ( 195). EXERCISE XXI ( 197). OP IRRATIONALS. Sections of the System of Rationals, Dedekind's Definition ( 200, 201). Equality and Inequality ( 202). Use of Sequences in defining a Real Number, Endless Decimals ( 203, 204). The Fundamental Operations of Arithmetic, Powers, Roots and Surds ( 204-209). Irrational Indices, Logarithms ( 209, 210). Definitions, Interval, Steadily Increasing Functions ( 210). Sections of the System of^ Real Numbers, the Continuum ( 211, 212). Ratio and Proportion, Euclid's Definition ( 212, 213). EXERCISE XXII ( 214). x CONTENTS CHAPTER XIV/ INEQUALITIES. Weierstrass' Inequalities ( 216). Elementary Methods ( 210, 217) For n Numbers a l9 a 2 a > * JACJJ n n n ( a* -!)/* ( a - I)/*,, ( 219). xa x ~ l ( a-b)$ a x - b x ^ xb x ~ l ( a - 6), ( 219). ( l+ x) n ^ l+ nx, ( 220). Arithmetic and Geometric Means ( 221, 222). - - V ^ n and Extension ( 223). Maxima and Minima ( 223, 224). EXERCISE XXIII ( 224). XV. SEQUENCES AND LIMITS. Definitions, Theorems, Monotone Sequences ( 228-232). E* ponential Inequalities and Limits, l m / i n / l-m / 1 ~ n 1) >(!+-) and ( 1--) <( l--) if m> n, m/ n/ mj nj / 1 n / l w lim ( 1-f-= lim( l--) = e, ( 232,233). n _ > 00 V nj nj EXERCISE XXIV ( 233). General Principle of Convergence ( 235-237). Bounds of a Sequent Limits of Inde termination ( 237-240). Theorems : ( 1) Increasing Sequence ( u n ), where u n - u n ^ l < k( u n _^^ n _ z ^ 1 ( 2) If u n > 0 and u n+ l lu n -* l, then u n n -* L ( 3) If lim u n l, then lim ( U