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TX 512.1.6478h
Wentworth,G. A.
Higher algebra /
Stanford University Libraries
3 6105 04932 6403
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STANFORD JUNIOR
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EnteredaccordingtoActofCongress,intheyear1891,by
G. A. WENTWORTH,
intheOfficeoftheLibrarianofCongress,atWashington.
ALL RIGHTS RESERVED.
3
TYPOGRAPHY BY J. 8. CUSHING & Co., Boston, U.S.A.
PRESSWORK BY GINN & Co., BOSTON, U.S.A.
PREFACE.
THISwork isintendedto give inonebookathorough prepara
tory coursefor Collegesand Scientific Schools, and in addition
asufficiently full treatment of the subjects usually read by students
in general in such institutions. In short, it provides a course
parallel to the course covered by the author's School and College
Algebras together. The elementary part is as full as the School
Algebra; the advanced part, however, is briefer than the College
Algebra. The book is substantially equivalent to the author's
Complete Algebra, but is greatly superior to that work in the
arrangementoftopics and in themethodsofpresenting them.
Preparatory Schools and Academies, if their pupils have had a
thorough drillin Arithmeticbefore they begin thestudyofAlgebra,
will find this book specially suited to their needs. The brighter
boys of the class can read the advanced chapters while the duller
boysreviewthe elementary chapters.
Colleges and Scientific Schools, if their pupils, owing to lack
of previous drill, have to review carefully the preparatory work
oftheschoolsbeforeenteringupon thecollegeworkproper, will find
inthisaaconvenientand sufficiently full book for theirrequirements.
Answers to the problems are bound separately, in paper covers,
andwillbefurnishedfree forpupils when teachersapply to the pub
lishersforthem.
Any corrections or suggestions relating to the work will be
thankfully received.
G. A. WENTWORTH.
PHILLIPS EXETER ACADEMY,
EXETER, N.H., May, 1891.
CONTENTS.
CHAPTER PAGE
I. DEFINITIONS 1
II. ADDITION AND SUBTRACTION 12
III. MULTIPLICATION 26
IV. DIVISION 39
V. SIMPLE EQUATIONS . 48
VI. MULTIPLICATION AND DIVISION 61
VII. FACTORS. 70
VIII. Common FACTORS AND MULTIPLES 93
IX. FRACTIONS 109
.
X. FRACTIONAL EQUATIONS 134
XI. SIMULTANEOUS EQUATIONSOFTHEFIRST DEGREE 154
XII. PROBLEMS INVOLVING TwoUNKNOWNNUMBERS . 169
XIII. SIMPLE INDETERMINATE EQUATIONS 186
XIV. INEQUALITIES 192
XV. INVOLUTION AND EVOLUTION 194
XVI. THEORY OF EXPONENTS 207
XVII. RADICAL EXPRESSIONS. 214
XVIII. IMAGINARY EXPRESSIONS . 229
XIX. QUADRATIC EQUATIONS 234
XX. SIMULTANEOUS QUADRATIC EQUATIONS 260
XXI. PROPERTIES OF QUADRATICS . 271
XXII. Ratio, PROPORTION, AND VARIATION 277
vi CONTENTS.
CHAPTER PAGE
XXIII. PROGRESSIONS 295
XXIV. INDETERMINATE COEFFICIENTS 312
.
XXV. BINOMIAL THEOREM 319
XXVI. COMMON LOGARITHMS 331
XXVII. INTEREST AND ANNUITIES 346
XXVIII. CHOICE 356
.
XXIX. CHANCE . 377
XXX. CONTINUED FRACTIONS. 381
XXXI. SCALES OF NOTATION 392
XXXII. THEORY OF NUMBERS . 397
XXXIII. VARIABLES AND LIMITS 403
XXXIV. SERIES 412
XXXV. GENERAL PROPERTIES OF EQUATIONS 441
XXXVI. NUMERICAL EQUATIONS 473
XXXVII. DETERMINANTS 499
XXXVIII. COMPLEX NUMBERS: 515
HIGHER ALGEBRA.
CHAPTER I.
DEFINITIONS.
1. Units. In counting separate objects the standards by
which wecount are called units; and in measuring contin
uous magnitudes the standards by which we measure are
called units.
Thus, in countingthe boys in a school, the unitis aboy; in sell
;
ing eggs by thedozen, the unitis adozen eggs; in sellingbricks by
a
the thousand, the unitisa thousand bricks; in measuringshort dis
tances, the unit is an inch, a foot, or a yard; in measuring long
distances, the unitis arod or amile.
2. Numbers. Repetitions of the unit are expressed by
numbers. If a man, in sawing logs into boards, wishes to
keep a count of the logs, he makes a straight mark for
every log sawed, and his record at differenttimes will be
as follows:
// /// //// Ne NW
MV // MU /// NU MII NUNNU
These representative groups are named one, two, three,
four, five, six, seven, eight, nine, ten, etc., and are known
collectively under the general name of numbers. It is
obvious that these representative groups will have the
samemeaning, whateverthe nature ofthe unit counted.
