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CHAPTER VIII.
FRACTIONS.
A fraction is not altered by multiplying its numerator and denomi-
nator by the same quantity
· Reduction of fractions to a common denominator
Addition and subtraction of fractions
Multiplication and division of fractions
.
Important theorems concerning fractions formed from given fractions
Examples IX..
CHAPTER IX.
PAGE
87
89
90
92
95
98
finding the factors of an expression
The problem of solving an equation the same as the problem of
Quadratic equations.
Discussion of roots of a quadratic equation
Zero and infinite roots
Relations between the roots and the coefficients of a quadratic
Relations between the roots and the coefficients of any equation
Equations with given roots
123
124
CHAPTER X.
SIMULTANEOUS EQUATIONS.
Equations of the first degree with two unknown quantities
Discussion of solution
Equations of the first degree with three unknown quantities
Method of undetermined multipliers
Equations with more than three unknown quantities
Equations with more than two unknown quantities
Examples XIV.
CHAPTER XI.
PROBLEMS.
Problems not always satisfied by the solutions of the corresponding
equations.
Examples XV..
CHAPTER XII.
MISCELLANEOUS THEOREMS AND EXAMPLES.
144
148
149
150
153
154
- 156
162
164
169
172
176
Examples of elimination
181
Equations with restrictions on the values of the letters
Identities deduced from the factors of a3 + b3 + c3 – 3abc
Examples XVI.
CHAPTER XIV.
SURDS. IMAGINARY AND COMPLEX QUANTITIES.
Properties of Surds
If a+√b=c+√d, where a and c are rational and ✅b and ✅d are
irrational, then a=c and b=d.
If either of two conjugate quadratic surds is a factor of a rational
expression, so also is the other
Square root of a+b
Examples XVIII.
Imaginary and complex quantities
213-215
216
217
219
220
222
Complex quantities obey the Fundamental Laws of Algebra
Definition and properties of the modulus of a complex quantity 223-225
If either of two conjugate complex quantities is a factor of a real
expression, so also is the other.
CHAPTER XV.
225
When any number of terms of a square root have been found, as
many more terms can be found by ordinary division
231
Cube root.
232
Method of finding the nth root of any algebraical expression
Examples XIX.
A ratio is made more nearly equal to unity by adding the same
The difference between any number (expressed in the scale r) and
the sum of its digits is divisible by r-1 .
Rule for casting out the nines .
Examples XXII.
CHAPTER XIX.
PERMUTATIONS AND COMBINATIONS.
Permutations of different things
Permutations all together of things which are not all different
266
267
CHAPTER XX.
THE BINOMIAL THEOREM.
Proof of the binomial theorem for a positive integral exponent
291
Continued product of n binomial factors of the form x+a, x+b, &c. 302
CHAPTER XXI.
CONVERGENCY AND DIVERGENCY OF SERIES.
Convergency and divergency of series, all of whose terms have the
same sign
314-320
Series whose terms are alternately positive and negative .
Application to the Binomial, the Exponential and the Logarithmic
320
321-323
323
325
326
If Za„x2 = 2b,x2, for all values of x for which the series are con-
vergent, then ar=b,
Examples XXVI.
Sum of the first r+1 coefficients of a + a1x+a‚x2+
337