Notes on the First Part of Algebra

(i. e. To Simple Equations Inclusive.)

Designed for Candidates for Responsions.

Preface

The following List of Subjects with its accompanying Cycle has been drawn up as a guide to the Student in choosing examples to work. They are so arranged as to secure his giving the most attention to those which are most important. Space has been left at the end of the Cycle for recording, at the close of each day’s work, the point reached in it.

To illustrate the use of this Cycle, we will suppose 40 to be the last entry made in this space. Looking to No. 41 in the Cycle, the Student finds 8 ξ, which the List explains as “simplifying continued fractions:” he turns to his book of examples for some of this kind, and, after working one or two, goes on to No. 42 of the Cycle, against which he finds 13 α, which the List explains as “Involution of single term: integral coefficient,” and so on till he has reached (say) No. 47 of the Cycle. He then enters 47 in the space at the end, to serve as a guide in beginning work next time.

If this List and Cycle be thought to be too complicated for convenient use, they may be simplified by disregarding the small Greek letters, and attending to the numerals only. In this case the Student may begin the Cycle again after reaching No. 113.

A few Rules, which ought to be committed to memory, are added at the end.

N.B. The Student is recommended to draw a line down the margin of the “List of Subjects,” to mark how far he has got in learning the subject; and in using the Cycle, he will of course select only those examples which fall within the range so marked.

Rules to be Commited to Memory

N.B. By covering the lower part of the page, the Student may test for himself his recollection of these rules.

Addition,
when all the signs are alike. (¹)
when some are +, and some −. (²)
Subtraction,
when the signs are alike, and the upper quantity largest. (³)
when this is not the case. (⁴)
Brackets, to put on or take off.
when the sign outside the bracket is +. (⁵)
when it is −. (⁶)
Multiplication and Division,
as to indices:
$a^{m} \times a^{n} =$ (⁷)
$a^{m} \div a^{n} =$ (⁸)
as to signs:
when the signs of the 2 quantities are alike. (⁹)
when they are different. (¹⁰)
Resolving into factors,
when the quantity is of the form $(a^{2} - b^{2})$ . (¹¹)
$(a^{2} + b^{2})$ . (¹²)
$(a^{3} - b^{3})$ . (¹³)
$(a^{3} + b^{3})$ . (¹⁴)
$(a^{2} + 2 a b + b^{2})$ . (¹⁵)
$(a^{2} - 2 a b + b^{2})$ . (¹⁶)
G. C. M. (particular cases,)
when a factor is observed which will divide one of the quantities, but not the other. (¹⁷)
when a factor is observed which will divide both quantities. (¹⁸)
when the first term of the divisor will not exactly divide the first term of the dividend. (¹⁹)
Theory of negative indices, and of $a^{0}$
$a^{- m} =$ (²⁰)
$a^{0} =$ (²¹)
Involution and Evolution,
${a^{m}}^{n} =$ (²²)
the nth root of $a^{m}$ , i. e. $\sqrt[n]{a^{m}} =$ (²³)
hence, the nth root of a, i. e. $\sqrt[n]{a}$ (²⁴)

add all the terms together, and bring down the sign. ↩
add all the + terms together, and add all the − terms together: and set down the difference of the 2 results, with the sign of the greater. ↩
subtract, and bring down the sign. ↩
change the sign of the lower quantity, and proceed as in Addition. ↩
leave the signs of all terms within the bracket unchanged. ↩
change the signs of all terms within the bracket. ↩
$= a^{m + n}$ . ↩
$= a^{m - n}$ . ↩
the sign of the result is +. ↩
the sign of the result is −. ↩
it $= (a + b) . (a - b)$ . ↩
it cannot be resolved. ↩
it will divide by $(a - b)$ . ↩
is will divide by $(a + b)$ . ↩
it $= {(a + b)}^{2}$ . ↩
it $= {(a - b)}^{2}$ . ↩
divide it out of that quantity. ↩
divide it out of both, and multiply the answer by it. ↩
find the L. C. M. of coefficients of these 2 terms, and multiply the divident by such a quantity as will raise the coefficient of its first term to this L. C. M. ↩
$= \frac{1}{a^{m}}$ . ↩
$= 1$ . ↩
$= a^{m n}$ . ↩
$= a^{\frac{m}{n}}$ . ↩
$= a^{\frac{1}{n}}$ . ↩