A

TREATISE

OF

ALGEBRA.

SECTION I

of Notation.

ALGEBRA is that science which teaches, in a general manner, the relation and comparison of abstract quantities: by means whereof such questions are resolved whose solutions would be sought in vain from common arithmetic.

In algebra, otherwise called specious arithmetic, numbers are not expressed as in the common notation. but every quantity, whether given or required, is commonly represented by some letter of the alphabet; the given ones for distinction sake, being, usually, denoted by the initial letters a, b, c, d, &c.; and the unknown, or required ones, by the final letters u, w, x, y, &c. There are, moreover, in algebra, certain signs or notes made use of, to show the relation and dependence of quantities one upon another, whose signification the learner ought, first of all, to be made acquainted with.

The sign, signifies that the quantity, which it is prefixed to, is to be added. Thus ab shows that the

B

number represented by b is to be added to that represented by a, and expresses the sum of those numbers; so that if a was 5, and b 3, then would a + b be 5 + 3, or 8. In like manner a + b + c, denotes the number arising by adding all the three numbers a, b, and c, together.

Note. A quantity which has no prefixed sign (as the leading quantity a in the above examples) is always understood to have the sign + before it; so that a signifies the same as a; and a + b, the same as + a + b.

The sign, signifies that the quantity which it precedes is to be subtracted. Thus ab shows that the quantity represented by b is to be subtracted from that represented by a, and expresseth the difference of a and b; so that, if a was 5, and b 3, then would a · b be 5

3 or 2. In like manner a+b-c-d represents the quantity which arises by taking the numbers c and d from the sum of the other two numbers a and b; as, if a was 7, b 6, c 5, 'and d 3, then would a+b-c―d be 7 + 6 or 5.

5

S

The notes and are usually expressed by the words plus (or more) and minus (or less.) Thus, we read, a+b, a plus b; and a- - b, a minus b.

Moreover, those quantities to which the sign + is prefixed are called positive (or affirmative;) and those to which the sign-is prefixed, negative.

The sign, signifies that the quantities between which it stands are to be multiplied together. Thus a × b denotes that the quantity a is to be multiplied by the quantity b, and expresses the product of the quantities so multiplied; and a xbx c expresses the product arising by multiplying the quantities a, b, and c, continually together: thus, likewise, a + bx c denotes the product of the compound quantity a + b by the simple quantity c; and a+b+c x a − b + cx a + c represents the product which arises by multiplying the three compound quantities a+b+c, ab+c, and a + c continually together; so that, if a was 5, b 4, and c 3; then

would a+b+cxa − b + cxa + c be 12 x 4 x 8, which is 384.

But when quantities denoted by single letters are to be multiplied together, the sign x is generally omitted, or only understood; and so ab is made to signify the same as a × b; and abc, the same as a × b

x c.

It is likewise to be observed, that when a quantity is to be multiplied by itself, or raised to any power, the usual method of notation is to draw a line over the given quantity, and at the end thereof place the exponent of the power. Thus a + b2 denotes the same as a + b x a + b, viz. the second power (or square) of a + b considered as one quantity: thus also, ab + bc3 denotes the same as abbc × ab + bc × ab + bc, viz. the third power, (or cube) of the quantity ab + bc.

But in expressing the powers of quantities represented by single letters, the line over the top is commonly omitted; and so a2 comes to signify the same as aa or a × a, and b3 the same as bbb or b xbx b: whence also it appears that a2b3 will signify the same as aabbb; and ac2 the same as aaaaacc; and so of others.

The note. (or a full point) and the word into, are likewise used instead of x, or as marks of multiplication.

Thus a+b. a + c and a + b into a + c both signify the same things as a + b × a + c, namely, the product of a + b by a + c.

The sign is used to signify that the quantity preceding it is to be divided by the quantity which comes after it: Thus cb signifies that c is to be divided by b; and a+ba-c, that a + b is to be divided by ab

Also the mark) is sometimes used as a note of division; thus, a + b) ab, denotes that the quantity ab is to be divided by the quantity a+b; and so of others. But the division of algebraic quantities is most com

1

monly expressed by writing down the divisor under the dividend with a line between them (in the manner of Thus represents the quantity

a vulgar fraction.)

b

arising by dividing c by b; and

a+b

a-c

denotes the quan

tity arising by dividing a + b by a-c. Quantities thus expressed are called algebraic fractions; whereof the upper part is called the numerator, and the lower the denominator, as in vulgar fractions.

The sign, is used to express the square root of any quantity to which it is prefixed: thus 25 signifies the square root of 25 (which is 5, because 5 × 5 is 25) thus also ab denotes the square root of ab; and ab + bc ab+bc d d

denotes the square root of

- or of

the quantity which arises by dividing ab + bc by d: ✔ab + bc but (because the line which separates the d

numerator from the denominator is drawn below √) signifies that the square root of ab+bc is to be first taken, and afterwards divided by d: so that, if a was 2,

b 6, c 4, and d 9, then would

✔ab + be
d

36

be

or

9

6

The same mark ✔, with a figure over it, is also used to express the cube,, or biquadratic root, &c. of any quantity thus 64 represents the cube root of 64, (which is 4, because 4 × 4 × 4 is 64,) and ab+cd

the cube root of ab + cd; also /16 denotes the biquadratic root of 16 (which is 2, because 2 × 2 × 2 × 2 is 16;) and ab+cd denotes the biquadratic root of ab + cd; and so of others. Quantities thus expressed are called radical quantities, or surds; where

of those, consisting of one term only, as a and ab, are called simple surds; and those consisting of several terms, or members, as ✔a2-b2 and a2-b2 + bc, compound surds.

*

the

Besides this way of expressing radical quantities, (which is chiefly followed) there are other methods made use of by different authors; but the most commodious of all, and best suited to practice, is that where the root is designed by a vulgar fraction, placed at the end of a line drawn over the quantity given. According to this notation, the square root is designed by the fraction, the cube root by, and the biquadratic root by 4, &c. Thus a] expresses the same thing with ✔a, viz. the square root of a; and a2 + ab same as a2 + ab, that is, the cube root of a2 + ab : also adenotes the square of the cube root of a; and a+ the seventh power of the biquadratic root of a+; and so of others. But it is to be observed, that, when the root of a quantity represented by a single letter is to be expressed, the line over it may be neglected; and so a will signify the same as a], and by the same as b] orb. The number, or fraction, by which the power or root of any quantity is thus designed, is called its index, or exponent.

=

The mark (called the sign of equality) is used to signify that the quantities standing on each side of it are equal. Thus 2+3=5, shows that 2 more 3 is equal to 5; and x-a-b, shows that x is equal to the difference of a and b.

The note: signifies that the quantities between which it stands are proportional: as abcd, denotes that a is in the same proportion to b, as c is to d; or that if a be twice, thrice, or four times, &c. as great as b.