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ALGEBRA is the science which treats of a general method of performing calculations, and resolving mathematical problems, by means of the letters of the alphabet.
Its leading rules are the same as those of arithmetic; and the operations to be performed are denoted by the following characters :
+ plus, or more, the sign of addition ; signifying that the quantities between which it is placed are to be added together.
Thus, a + b shows that the number, or quantity, represented by b, is to be added to that represented by a ; and is read a plus .
minus, or less, the sign of substraction; signifying that the latter of the two quantities between which it is placed is to be taken from the former.
b shows that the quantity represented by b is to be taken from that represented by a : and is read a minus b.
Also, arb represents the difference of the two quantities a and b, when it is not known which of them is the greater.
x into, the sign of multiplication; signifying that the quantities between which it is placed are to be multiplied together.
Thus, a x b shows that the quantity represented by a is to be multipled by that represented by b; and is read a into b.
The multiplication of simple quantities is also frequently denoted by a point, or by joining the letters together in the form of a word.
Thus, a xb, aob, and ab, all signify the product of a and b; also, 3 X d, or 3a, is the product of 3 and a; and is read 3 times a.
by, the sign of division ; signifying that the former of the two quantities between which it is placed is to be divided by the latter.
Thus, a -- b, shows that the quantity represented by a is to be divided by that represented by b; and is read a by b. or a divided by b.
Division is also frequently denoted by placing one of the two quantities over the other, in the form of a fraction
Thus, ba and - both signify the quotient of b divided
-6 by a; and
signifies that a -- b is to be divided by
ato. a tc.
equal to, the sign of equality ; signifying that the quantities between which it is placed are equal to each other.
Thus, x = a + b shows that the quantity denoted by x is equal to the sum of the quantities a and b; and is read x equal to a plus b.
Any two algebraic expressions are said to be identical, when they are of the same value , for all the values of the letters of which they are composed.
* Thus (a + a) x (x − a) 22 -- a, whatever numeral values may be given to the quantities represented by a
> greater than, the sign of majority ; signifying that the former of the two quantities between which it is placed is greater than the latter,
Thus a > b shows that the quantity represented by a is greater than that represented by b; and is read a greater
< less than, the sign of minority ; signifying that the former of the two quantities between which it is placed is less than the latter.
Thus, a < b shows that the quantity represented by a is less than that represented by b; and is read a less than b.
as, or to, and :: so is, the signs of an equality of ratios; signifying that the quantities between which they are placed are proportional.
Thus, a:b::C:d denotes that a has the same ratio to b that c has to d, or that a, b, c, d, are proportionals; and is read, as a is to b so is c to d, or, a is to b as c is to d.
✓ the radical sign, signifying that the quantity before which is placed is to have some root of it extracted.
* Woodhouse, in his Principles of Analytical Calculation, says that
na is not generally = (-a): (xta); for instance, the particular case of x = a is to be excluded; the proof essentially demanding this circumstance, to wit, that cabe a quantity, or thať z be greater than a. Euler calls x 1 1 an identical equation ; and shows that x is indeterminate, or that any number whatever may be substituted for it. See Euler's Algebra, page 289, Vol. I.-ED.
Thus, va is the square root.of a; Va is the cube root of a; and va is the fourth root of a ;. &c.
The roots of quantities are also represented by figures placed at the righthand corner of them, in the form of a fraction.
Thus, a' is the square root of a; ai is the cube root of a; and an is the nth root of a, or a root denoted by any num
In like manner, a is the square of a ; a' is the cube of a; and an is the mth power of a, or any power denoted by the number m.
o is the sign‘of infinity, signifying that the quantity standing before it is of an unlimited value, or greater than any quantity that can be assignede
The coefficient of a quantity is the number or letter which is prefixed to it.
Thus, in the quantities 30, - 3b, 3 and - are the coefficients of b; and a is the coefficient of x in the quantity ax.
A quantity without any coefficient prefixed to it is supposed to have 1 or unity, and when a quantity has no sign before it, + is always understood. Thus, a is the same as + a, or + la; and
a is the la. A term is any part or member of a compound quantity, which is separated from the rest by the signs + or Thus a and b are the terms of a +b; and 3at,
2b, and + 5cd, are the terms of 3a 2b + 5cd.
In like manner, the terms of a product, fraction, or proportion, are the several parts or quantities of which they are composed.
Thus, a and b are the terms of ab, or of
and a, b, c, d,
2 are the terms of the proportion a:b::c:d.
A factor is one of the terms, or multipliers which form the product of two or more quantities.
Thus, a and b are the factors of ab; also, 2, a, and , are the factors of 2ab2; and a and o
- X are the factors of the product (a - x) x (b − x).
A composite number, or quantity, is that which is produced by the multiplication of two or more terms or factors.
Thus, 6 is a composite number, formed of the factors 2 and 3, or 2 X 3; and 3abc is a composite quantity, the factors of which are 3, a, b, c.
Like quantities, are those which consist of the same letters or combinations of letters; as a and 3a, or 5ab and 7ab, or 2a-6 and 9c2b.
Unlike quantities, are those which consist of different letters, or combinations of letters; as a and b, or 3a, and a-, or 5ab2 and 7a26.
Given quantities, are such as have known values, and are generally represented by some of the first letters of the alphabet; as a, b, c, d, &c.
Unknown quantities, are such as have no fixed values, and are usually represented by some of the final letters of the alphabet; as x, y, 2.
Simple quantities are those which consist of one term only; as 3a, 5ab, 8aob, &c.
Compound quantities, are those which consist of several terms; as 20 + b, or 3a - 2c, or a + 2b - 3c, &c.
Positive, or affirmative quantities, are those which are to be added; as a, or to ab, or + 3ab, &c. Negative quantities are those which are to be subtracted : 3ab, or
&c. Like signs, are such as are all positive, or all negative; as + and +, or and
Unlike signs, are when some are positive and others negative; as + and
A monomial, is a quantity consisting of one term only; as d, 2b, — 3a4b,
3a2b, &c. A binomial, is a quantity consisting of two terms, as a + b, or a -- b; the latter of which is, also, sometimes called a residual quantity
A trinomial, is a quantity consisting of three terms, as a + 25
3c; a quadrinomial of four, as a 2b + 3c-d, and a polynomial, or multinomial, is that which has many
The power of a quantity, is its square, cube, biquadrate, &c.; called also its second, third, fourth power, &c.; as a, a, a, &c.
The index, or exponent of a quantity, is the number which denotes its power or root. Thus, 1 is the index of a-1, 2 is the index of a>, and 1
- 름 of á or V a.
When a quantity appears without any index, or exponent, it is always understood to have unity, or 1.
Thus, à is the same as a', and 2x is the same as 2x2; the 1, in such cases, being usually omitted.