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A L G E B R A.
DEFINITIONS AND NOTATION. 1. Quantity is any thing that can be increased or diminished, and that can be measured.
A line, a surface, a solid, a weight, etc., are quantities; but the operations of the mind, such as memory, imagination, judg. ment, etc., are not quantities.
A quantity is measured by finding how many times it contains some other quantity of the same kind taken as a standard. The assumed standard is called the unit of measure.
2. Mathematics is the science of quantity, or the science which treats of the properties and relations of quantities. It employs a variety of symbols to express the values and relations of quantities, and the operations to be performed upon these quantities, or upon the numbers which represent these quantities.
3. Mathematics is divided into pure and mixed. Pure math. ematics comprehends all inquiries into the relations of quantity in the abstract, and without reference to material bodies. It embraces numerous subdivisions, such as Arithmetic, Algebra, Geometry, etc.
In the mixed mathematics, these abstract principles are applied to various questions which occur in nature. Thus, in Surveying, the abstract principles of Geometry are applied to the measurement of land; in Navigation, the same principles are applied to the determination of a ship's place at sea; in Optics, they are employed to investigate the properties of light; and in Astronomy, to determine the distances of the heavenly bodies.
4. Algebra is that branch of mathematics in which quantities are represented by letters, and their relations to each other, as well as the operations to be performed upon them, are indicated by signs or symbols. The object of algebraic notation is to abridge and generalize the reasoning employed in the solution of all questions relating to numbers. Algebra may there- . fore be called a species of Universal Arithmetic.
5. The symbols employed in Algebra may be divided into three classes :
1st. Symbols which denote quantities.
2d. Symbols which indicate operations to be performed upon quantities.
3d. Symbols which indicate the relations subsisting between different quantities, with respect to their magnitudes, etc.
Symbols which denote Quantities. 6. In order to generalize our reasoning respecting numbers, we represent them by letters, as a, b, c, or X, Y, Z, etc., and these may represent any numbers whatever. The quantities thus represented may be either known quantities—that is, quantities whose values are given; or unknown quantities—that is, quantities whose values are to be determined.
Known quantities are generally represented by the first letters of the alphabet, as a, b, c, d, etc., and unknown quantities by the last letters of the alphabet, as x, y, z, U, etc. This, however, is not a necessary rule, and is not always observed.
7. Sometimes several quantities are represented by a single letter, repeated with different accents, as a', a", a'", a''", etc., which are read a prime, a second, a third, etc.; or by a letter repented with different subscript figures, as ay, , az, a,, etc., which may be read a one sub, a two sub, a three sub, etc. All these symbols represent different quantities, but the accents or numerals are employed to indicate some important relation between the quantities represented.
8. Sometimes quantities are represented by the initial letters of their names. Thus s may represent sum; d, difference or diameler; r, radius or ratio ; c, circumference; h, height, etc. All these letters may be used with accents. Thus, in a problem
a relating to two circles, d may represent the diameter of one cir. cle, and d' the diameter of the other; c the circumference of one, and c' the circumference of the other, etc.
Symbols which indicate Operations. 9. The sign of addition is an erect cross, +, called plus, and when placed between two quantities it indicates that the second is to be added to the first. Thus, 5+3 indicates that we must add 3 to the number 5, in which case the result is 8. We also make use of the same sign to connect several numbers together. Thus, 7+5+9 indicates that to the number 7 we must add 5 and also 9, which make 21. So, also, 8+5+13+11+1 +3+10 is equal to 51.
The expression a+b indicates the sum of two numbers, which we represent by a and b. In the same manner, m+n+a+y indicates the sum of the numbers represented by these four letters. It we knew, therefore, the numbers represented by the letters, we could easily find by arithmetic the value of such expressions.
10. The sign of subtraction is a short horizontal line, -, called minus. When placed between two quantities, it indicates that
. the second is to be subtracted from the first. Thus, 8-5 indicates that the number 5 is to be taken from the number 8, which leaves a remainder of 3. In like manner, 12-7 is equal to 5, etc.
Sometimes we may have several numbers to subtract from a single one. Thus, 16-5-4 indicates that 5 is to be subtracted from 16, and this remainder is to be further diminished by 4, leaving 7 for the result. In the same manner, 50–1–5–3– 9-7 is equal to 25.
The expression a-b indicates that the number designated by a is to be diminished by the number designated by b.
11. The double sign is sometimes written before a quantity to indicate that in certain cases it is to be added, and in others it is to be subtracted. Thus, bc is read be plus or minus c, and denotes either the sum or the difference of these two quantities.
12. The sign of multiplication is an inclined cross, X. When placed between two quantities, it indicates that the first is to be multiplied by the second. Thus, 3 x 5 indicates that 3 is to be multiplied by 5, making 15. In like manner, a xb indicates that a is to be multiplied by b; and axbxc indicates the continued product of the numbers designated by a, b, and c, and so on for any number of quantities.
Multiplication is also frequently indicated by placing a point between the successive letters. Thus, a.b.c.d signifies the same thing as a xbxcxd.
Generally, however, when numbers are represented by letters, their multiplication is indicated by writing them in suc. cession without any intervening sign. Thus, abc signifies the same as a xbxc, or a.b.c.
The notation a.b or ab is seldom employed except when the numbers are designated by letters. If, for example, we attempt to represent in this manner the product of the numbers 5 and 6, 5.6 might be confounded with 500; and 56 would be read fifty-six, instead of five times six.
The multiplication of numbers may, however, be denoted by placing a point between them in cases where no ambiguity can arise from the use of this symbol. Thus, 188.8.131.52.5 is sometimes used to represent the continued product of the numbers 1, 2, 3, 4, 5.
13. When two or more quantities are multiplied together, each of them is called a factor. Thus, in the expression 7x5, 7 is a factor, and so is 5. In the product abc there are three factors, a, b, c.
When a quantity is represented by a letter, it is called a literal factor. When it is represented by a figure or figures, it is called a numerical factor. Thus, in the expression 5ab, 5 is a numerical factor, while a and b are literal factors.
14. The sign of division is a short horizontal line with a point above and one below, -. When placed between two quantities, it indicates that the first is to be divided by the second.
Thus, 24-6 indicates that 24 is to be divided by 6, making 4. So, also, a-6 indicates that a is to be divided by b.
Generally, however, the division of two numbers is indi. cated by writing the divisor under the dividend, and drawing a line between them. Thus, 24-6 and a-b are usually writ
15. The products formed by the successive multiplication of the same number by itself, are called the powers of that number. Thus, 2 x2=4, the second power or square of 2.
2x2x2=8, the third power or cube of 2.
2x2x2x2=16, the fourth power of 2, etc. So, also, 3X3=9, the second power of 3.
3x3x3=27, the third power of 3, etc. Also, axa=aa, the second power of a.
axaxa=aaa, the third power of a, etc. In general, any power of a quantity is designated by the number of equal factors which form the product.
16. The sign of involution is a number written above a quantity, at the right hand, to indicate how many times the quantity is to be taken as a factor.
A root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity.
The figure which indicates how many times the root or factor is taken, is called the exponent of the power.
Thus, instead of aa, we write a’, where 2 is the exponent of the power; instead of aaa, we write a’, where 3 is the expo