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RAY'S

ALGEBRA:

PART SECOND.

CHAPTER I.

DEFINITIONS AND NOTATION.

ARTICLE 1. Quantity is anything capable of being increased
or diminished; such as numbers, lines, space, time, motion, &c.

REMARK. If the pupil does not already know, let the instructor here
explain to him what is meant by unit of quantity, the numerical value
of quantity, &c. See Ray's Algebra, Part First, Arts. 2 to 10.

ART. 2. Mathematics is the science of quantity, and of the
symbols by which quantity is represented.

ART. 3. Algebra is that branch of Mathematics which relates
to the solution of problems and the demonstration of theorems,
when any of the quantities employed are designated by letters.
ART. 4. A problem is a question proposed for solution; a theo-
rem is a proposition to be proved by a chain of reasoning.

ART. 5. The operations of Algebra are conducted by means of
figures, letters, and signs. The letters and signs are often called
symbols.

ART. 6. Known quantities are generally represented by the
first letters of the alphabet, as a, b, c, &c.; and unknown quanti-
ties by the last letters, as t, v, x, y, z.

= +,

ART. 7. The principal signs used in Algebra are the following:
X, ÷ ( ), >, √.
Each sign is the representative of certain words; and is used for
the purpose of expressing the various operations in the most clear
and brief manner.

ART. 8. The sign, is termed the sign of equality. It is read equal to, or equals, and denotes that the quantities between which it is placed are equal to each other. Thus, x=5, denotes that the quantity represented by x is equal to 5.

ART. 9. The sign+, is termed the sign of addition. It is read plus, and denotes that the quantity to which it is prefixed is to be added.

Thus, a+b denotes that b is to be added to a. If a=3, and b=5, then a+b=3+5, which are = =8.

ART. 10. The sign- is termed the sign of subtraction. It is read minus, and denotes that the quantity to which it is prefixed is to be subtracted. Thus, a-b denotes that b is to be subtracted from a. If a 7 and b=4, then 7-4-3.

ART. 11. The signs + and are called the signs; the former is called the positive, and the latter the negative sign; they are said to be contrary, or opposite.

ART. 12. Every quantity is supposed to have either the positive or negative sign. Quantities having the positive sign are called positive; those having the negative sign are called negative. When a quantity has no sign prefixed to it, the sign is understood, and it is to be considered positive.

ART. 13. Quantities having the same sign are said to have like signs; those having different signs are said to have unlike signs. Thus, +a and +b, or —a and —b have like signs; while +c and -d have unlike signs.

ART. 14. The sign X, is termed the sign of multiplication. It is read into, or multiplied by, and denotes that the quantities between which it is placed are to be multiplied together.

A dot, or period, is sometimes used to denote multiplication. Thus, ab, and a.b, each denote that a and b are to be multiplied together. The dot is not often employed to denote the multiplication of figures, since it is used, by some authors, to separate whole numbers and decimals.

The product of two or more letters is generally denoted by writing them in close succession. Thus, ab denotes the same as ab, or a.b; and abc means the same as aXbXc, or a.b.c; each signifying the continued product of the numbers designated by a, b, and c.

ART. 15. When two or more quantities together, each of them is called a factor.

are to be multiplied Thus, in the product

ab there are two factors, a and b; in the product 3×5×7 there

are three factors, 3, 5, and 7.

ART. 16. The sign, is termed the sign of division. It is read divided by, and denotes that the quantity preceding it is to be divided by that following it. The most common method of representing the division of two quantities is to place the dividend as the numerator, and the divisor as the denominator of a fraction. Thus, ab, or signifies that a is to be divided by b.

a

b

Division is also represented thus, ab, where a denotes the dividend, and b the divisor.

ART. 17. The sign >, is termed the sign of inequality. It denotes that one of the two quantities between which it is placed is greater than the other, the opening of the sign being turned toward the greater quantity.

Thus, ab, denotes that a is greater than b. It is read, a greater than b. If a=7 and b=2, then 7>2.

Also, cd, denotes that c is less than d. It is read, c less than d. If c-3 and d=5, then 3<5.

ART. 18. A coëfficient is a number or letter prefixed to a quantity, to show how often it is taken.

Thus, if a is to be taken 4 times, instead of writing a+a+a +a, we write 4a; in like manner, az+az+az=3az. The coëfficient is called numeral, or literal, according as number or a letter. Thus, in the quantities 5x and mx, numeral and m a literal coëfficient.

it is a 5 is a

In 3az, 3 may be considered as the coëfficient of az, or 3a may be considered as the coëfficient of z.

When no numeral coëfficient is expressed, 1 is always understood. Thus, a is the same as la, and ax is the same as lax.

ART. 19. A power of a quantity, is the product arising from multiplying the quantity by itself one or more times. When the quantity is taken twice as a factor, the product is called the second power; when taken three times, the third power, and so on. Thus, 2x2=4, is the second power of 2.

2x2x2=8, is the third power of 2. 2×2×2×2=16, is the fourth power of 2. Also, aXa=aa, is the second power of a.

aXaXa=aaa, is the third power of a; and so on. The second power is often termed the square, and the third power, the cube. To avoid repeating the same quantity as a factor, a small figure, termed an exponent, is placed on the right, and a little above it, to denote the number of times the quantity is taken as a factor, Thus, aa is written a2; aaa is written a3; aabbb is written a2b3.

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