ALGEBRA.

I. DEFINITIONS AND NOTATION.

1. Algebra is that branch of mathematics in which the relations of numbers are investigated, and the reasoning abridged and generalized by means of symbols.

Note. Writers on Algebra employ the word "quantity" as synonymous with "number"; this definition of the word will be understood throughout the present work.

2. The Symbols of Algebra are of four kinds :

1. Symbols of Quantity.

2. Symbols of Operation.

3. Symbols of Relation.

4. Symbols of Abbreviation.

SYMBOLS OF QUANTITY.

3. The symbols of quantity generally used are the figures of Arithmetic, and the letters of the alphabet.

Figures are used to represent known quantities and determined values; while letters may represent any quantities whatever, known or unknown.

4. Known Quantities, or those whose values are given, when not expressed by figures, are usually represented by the first letters of the alphabet, as a, b, c.

5. Unknown Quantities, or those whose values are to be determined, are usually represented by the last letters of the alphabet, as x, y, z.

6. Quantities occupying similar relations in the same problem, are often represented by the same letter, distinguished by different accents; as a', a", a"", read "a prime," 99.66 a second," a third," etc.

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They may also be distinguished by different subscript figures; as a1, a2, ag, read "a one," "a two," "a three," etc.

7. Zero, or the absence of quantity, is represented by the symbol 0.

SYMBOLS OF OPERATION.

8. The Sign of Addition, +, is called "plus."

Thus, a+b, read "a plus b,” indicates that the quantity b is to be added to the quantity a.

9. The Sign of Subtraction, -, is called "minus.” Thus, ab, read "a minus b," indicates that the quantity b is to be subtracted.from the quantity a.

Note. The sign ~ indicates the difference of two quantities; thus, ab denotes that the difference of the quantities a and b is to be found.

10. The Sign of Multiplication, x, is read "times," "into," or "multiplied by."

Thus, ab indicates that the quantity a is to be multiplied by the quantity b.

The sign of multiplication is usually omitted in Algebra, except between arithmetical figures; the multiplication of quantities is therefore indicated by the absence of any sign between them. Thus, 2 ab indicates the same as 2 × a × b. A point is sometimes used in place of the sign × between two or more figures; thus, 2.3.4 denotes 2 × 3 × 4.

11. Quantities multiplied together are called factors, and the result of the multiplication is called the product.

Thus, 2, a, and b are the factors of the product 2 ab.

12. A Coefficient is a number prefixed to a quantity to indicate how many times the quantity is to be taken.

Thus, in 4 ax, 4 is the coefficient of ax, and indicates that ax is to be taken 4 times; that is, 4ax is equivalent to ax+ax+ax tax.

When no coefficient is expressed, 1 is understood to be the coefficient. Thus, a is the same as 1a.

When any number of factors are multiplied together, the product of any of them may be regarded as the coefficient of the product of the others. Thus, in abcd, ab is the coefficient of cd; b of acd; abd of c; etc.

13. An Exponent is a figure or letter written at the right of, and above a quantity, to indicate the number of times the quantity is taken as a factor.

Thus, in a3, the 3 indicates that x is taken three times as a factor; that is, a is equivalent to xxx.

14. The product obtained by taking a factor two or more times is called a power. A single letter is also often called the first power of that letter. Thus,

a2 is read "a to the second power," or indicates aa;

a3 is read "a to the third power," or cates aaa;

66 a square," and

66 a cube," and indi

at is read "a to the fourth power," or indicates aaaa; etc.

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a fourth," and

When no exponent is written, the first power is understood; thus, a is the same as a1.

15. The Sign of Division,, is read "divided by."

Thus, ab denotes that the quantity a is to be divided by the quantity b.

Division is also indicated by writing the dividend above, and the divisor below, a horizontal line. Thus, indicates the same as a ÷ b. When thus written, is often read 66 α over b."

a

SYMBOLS OF RELATION.

16. The symbols of relation are signs used to indicate the relative magnitudes of quantities.

17. The Sign of Equality,, read "equals," or "is equal to," indicates that the quantities between which it is placed are equal.

Thus, x = y

indicates that the quantities x and y are equal.

A statement that two quantities are equal is called an equation.

Thus, x+4=2x 4 equals 2x minus 1.”

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- 1 is an equation, and is read x plus

18. The Sign of Inequality,> or <, read "is greater than” and “is less than" respectively, when placed between two quantities, indicates that the quantity toward which the opening of the sign turns is the greater.

Thus, x>y is read " is greater than y"; x 6 <y is read" minus 6 is less than y."

SYMBOLS OF ABBREVIATION.

19. The Sign of Deduction, ..., stands for therefore or hence.

20. The Signs of Aggregation, the parenthesis (), the brackets [], the braces {}, and the vinculum indicate that the quantities enclosed by them are to be taken collectively. Thus,

(a+b)x, [a+b]x, {a+b}x, a+b×x,

all indicate that the quantity obtained by adding a and b is to be multiplied by x.

reads "a, a plus b, a plus 2b, a plus 3b, and so on."

ALGEBRAIC EXPRESSIONS.

22. An Algebraic Expression is any combination of algebraic symbols; as 2x2 - 3 ab+c3.

23. A Term is an algebraic expression whose parts are not separated by the signs or -; as 2x2, -3 ab, or + c3.

2 x2, - 3 ab, and c3 are called the terms of the expression 2x2-3 ab+c3.

24. Positive Terms are those preceded by a plus sign; as

+2x2, or +c3.

For this reason, the sign + is often called the positive sign. If no sign is expressed, the term is understood to be positive; thus, a is the same as +a.

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25. Negative Terms are those preceded by a minus sign;

For this reason, the sign is often called the negative sign; it can never be omitted before a negative term.

Note. In a negative term, the numerical coefficient indicates how many times the quantity is to be taken subtractively. (Compare Art. 12.)

Thus, 3ab is equivalent to ab ab - ab.

26. In Arithmetic, if the same number be both added to and subtracted from another, the value of the latter will not be changed. Thus,

5+3-35.
=