NEW SCHOOL ALGEBRA..

CHAPTER I.

DEFINITIONS AND NOTATION.

Numbers and Number-Symbols.

1. Algebra. Algebra, like Arithmetic, treats of numbers.

2. Units. In counting separate objects or in measuring magnitudes, the standards by which we count or measure are called units.

Thus, in counting the boys in a school, the unit is a boy; in selling eggs by the dozen, the unit is a dozen eggs; in selling bricks by the thousand, the unit is a thousand bricks; in expressing the measure of short distances, the unit is an inch, a foot, or a yard; in expressing the measure of long distances, the uniti

4. Quantities. A number of specified units of any kind is called a quantity; as 4 pounds, 5 oranges.

NOTE. Quantities are often called concrete numbers, the adjective concrete being transferred from the units counted to the numbers that count them; but a number signifies the times a unit is taken, whether the unit is expressed or understood, and is always abstract.

Thus, 4 barrels of flour means 4 times 1 barrel of flour; and 10 cords of wood means 10 times 1 cord of wood.

5. Number-Symbols in Arithmetic. Instead of groups of straight marks, we use in Arithmetic the arbitrary symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, called Arabic numerals, for the numbers one, two, three, four, five, six, seven, eight, nine.

The next number, ten, is indicated by writing the figure 1 in a different position, so that it shall signify not one, but ten. This change of position is effected by introducing a new symbol, 0, called nought or zero, and signifying none.

All succeeding numbers up to the number consisting of 10 tens are expressed by writing the figure for the number of tens they contain in the second place from the right, and the figure for the number of units besides in the first place. The hundreds of a number are written in the third place from the right. The thousands are written in the fourth place from the right; and so on.

6. Number-Symbols in Algebra. Algebra employs the letters of the alphabet in addition to the figures of Arithmetic to represent numbers. The letters of the alphabet are used as general symbols of numbers to which any particular values may be assigned. In any problem, however, a letter is understood to have the same value throughout the problem.

7. Terms Common to Arithmetic and Algebra. Terms common to Arithmetic and Algebra, as addition, sum, subtraction, minuend, subtrahend, difference, etc., have the same meaning in both; or an extended meaning in Algebra consistent with the sense attached to them in Arithmetic.

The Principal Signs of Operations.

The principal signs of operations are the same in Algebra as in Arithmetic.

8. The Sign of Addition, +. The sign + is read plus. Thus, 4+3 is read 4 plus 3, and indicates that the number 3 is to be added to the number 4; a + b is read a plus b, and indicates that the number b is to be added to the number a.

9. The Sign of Subtraction, —. The sign is read minus. Thus, 4 3 is read 4 minus 3, and indicates that the number 3 is to be subtracted from the number 4; α -b is read a minus b, and indicates that the number b is to be subtracted from the number a.

10. The Sign of Multiplication, X. The sign X is read times, or multiplied by.

Thus, 4 × 3 is read 4 times 3, and indicates that the number 3 is to be multiplied by 4; a × b is read a times b, and indicates that the number b is to be multiplied by the number a.

A dot is sometimes used for the sign of multiplication. Thus, 2.3.4.5 means the same as 2×3×4×5. Either sign is read multiplied by when followed by the multiplier. $a × b, or $ab, is read a dollars multiplied by b.

11. The Sign of Division, ÷ The sign is read divided by. Thus, 42 is read 4 divided by 2, and indicates that the number 4 is to be divided by 2; a÷bis read a divided by b, and indicates that the number a is to be divided by the number b.

Division is also indicated by writing the dividend above the divisor with a horizontal line between them; or by separating the dividend from the divisor by an oblique line, called the solidus.

Thus, or a/b, means the same as a÷b.

NOTE. The operation of adding b to a, of subtracting b from a, of multiplying a by b, or of dividing a by b is algebraically complete when the two letters are connected by the proper sign.

12. The Radical Sign, √. The sign is called the radical sign, and denotes that a root of the number before which it is placed is to be found.

Other Signs Used in Algebra.

13. The Sign of Equality, =. The sign = is read is equal to, and when placed between two numbers indicates that these two numbers are equal.

Thus, 8 + 4 = 12 means that the sum of 8 and 4 is equal to 12; x + y = 20 means that the sum of x and y is equal to 20; andx=a+b means that x is equal to the sum of a and b.

14. The Sign of Deduction, ... The sign... is read hence or therefore.

15. The Sign of Continuation, ..... The sign ..... is read and so on.

Thus, 1, 2, 3, 4,

is read one, two, three, four, and so on. α1, an is read a sub one, a sub two, a sub three, and so on to a sub n. a', α", α'", is read a prime, a second, a third, and so on.

16. The Signs of Aggregation. The signs of aggregation are the parenthesis (), the bracket ( ), the brace {}, the vinculum, and the bar |.

These signs mean that the indicated operations in the expressions affected by them are to be performed first, and the result treated as a single number.

Thus, (a + b) × c means that the sum of a and b is to be multiplied by c; (a - b) × cmeans that the difference of a and b is to be multiplied by c.

The vinculum is written over the expression that is to be treated as a single number.

Thus, a-b+c means the same as a (b+c), and signifies that the sum of b and cis to be subtracted from a; and Va-b means the same as √(a - b), and signifies that bis to be subtracted from a, and the square root of the remainder found.

Factors, Powers, Roots.

17. Factors. When a number is the product of two or more numbers, each of these numbers, or the product of two or more of them, is called a factor of the given number. Thus, 2, a, b, 2a, 2b, ab are factors of 2 ab.

18. Factors expressed by letters are called literal factors; factors expressed by figures are called numerical factors.

19. The sign × is omitted between factors, if the factors are letters, or a numerical factor and a literal factor. Thus, we write 63 ab for 63 ×a×b; we write abc for a × b × c. 20. The expression abe must not be confounded with a+b+c. abc is a product; a+b+cis a sum.

If

then

but

a = 2, b = 3, c = 4,

abc = 2 × 3 × 4 = 24;

a+b+c=2+3+4 = 9.

NOTE. When a sign of operation is omitted in the notation of Arithmetic, it is always the sign of addition; but when a sign of operation is omitted in the notation of Algebra, it is always the sign of multiplication. Thus, 456 means 400+50+6, but 4 ab means 4 xaxb.

21. If one factor of a product is equal to 0, the product is equal to 0, whatever the values of the other factors. Such a factor is called a zero factor.

Thus, abcd = 0, if a, b, c, or d = 0.

22. Coefficients. Any factor of a product may be considered as the coefficient of the remaining factors; that is, the co-factor of the remaining factors. Coefficients expressed by letters are called literal coefficients; expressed by Arabic numerals, numerical coefficients.

Thus, in 7x, 7 is the numerical coefficient of x; in ax, a is the literal coefficient of x.

If no numerical coefficient is written, 1 is understood.