SCHOOL ALGEBRA

INTRODUCTION

1. In passing from arithmetic to algebra the meaning of number and the method of representing it are extended, but there is nothing contradictory to what has already been learned in arithmetic. Algebra, like arithmetic, treats of number, and may be regarded as generalized arithmetic.

The symbols, 1, 2, 3, etc., are retained in algebra with their arithmetical meanings, and the same symbols, +, ×, ÷, ( ), =, are used in each. Fractions, powers, and roots have the same as well as an extended meaning, and are written in the same form.

LITERAL OR GENERAL NUMBER

2. An important difference between arithmetic and algebra comes from the frequent and extended use in the latter of letters to represent numbers. Just as, in interest problems, p may stand for principal, for rate per cent, t for the time, etc., so in any case such symbols as a, b, x, y, may be used to represent any numbers whatever.

In arithmetic we speak of 6 feet, meaning a certain number of feet, or of $10, meaning a certain number of dollars; in algebra we speak of n feet, meaning any number or an unknown number of feet, or of a dollars, meaning any number or an unknown number of dollars.

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3. Numbers represented by letters are called Literal or General Numbers. The reasoning is the same whether numbers are represented by letters or by figures.

Thus, if a stands for the number of pupils in a room, then 2 a stands for twice this number, 3 a for three times this number, etc.

EXERCISES

4. 1. If stands for the number of books in my library, what is the meaning of 3n? Of 5n? Of 6n? Of in?

2. If x, y, and z stand for the cost of a horse, a cow, and a sheep respectively, for what does + y stand?

x + y + z?

7, what is the value of + y ?

4. If in a number of two digits, the digit in the ones' place is 5, and the digit in the tens' place is 2, the number is 2 × 10 +5. Write a number containing a ones and tens. What

is the number if a 5 and b = 3?

=

If a = 2, b = 3, c= 4, d= 5, find the values of the following:

5. As in arithmetic, the Symbol of Addition, +, called plus, is placed before the number to be added.

Thus, a + b, read a plus b, means that is to be added to a, just as 4+3 means that 3 is to be added to 4.

6. The Symbol of Subtraction, -, called minus, is placed before the number to be subtracted.

b, read a minus b, means that b is to be subtracted from ɑ.

7. The Symbol of Multiplication, ×, read multiplied by, when placed between two numbers or expressions, indicates that the one on the left is to be multiplied by the one on the right. A dot () is sometimes used instead of the sign ×, but more frequently multiplication is indicated by the absence of all symbols, especially where only letters or a numeral and one or more letters are used.

Thus, a × b, a · b, and ab all indicate the product of a multiplied by b. Twice the product of a by b is usually written 2 ab, and read two ab. In indicating the multiplication of numerals, however, we cannot omit the sign ×, as that would change the meaning of the expression. Thus, 4 x 7 cannot be written 47. Why not?

8. The Symbol of Division, ÷, read divided by, when placed between two numbers, shows that the one on the left is to be divided by the one on the right. The colon and the fractional form are also used to indicate division.

Thus, ab, a:b, and all mean the quotient of a divided by b.

9. The expression (a + b) k, or [a + b] k, means that a + b is to be multiplied by k. The parentheses and the brackets inclosing a + b are called Symbols of Aggregation. The bar and braces {} are also used.

(4+3) a = 7a; (5-2) a=3a; (4+2) − (85) = 6 −3.

10. Custom has established the following Conventional Order for performing the operations where several are involved, as in the expression a+bxc÷d-e: First, the multiplications and divisions are to be performed successively from left to right; second, the additions and subtractions in the same order.

The order will be seen in the following:

(a) 5+2×6÷3¬4 = 5 + 12 ÷3-4=5+4-− 4 = 5. (b) 5+12÷2×3-6=5+6 × 3-6 = 5 + 18-617. (e) 512(2 × 3) - 6 = 5+12÷6-6=5+26=1.

If parentheses appear in the expression, the indicated operations within the parentheses must first be performed, as in (c) above.

EXERCISES

11. 1. The sum of a and b is indicated thus: a+b. What is the sum of a and c ? Of b and d?

2. One pail holds a quarts, another 3 a quarts. They both hold

quarts.

3. May has b yards of ribbon, Helen has a yards. How many yards have they both?

4. 3b years is 2d years less than my age. I am years of age.

5. A table is m feet long and n feet wide. What is its perimeter, or distance around the table?

6. Taking from a is indicated thus: ab. How can you express the taking of c from ɑ ? d from b?

7. John took a quarts of milk out of a pail containing b quarts. There remained. quarts.

8. A pencil was m inches long. I used n inches of it. What does m n inches indicate?

9. Homer had $a and Frank had $b. They spent $c. What indicates the sum they had left?

10. The product of a by b is ab. What is the product of a by c? Of b by d? Of 2 by a?

11. At $c each, what will d sheep cost?

12. How many shoes will be needed for b boys if each boy gets 2 shoes?

13. A field is m rods long and n rods wide. It contains square rods.

14. A dog ran 8 a rods and a rabbit ran 5b rods.

rods farther than the rabbit.

a = 3 and b = 5, which ran the farther?

15. If m÷n or indicates the quotient of m divided by

m
n

n, what is the quotient of a divided by b? Of 3x divided by 3? By x?

16. A man gave $b to c boys. If each boy got the same sum, what was that sum?

17. The area of a field is a square rods; its length is d rods. Its width is

rods.

12. A collection of letters, or of letters and other number symbols, connected by one or more of the signs of operation (+, -, x,, etc.) is called an Algebraic Expression.

Thus, 2 x + a and 3 abc are algebraic expressions. It should be observed that a means a × 1, or one a.

13. The parts of an algebraic expression connected by the signs and - (neither sign being within parentheses) are called its Terms. If the expression contains neither sign, it has but one term and is called a Monomial.

Thus, 4 ab, x, and 5 m are monomials. In the expression 2 a + 3 b − 5 c, the three terms are 2 a, 3b, and 5 c, and each taken by itself is a monomial. In ax + 2(a + b) c, there are three terms, 2 (a+b) being treated as a single term.