ELEMENTARY ALGEBRA.

1. Algebra is a branch of mathematics in which quantities are represented by letters and processes by signs. It may be con sidered an extension of Arithmetic.

SYMBOLS.

The quantities and processes used in algebra are represented by symbols.

2. Symbols of Quantity are the letters of the alphabet and the figures used in Arithmetic.

The first letters of the alphabet and figures are used to represent those quantities whose values are known.

Unknown Quantities are those whose values are to be found and are represented by the last letters of the alphabet.

3. Symbols of Operation indicate that some process is to be performed.

4. The sign of addition + is called plus; thus, 3+4 is read "3 plus 4" and means that 4 is to be added to 3. Similarly, a+b means that the quantity represented by b is to be added to the quantity represented by a.

5. The sign of subtraction

is called minus; thus, e -ƒ

is read "e minus f" and means that fis to be subtracted from e.

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6. Multiplication is expressed by the sign X which is read multiplied by" or "times."

7. Division may be expressed by the signor by writing the dividend above and the divisor below a horizontal line. Thus,

8. These signs are the same as those used in Arithmetic.

In multiplication, however, there is a difference.

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In Arithmetic,

the sign must always be used, but in Algebra it is frequently omitted.

Thus,

ab my is the same as

x y

a x b x m x y

х Ху

If, however, we omit the sign between the figures in the

We can write 6 X a X b as 6 a b. Also, 12 X 4 X m × nas 12 x 4 mn but not 124 mn.

9. We may say: The sign of multiplication may be omitted between letters and between letters and figures but not between figures. When there is no sign between letters multiplication is understood. For this reason the other signs must be used.

The only exception to the above statement is found in the case of the sign; it may be omitted at the beginning. Thus, 2a+3b is the same as +2a+3b. The sign of 2 a is understood to be +.

10. Sometimes a point or period is used in place of the multiplication sign; thus,

a. b. c. d. a xbx ex d=abed.

11. A Coefficient is a number prefixed or placed before a quantity to indicate how many times the quantity is to be taken. Thus, in 3 a b, 3 is the coefficient and shows that ab is to be taken 3 times. In other words 3 a b is the same as a b+ ab + ab. Let us consider the expression 6 m n. The coefficient of mn is 6 and 6 m is considered the coefficient of n. Similarly, 6 n is the coefficient of m.

When no coefficient is expressed, it is understood to be 1. Thus c is the same as 1 c because e multiplied by 1 is c.

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12. An Exponent is a figure or letter written at the right and a little above a quantity to indicate the number of times the quantity is to be taken as a factor. For example, in c1 the 4 shows that e is to be taken 4 times as a factor. c is equivalent to eXcX CX c.

13.

When no exponent is written, the first power is understood; bis

the same as b1.

b2 is "b to the second power" or b square; it is equal to b b.

b3 is "b to the third power" or b cube; it is equal to b b b.

b is "b to the fourth power" and is equal to b b b b.

The distinction between coefficient and exponent must be thoroughly understood. In the case of the coefficient, the coefficient shows how many times the quantity is taken to make a given sum. The exponent indicates how many times the quantity is to be taken to make a given product. Thus,

4 a = a + a + a + a.

a4 = а ХаХаХа.

14. Symbols of Relation are the signs which show equality and inequality.

The sign is read "equals." The expression a = b means that the quantity represented by a is equal to the quantity represented by b.

When two quantities are connected by the = sign, the expression is called an equation.

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The sign is read "is greater than" and the sign <"is less than." Thus, ab means that the quantity represented by a is greater than the quantity represented by b. that m is less than n.

The sign means "is equivalent to."

mn means

15. Symbols of Abbreviation. The following are the most common symbols of abbreviation:

In Algebra the parentheses, brackets, braces and vinculum are used in the same way as in Arithmetic and have exactly the same meaning.