denotes that a quantity preceding it is less than another following it.

Thus, ab means that a is less than b. If a = 5 and b=6, then 5 ▲ 6.

(27.) The sign co or, denotes a quantity greater than any that can be assigned; that is, a quantity indefinitely great, or infinity.

(28.) The numeral coefficient of a quantity is a number prefixed to it, which shows how often the quantity is to be taken.

2

Thus, 4 a means a taken 4 times; x means

3

2

3

of x;

so 5 ax means that the product ax is to be taken 5 times. If a = = 2, and x = 3, then 5 ax=5×2×3=5×6=30. (29.) The literal coefficient of a quantity is a quantity by which it is multiplied.

Thus, in the quantity az, a may be considered to be a coefficient of z, or z a coefficient of a. Quantities having literal coefficients are commonly considered to be unknown, and the coefficients to be known, quantities.

(30.) The coefficient of a quantity may consist of a number, and also a literal part.

Thus, in 5 abx, the quantity 5 ab may be considered to be the coefficient of x. If a=2, and 63, then 5 ab = 5×2×3=5×6=30, and 5 abx=30 x.

When no numeral coefficient is prefixed to a quantity, unity is understood to be its coefficient. Thus, the coefficient of x, or of ax, or of abx, is 1; that is, a = la, and ax lax.

(31.) A power of a quantity is a quantity resulting from the multiplication of a quantity into itself one or more times.

When the quantity is repeated twice as a factor, the product is called its square or second power; when it is repeated three times, the cube or third power; when four times, the fourth power; and so on.

Instead of repeating the same quantity as a factor, a small figure is placed over it, to point out the number of times that the quantity is repeated. This figure is called the ex

ponent; and hence a small 2 is used to denote the square; 3, the cube; 4, the fourth power; 5, the fifth power; and

and generally the nth power = xxx .

...

****

where x is supposed

to be repeated n times, and is expressed by a".

When a letter has no exponent, it is considered to be the first or simple power of the quantity, and unity is considered to be its exponent; thus, x or x is the simple power of x. (32.) To involve a quantity to any power, is to find that power of the quantity.

(33.) The root of any quantity is a quantity, some power of which is equal to the quantity.

Thus, the fourth power of a is a; hence x is the fourth root of 24, for the continued product of a repeated four times as a factor is a1.

(34.) To extract any root of a quantity, is to find that

root.

(35.) The sign, called the radical sign, placed before a quantity, indicates that some root of it is to be taken; and a small figure placed over the sign, called the exponent of the root, shows what root is to be extracted.

Thus,

square

root of a.

and generally

a, ora, means the

3 a

a

nth

(36.) A simple quantity consists of a single letter, or the product of two or more letters, or powers of letters, with or without a coefficient.

Thus, a, 5 a, — a2x, 3 ax2y, are simple quantities.

(37.) A compound quantity consists of two or more simple quantities called its terms, which are connected by the signs of addition or subtraction.

Thus, a + b, 3a -2x, 2 a -3x2y+cz, are compound quantities.

(38.) A simple quantity is also called a monomial; a compound quantity, consisting of two terms, is called a binomial; one of three terms, a trinomial; of four terms, a quadrinomial; and of more than four terms, a multinomial or polynomial. A binomial, whose second term is negative, is called a residual quantity.

(39.) When any operation is to be performed on a compound quantity, it is expressed by enclosing the quantity in parentheses (), and adding the sign of the operation. Thus, (a-2 ax) means that a- 2 ax is to be subtracted; 5 (3a-2x) means that 3 a 2x is to be multiplied by 5; (3 a—2x) (a+b) means that 3 a 2 x is to be multiplied by a+b; (a + x) ÷ (a—x) means that a+x is to be divided by a — x; (ax)2 means that a—a is to be raised to the square; and (a2x2) means that the cube root of a2 - 22 is to be taken.

A vinculum theses.

is sometimes used instead of paren

Thus, a -x.y2 means that a x is to be multiplied by y2.

A bar is sometimes used to denote that a compound quantity is to be multiplied by some quantity.

Thus, a 3 has the same meaning as (a-2c+d) x3.

2 c

+ d

(40.) The dimension or degree of any simple quantity or term is the number of first or simple powers of factors of which it is composed.

Thus, ax consists of two simple powers a and a, and is of the second dimension; 3 a2x3 consists of five simple powers, namely, a, a, and x, x, x, and is of the fifth dimension; and so on.

(41.) A compound quantity is said to be homogeneous when its terms are all of the same dimension.

Thus, 2 a2x35xyz2+az is a homogeneous quantity, its terms being of the fifth dimension.

* (42.) An insulated negative quantity is a quantity with a negative sign, which is considered to exist unconnected with a positive quantity. (Note A.)

Thus, -a, when it is not supposed to be subtracted from any positive quantity, is called an insulated negative quantity, and, for distinction, it may be enclosed in brackets, thus [-a]

(43.) The sign of an insulated negative quantity is called a sign of relation or affection, whereas in the case of an ordinary negative or subtractive quantity, it is the sign of an operation, namely, that of subtraction.

(44.) When a negative quantity is considered to be insulated, it is taken in a sense directly the opposite of that in which it would be understood if it were a positive quantity.

Thus, if a represent a number of miles in one direction, then [a] means an equal number in an opposite direction. So if + a mean a number of pounds of property, [a] means an equal quantity of debt. If 10 means 10 miles west or of westing, [10] means 10 miles east or of easting, or of negative westing. So if + a means a number of men who pay away money, [a] means the same number who receive money. The expression that—a is a less than nothing, merely means that it is an insulated negative quantity.

Since all algebraical quantities represent numbers (7), and numbers may be represented by lines, therefore if + a be represented by the line OA, [-a] will be represented by the equal line O A' in the opposite direction; and thus may all insulated negative quantities be represented, even when they denote abstract numbers.*

α

A'

NUMERICAL EVALUATION OF ALGEBRAICAL

EXPRESSIONS.

+ a

A

(45.) When any numerical values are assigned to algebraical quantities, the numerical value of any algebraical expression containing them may be found by substituting for the letters their values in numbers, and then performing on these numbers the various operations indicated by the algebraical signs contained in the expression.

EXAMPLES.

1. Find the numerical value of the expression a—2b+ 3 c2, when a = 3, b = 4, and c = 6.

a-2b+3c2=3-2x4+3x 623-8+ 108=

111-8-103.

=

(46.) Any other numerical values may be assigned to the letters a, b, and c, and then the expression may be converted into its numerical value; which will generally, but not always, be different for different values of the letters.

2. Find the value of a4+2 (ax) (a + x) — 4 xy3, when a = 3, x = 2, and y 5.

This quantity=34 +2 (3-2) (3+2)-4 × 2 × 5° = 81+2 × 1 × 5-4 × 2 × 12581 + 10—1000= 91 -1000 = 909.

3. Find the value of (a) (a2 + x2) —

+3/ax, when a = 9 and x = 4.

It becomes = (81 — 16) (81 +16)

3 (a-x)2

a + x

3 x 52

+3√9x4

9+4

10

3

13

13

13.

10

6305+ 18-5 =6323-5 =6317

4. For any value whatever of and y, the expression (x + y) (x − y) is equal to x2—y2 or (x+y) (x − y) = x2-y2.

Let x5 and y = 3,

(5—3)

then (x+y) (x − y) = (5+3) (5-3)=8 x 2 = 16, and 22. ·y2 = 52. -32-25-9= 16, the same result as

before.

Let x=6 and y = 4,

then (x+y) (—y)=(6+4) (6-4)= 10 × 2=20, and a-y-62-42-36-16-20.

EXERCISES.

In the following exercises, the values given to the different letters are, a = = 4, b = 3, c = 5, d = 10, x = 2, and y 6. Any other values, however, may be assigned, but then