p explain the method of performing the operations of addition, subtraction, multiplication, division, etc., upon quantities represented by letters. These methods are essentially the same as practiced in arithmetic, but there is some apparent difference arising from the difficulty we experience in algebra in uniting many terms into one single term, as we do in arithmetic.

Some peculiarities of notation are adopted when numbers are represented by letters.

(20.) The first letters of the alphabet are commonly used to represent known quantities, and the last letters those which are unknown.

(21.) Quantities preceded by the sign + are called positive quantities; those preceded by the sign-, negative quantities. When no sign is prefixed to a quantity, + is to be understood. Thus, a+b-c, is the same as +a+b-c.

(22.) When numbers are represented by letters, multiplication is usually indicated by writing the letters in succession without the interposition of any sign. Sometimes it is indicated by placing a point between the successive letters. Thus, abcd is equivalent to axbxcxd, or a.b.c.d.

Thus, if we suppose a=2, b=3, c=4, and d=5, we have abcd=2x3x4x5=120.

(23.) When two or more quantities are multiplied together, each of them is called a factor. Thus, in the expression 7X5, 7 is a factor, and so is 5. In the product abc there are three factors a, b, and c.

QUEST.-How are known quantities represented? What are posi tive quantities? What are negative quantities? How may multipli cation be denoted?

What is a factor?

When a quantity is represented by a letter, it is called a literal factor, to distinguish it from a numerical factor, which is represented by an Arabic numeral. Thus, in the expression 8ab, 8 is a numerical factor, while a and b are literal factors.

(24.) The symbol > is called the sign of inequality, and when placed between two quantities, denotes that one of them is greater than the other, the opening of the sign being turned toward the greater number.

Thus, 4<7 signifies that 4 is less than 7, and 12>9 denotes that 12 is greater than 9. So, also, a>b shows that a is greater than b, and c<d shows that c is less than d.

(25.) The coefficient of a quantity is the number or letter prefixed to it, showing how often the quantity is to be taken.

Thus, instead of writing a+a+a+a+a, which represents 5 a's added together, we write 5a where 5 is the coefficient of a. In like manner, Sab signifies eight times the product of a and b. When no coefficient is expressed, 1 is always to be understood. Thus, la and a signify the same thing.

The coefficient may be a letter as well as a figure. In the expression mx, m may be considered as the coefficient of x, because x is to be taken as many times as there are units in m. If m stands for 4, then mx is four times x.

(26.) The products formed by the successive multiplication of the same number by itself are called the powers of that number.

QUEST-What is a literal factor? What is a numerical factor? What is the sign of inequality? What is a coefficient? What are powers?

Thus, 2x2= 4, which is the second power of 2.

2×2×28, the third power of 2.

2×2×2×2=16, the fourth power of 2, etc. So, also, 3x3= 9, the second power of 3. 3x3x3=27, the fourth power of 3, etc.

Also,

axa=aa, the second power of a.

a×a×a=aaa, the third power, etc.

(27.) For the sake of brevity, powers are usually expressed by writing the root once, with a number above it at the right hand, showing how many times the root is taken as a factor.

the exponent of the power.

Thus, instead of

This number is called

aa, we write a2, where 2 is the exponent of the power. aaa, we write a', where 3 is the exponent of the

power.

aaaa, we write a', where 4 is the exponent of the power, etc.

When no exponent is expressed, 1 is always understood. Thus, a' and a signify the same thing.

Exponents may be attached to figures as well as letters.

Thus, the product of 3 by 3 may be written 32, which equals 9.

The product of 3×3×3 may be written 33, which equals 27.

The product of 3×3×3×3 may be written 3', which equals 81, etc.

(28.) A root of a quantity is a factor which, multiplied by itself a certain number of times, will produce the given quantity.

QUEST.-What is an exponent? What is a root of a quantity?

The symbol is called the radical sign, and, when prefixed to a quantity, denotes that its root is to be extracted. Thus,

9, or simply √9, denotes the square root of 9, which is 3.

V64 denotes the cube root of 64, which is 4.
V16 denotes the fourth root of 16, which is 2.
So, also,

Va, or simply va, is the square root of a."

Va denotes the third or cube root of a.

α

Va denotes the nth root of a, when n may represent any number whatever.

(29.) The number placed over the radical sign is called the index of the root. Thus 2 is the index of the square root, 3 of the cube root, 4 of the fourth root, and n of the nth root. The index of the square root is usually omitted. Thus, instead of Vab, we usually write

ab.

(30.) A vinculum or a parenthesis ( ), indicates that several quantities are to be subjected to the same operation.

Thus, a+b+cxd, or (a+b+c)×d, denotes that the sum of a, b, and c is to be multiplied by d. But a+b+cxd denotes that c only is to be multiplied by d.

When the parenthesis is used, the sign of multiplication is generally omitted. Thus (a+b+c)xd is the same as (a+b+c)d, or d(a+b+c).

(31.) Every quantity expressed in algebraic an

QUEST.--What is the radical sign? What is the index of a ooơ* ? What is a vinculum?

guage, that is, by the aid of algebraic symbols, is called an algebraic quantity, or an algebraic expression Thus,

5a is the algebraic expression for five times the number a.

4a2 is the algebraic expression for four times the square of the number a.

6a2b3 is the algebraic expression for six times the square of a, multiplied by the third power of b.

(32.) An algebraic quantity composed of a single term is called a monomial.

Thus, 2a, 3bc, 5xy', are monomials.

An algebraic expression, consisting of two terms only, is called a binomial; one consisting of three terms is called a trinomial.

Thus, 3a+56 is a binomial, and a+2bc+5xy is a trinomial.

An algebraic expression which is composed of several terms is called a polynomial.

Thus, 2a+5b+7c-4d is a polynomial.

(33.) The numerical value of an algebraic expression is the result obtained when we attribute particular values to the letters.

Suppose the expression is 5ab2.

If we make a=2 and b=3, the value of this expression will be 5×2×3×3=90.

If we make a=3 and b=4, the value of the same expression will be 5x3x4x4=240.

QUEST.-What is an algebraic expression? What is a monomial? What is a binomial? What is a trinomial? What is a polynomial? What is the numerical value of an algebraic expression?