5. Subtraction is the taking as much from one quantity as is equal to another. Subtraction is denoted by a single line; as a-b, or a minus b, which is the part of a remaining, when a part equal to b has been taken from it; if a=9, and b=5; a—b expresses 9 diminished by 5, which is equal to 4, or a-b=9-5=4.

6. Also, the difference of two quantities a and b ; when it is not known which of them is the greater, is represented by the sign; thus, ab is a-b, or b-a; and ab signifies the sum or difference of a and b.

7. Multiplication is the adding together so many numbers or quantities equal to the multiplicand as there are units in the multiplier, into one sum called the product. Multiplication is expressed by an oblique cross, by a point, or by simple apposition; thus, a xb, a. b, or ab, signifies the quantity denoted by a, is to be multiplied by the quantity denoted by b; if a=5 and b=7; then a Xb=5×7= 35, or a . b=5.7=35, or ab=5×7=35.

Scholium. The multiplication of numbers cannot be expressed by simple apposition. A unit is a magnitude considered as a whole complete within itself, And a whole number is composed of units by continued additions; thus, one plus one composes two, 2+1=3, 3+1=4, &c.

8. Division is the subtraction of one quantity from another as often as it is contained in it; or the finding of that quotient, which, when multiplied by a given divisor, produces a given dividend."

Division is denoted by placing the dividend before the sign, and the divisor after it; thus ab, implies that the quantity a is to be divided by the quantity b. Also, it is frequently denoted by placing

one of the two quantities over the other, in the

α

form of a fraction; thus, ab; ifa=12,6=4;

α

b

12

then, 86: =12÷=4——3.

b

4

9. A simple fraction is a number which by continual addition composes a unit, and the number of such fractions, contained in a unit, is denoted by the denominator, or the number below the line; thus, ++=1. A number composed of such simple fractions, by continual addition, may properly be termed a multiple fraction; the number of simple fractions composing it, is denoted by the 3 In this sense, 3, 3, 37 upper figure or numerator. are multiple fractions; and 3=1; ÷=3+}=1+} =1}.

4

10. When any quantities are enclosed in a parenthesis, or have a line drawn over them, they are considered as one quantity with respect to other symbols; thus, a―(b+c), or a-b+c; implies the excess of a above the sum of b and c; Let a=9, b=3, and c=2; then, a-(b+c)=9-(3+2)=954, or a-b+c=9—3+2=9-5= 4. (a+b)x(c+d), or a+bXc+d, denotes that the sum of a and b is to be multiplied by the sum of c and d; thus, let a=4, b=2, c=3, and d=5; then (a + b) x (c + d)=(4+2) × (3+5)=6×8=48, or a + bx c + d≈4 + 2×3 +5=6×8 = 48. And a-b

Also,

(a−b)÷(c+d), or c+d; implies the excess of a above b, is to be divided by the sum of c and d; if a=12, b=2, c=4, and d=1; then, (ab)—~

(c + d)=(12—2)÷(4+1)=10÷÷÷5=2, or

a-b

c+d

The line drawn over the quantities is sometimes called a vinculum.

=

11. A quantity multiplied by itself, is the square of the quantity; thus, a Xa= the square of a; if a=5, then a×a = 5 X 5 25 the square of 5. Also, a×a×a= the cube of a, or 5 × 5 × 5=125= the cube of 5; xXxXxXx = the fourth power of x; and so on.

12. The square, the cube, the biquadrate, &c. called the powers of algebraic quantitics, are expressed by placing a small figure, (equivalent to the number of factors, and called the index or exponent of the power;) at the right-hand of the quantity; thus, a, is called the first power of a, and is =a; a2, the second power, or square of a, and is a×a; a3, the third power, or cube of a, and is a×a×a;· a', the fourth power of a, equal to a XaXaXa; as, the fifth power of a, equal to axaxaxaxa; (a+b)3, the third power of a+b, equal to (a+b)× (a+b)x(a+b);

(a-x), the fourth power of (a-x), equal to (a--x) ×(α-x)×(a—x)×(a—x).

13. Factors are the numbers or quantities, from the multiplication of which, the proposed numbers or quantities are produced; thus, the factors of 35 are 7 and 5, because 7×5=35; also, a and b, are the factors of ab; 3, a2, b and c2, are the factors of 3a2bc2; and a+b and a-b are the factors of the product (a+b)×(a—b).

When a number or quantity is produced by the multiplication of two or more factors; it is called

a composite number or quantity; thus, 35 is a composite number, being produced by the product of 7 and 5; also, 5acx is a composite quantity, the factors of which are 5, a, c, and x.

14. The coefficient of a quantity is the number or letter prefixed to it; being that which shows how often the quantity is to be taken; thus, in the quantities 36 and 5x2, 3 and 5 are the coefficients of b and x2. Also, in the quantities 3ay and 5a2 x, 3a and 5a2 are the coefficients of y and x.

When a quantity has no number prefixed to it, the quantity has unity for its coefficient, or it is supposed to be taken only once; thus, x is the same as 1x; and when a quantity has no sign before it, the sign is always understood; thus, 3a2b is the same as +3a2b, and 5a-3b is the same as +5α3b.

15. The root of any proposed quantity is that quantity which, by its continual multiplication, as often as there are units in the index, would produce the same quantity; thus, the square root of a2 is a; because a Xa=a2, the square root of 25 is 5; the cube root of 27=3; the fourth root of 81-3; and the square root of a is a2; because a Xa2=a Xa

xaxa=a*.

According to the above definition of the term root; it is to be understood that the first process in the multiplication is unity multiplied by the quantity; thus, a1 is equal to 1 Xa; a2 is equal to 1 Xax a; and a3=1×a×a×a.

16. Quantities which can be expressed in finite terms, or the roots of which can be accurately expressed, are rational quantities; thus, 3a, a, and the square root of 4a2, are rational quantities; for if a=10; then, 3a 3 X 10-30; fa=3x10=20= 4; and the square root of 4a2 the square root of

4 × 102 the square root of 4 × 10 × 10= the square root of 400=20.

17. An irrational quantity, or surd, is that of which the value cannot be accurately expressed in numbers, as the square root of 3, 5, 7, &c.; the cube root of 7, 9, &c.

ex

18. The roots of quantities are expressed by means of the radical sign, with the proper index annexed, or by fractional indices placed at the right-hand of the quantity; thus, a, or a3, presses the square root of a; (a+x), or (a+x)3, the cube root of (a+x); î/(a+x), or (a+)x3, the fourth root of (a+x). When the roots of quantities are expressed by fractional indices; thus, a3, (a+x), (a+x); they are generally read a in the power (), or a with (1) for an index; (a+x) in the power (1), or (a+x) with (1) for an index; and (a+x) in the power (4), or (a+x) with () for an

index.

19. Like quantities are such as consist of the same letter or the same combination of letters, or that differ only in their numeral coefficients; thus, 5a and 7a; 4ax and 9ax; +2ac and 9ac-5ca ; &c., are called like quantities; and unlike quantities are such as consist of different letters, or of different combination of letters; thus, 4a, 3b, 7ax, 5ay2, &c. are unlike quantities.

20. Algebraic quantities have also different denominations, according to the sign +, or -.

Positive, or affirmative quantities, are those that are additive, or such as have the sign + prefixed to them; as, +a, +6ab, or 9ax.

21. Negative quantities are those that are subtractive, or such as have the sign - prefixed to