INTRODUCTION TO ALGEBRA.

CHAPTER I.

PRELIMINARY REMARKS AND DEFINITIONS.

(1.) Quantity, or magnitude, is any thing whatever that will admit of increase or diminution; for example, a sum of money, the length of a line, and a given weight, or bulk, are quantities, since they can be increased or diminished. Mathematics is the science that investigates the means of measuring quantity, and hence is called the science of quantity.

A quantity cannot be great or small, much or little in itself; it is only by comparing it with another quantity of the same kind that it becomes so. Thus, a mile is a large quantity compared with an inch, but small compared with the diameter of the earth; and this again is a small quantity compared with the distance of the earth from the sun.

(2.) It is evident, therefore, that in order to measure a quantity, we must compare it with some known quantity of the same kind, assumed as the unit or measure of quantity. For example, if the quantity to be measured is a sum of money, and we assume one dollar as the unit, the number of times one dollar is contained in that quantity, is the magnitude of the quantity. Thus 300 dollars is conceived to be composed of 300 parts, each of which is equal in value to one dollar.

If the quantity be the length of a line, and we assume one foot as the unit of measure, the number of times a line one foot in length can be applied to this line, is its magnitude. If it can be applied 10 times, we say the given line is ten feet in length.

Quantities of every kind whatever can therefore be expressed by numbers, the number always expressing how many times the assumed unit of quantity is contained in the given quantity.

(3.) Arithmetic and Algebra both treat of the relations of quantity as expressed by numbers.

Arithmetic treats of numbers in particular, and their simplest combinations. Algebra is but a continuation of arithmetic, in which proper signs are used, to abridge and generalize the reasoning required in the resolution of questions relating to numbers.

ARITHMETIC.

NOTATION AND NUMERATION.

(4.) It would evidently be impossible to express each number by a separate term, since in that case the number of terms would be infinite. To express the numbers comprehended between one and one million, a million of terms would be necessary; and as any number, however large, may be increased without limit, by adding unity to it successively, the number of terms would also increase without limit.

To obviate this difficulty, a nomenclature has been devised in which, by the use of a small number of terms properly combined, and subject to regular forms, all possible numbers can be expressed.

We

This nomenclature is based upon the principle of successive orders of units, of such values that ten units of the first order shall be equal to one of the second order, ten of the second order equal to one of the third order, and so on. number to ten, and then commence again at one; eleven, twelve, thirteen, &c., being the same as ten and one, ten and two, ten and three, &c., till we have numbered another ten, when we commence at unity as before: twenty-one, twentytwo, &c. Proceeding in this manner till we have numbered ten tens, we pass to hundreds, and in like manner from hundreds to thousands.

(5.) Numbers were at first written separately, either in words at length, or as among the Romans, in characters, commonly the letters of the alphabet; and for large numbers, such letters were used as would make up, by the addition of their separate values, the number required. Thus the number four hundred and fifty-eight was written CCCCLVIII., the value of C being one hundred, L fifty, V five, and I one.

The system of notation now in use, and which originated in Arabia, is so contrived as to express all numbers with ten

characters only. Nine of these are significant, and represent numbers, and the other is used to denote nothing, or the absence of quantity.

To express numbers greater than nine, recourse is had to a law which assigns different values to the figures, according to the position which they occupy. According to this law, units of the first order occupy the first place on the right of the written expression, units of the second order, the second place, and so of the others.

The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, are each expressed by separate characters; but when we would write ten, which is a unit of the second order, we place unity in the second place, and a 0 in the first, thus, 10, which may either be considered as one unit of the second order, or ten units of the first. Proceeding with units of the second order in the same manner as with those of the first, we shall have, 10, 20, 30, 40, 50, 60, 70, 80, 90, and next a unit of the third order, which is written 100, the significant figure being written in the third place. In like manner we pass from the third to the fourth order.

A number may consist of units of more than one order, and then the places of the O's are supplied by significant figures. For example, the number three hundred and fifty-six has six units of the first order, five of the second, and three of the third; and writing these units in their proper places, we have the expression 356.

The following table can now be understood, which exhibits the successive orders of units, and the manner of writing and reading numbers.

This table contains fifteen places of figures, and, consequently, fifteen orders of units, and the number expressed is read thus: four hundred and twelve trillions, three hundred and seventy-four billions, nine hundred and sixty-one millions, two hundred and forty-three thousand, nine hundred and eighty-seven.

It will be observed in reading the number written in the table, that for each period of three figures, the same names are used as for the first at the right, with an additional term to express a higher order of units. This, in the second period, is thousands; in the third, millions; in the fourth, billions; and in the fifth, trillions; and hence in writing numbers, it is usual to divide the figures into periods of three each, as follows:

412,374,961,243,987.

To express a number which does not contain units of every order comprehended between the lowest order and the highest contained in the given number, the places of such as are wanting are to be filled up with O's; for example:

Two hundred trillions, twenty billions one million, twentytwo thousand and seven, is written

.

200,020,001,022,007.

EXAMPLES FOR EXERCISE.

1. Write thirty thousand one hundred and six.

2. Write one million one thousand one hundred and one.

3. Write thirty billions, two millions, one hundred and one thousand and twenty-seven.

4. Write sixteen billions, six millions one thousand and

seven.

5. Write eleven thousand eleven hundred and eleven.

EXPLANATION OF SIGNS.

To denote arithmetical operations and conditions, certain characters or signs are used; they are the following:

+, which is read plus, and indicates Addition.
-, which is read minus, and indicates Subtraction.
X, which indicates Multiplication.

Division.
Proportion,

✓ ✓ V, &c. which signifies 4th, 5th, 6th root, &c. Thus : 6+ 8 denotes that 6 is to be added to 8.

6 4 denotes that 4 is to be subtracted from 6. 7x3 or 7.3 denotes that 7 is to be multiplied by 3. 164 or 6 denotes that 16 is to be divided by 4. 9:6=12:8, shows that 9 is to 6 as 12 is to 8.

8+ 4 = 12 shows that the sum of 8 and 4 is equal to 12.

✓ 6 signifies the square root of the number 6.

3

✓ 6 signifies the cube root of the number 6.

4

5

6

√2, v2, v2, &c. signifies the 4th, 5th, 6th, &c. root of the number 2.

THE FOUR RULES OF ARITHMETIC.

(6.) As quantity admits of no other changes than increase and diminution, it is evident that all the operations of arithmetic must be based upon these only.

Under the first are comprehended addition and multiplication, and under the second, subtraction and division.

Addition consists in finding the sum of two or more num

bers.

Multiplication is the successive addition of a number to itself a given number of times.

Subtraction consists in finding the difference between two numbers, or diminishing one given number by another.

Division is the same as subtracting one number successively from another, in order to find how many times the smaller number is contained in the larger.

Addition, subtraction, multiplication, and division, are called the four rules of arithmetic; all arithmetical operations whatever are combinations of these.

ADDITION.

(7.) Addition consists in finding the sum of two or more numbers. The sign of addition is +, and is read plus; 5+6 signifies that 6 is added to 5.