LESSONS IN ARITHMETIC.—I.
The term Arithmetic, which is derived from the Greek verb αριθμεω (pronounced a-rith'-me-o), to count, is properly applied to the science of Numbers, and the art of performing calculations by them, and investigating their relations. To a certain extent, this science must have been coeval with the history of man. As an art, arithmetic is indispensable in daily business; and the man who is best acquainted with its practical details has always the preference in every mercantile establishment. Our object in these lessons shall be twofold—to develop its principles as a science, and to show the application of its rules as an art. For this purpose, it will be necessary to begin with the first principles of Numeration and Notation, and to give such rules as will enable any one to read and write a given number correctly.
Notation and Numeration.
1. Any single thing—as for instance, a pen, a sheep, a house—is called a unit: we say there is one such thing. If another single thing of the same kind be put with it, there are said to be two such things; if another, three; if another, four; if another, five; and so on.
Each of these collections of things of which we have spoken is a number of things; and the terms one, two, three, four, five, etc., by which we express how many single things or units are under consideration, are the names of numbers. A number therefore is a collection of units. This is also sometimes called an integer, or whole number.
It will be seen that the idea of number is quite independent of the particular kind of units, a collection of which is counted. Thus, if there are four pigs, the number of pigs is the same as if there were four pens. We can thus abstract a number from any particular unit or thing, and talk of the number four, the number five, etc. Numbers thus abstracted from their reference to any particular unit or thing are called abstract numbers. When a collection of things or objects is indicated, it is called a concrete number.
We shall treat first of abstract numbers.
2. The art of expressing numbers by symbols, or figures, is called Notation.
In the system of notation which we are about to explain, all numbers can be expressed by means of ten symbols (figures, or digits,* as they are called), representing respectively the first nine numbers, and nothing, i.e., the absence of number. These are—
1 representing the number one
2 " " two
3 " " three
4 " " four
5 " " five
6 representing the number six
7 " " seven
8 " " eight
9 " " nine
0 called a nought, a cipher, or zero.
N.B.—Ten times ten is called one hundred; ten times a hundred, a thousand.
3. Numbers are represented by giving t the figures employed what is called a local value—i.e., a value depending upon the positions in which they are placed.
Let a number of columns be drawn as below, that being called the first which is on the right, and reckoning the order of the columns from right to left.
If a figure—5, for instance—be placed in the first column, it denotes five units, or the number five; if it be placed in the second column, it denotes five tens; if in the third, five hundreds; if in the fourth, five thousands; if in the fifth, five times ten thousand; and so on, each column corresponding to a number ten times as great as the one immediately on its right.
* Digits. So called from digitus, a "finger." This decimal notation clearly took its origin from these natural counting instruments.
Thus | 7 | 9 | 4 | 3 | would denote seven thousands, nine hundreds, four tens, and three ones; or, as it would be expressed, seven thousand, nine hundred, and forty-three.
Similarly, | 8 | 3 | 0 | 5 | 4 | 7 | would denote eight times a hundred thousand, three times ten thousand, no thousands, five hundreds, four tens, and seven ones; or, as it would be more briefly expressed, eight hundred and thirty thousand, five hundred and forty seven.
We need not, however, draw the columns: it will be the same thing if we imagine them, and, instead of columns, talk of figures being in the first, second, third, fourth places, etc.
The symbol 0 put in any place, as already indicated in the previous example, denotes that the number corresponding to the particular column or place in which it stands is not to be taken at all: the 0 only fills up the place—thus, however, answering the important purpose of increasing the figure after which it stands tenfold.
Thus, 10 means that once ten and no units are taken—i.e., it denotes the number ten; 100 means that once a hundred but no tens and no units are taken—i.e., it denotes the number a hundred; 5001 means that five thousands, no hundreds, no tens, and one unit, are taken, or, as it would be more briefly expressed, five thousand and one.
4. Before proceeding further, we will give the names of the successive numbers:—
Ten ... ... ... 10
Eleven ... ... ... 11
Twelve ... ... ... 12
Thirteen ... ... ... 13
Fourteen ... ... ... 14
Fifteen ... ... ... 15
Sixteen ... ... ... 16
Seventeen ... ... ... 17
Eighteen ... ... ... 18
Nineteen ... ... ... 19
Twenty ... ... ... 20
Thirty ... ... ... 30
Forty ... ... ... 40
Fifty ... ... ... 50
Sixty ... ... ... 60
Seventy ... ... ... 70
Eighty ... ... ... 80
Ninety ... ... ... 90
Hundred (ten times ten) ... ... ... 100
Thousand (ten hundreds) ... ... 1000
Million (a thousand thousands) ... ... 1000000
Billion* ... ... 1000000,000000
Trillion (a million billions) ... ... 1000000,000000,000000.
The numbers between twenty and thirty are expressed thus: twenty-one, twenty-two, twenty-three, etc, up to twenty-nine, to which succeeds thirty; and similarly between any other two of the names above given, from twenty up to a hundred: thus, 95 is called ninety-five.
After one hundred, numbers are denoted in words, by mentioning the separate numbers of units, tens, hundreds, thousands, etc, of which they are made up. For example, 134 is one hundred and thirty-four; 5,342 is five thousand three hundred and forty-two; 92,547 is ninety-two thousand five hundred and forty-seven; 84,319,652 is eighty-four millions, three hundred and nineteen thousand, six hundred and fifty-two.
5. It is useful, in reading of into words a number expressed in figures, to divide the figures into periods of the three, commencing on the right, as the following example will indicated:—
Billions. Thousands of Millions. Millions. Thousands. Units.
561 234 826 479 365
Thus the figures 561,234,826,479,365 would denote five hundred and sixty-one billions, two hundred and thirty-four thousand eight hundred and twenty-six millions, four hundred and seventy-nine thousand, three hundred and sixty-five.
We have then the following
Rule for reading numbers which are expressed in figures:—"
Divide them into periods of three figures each, beginning at the right hand; then, commencing at the left hand, read the figures of each period are read, and at the end of period pronounces its name.
The art of indicating by words numbers expressed by figures is called Numeration.
Exercise 1.
Write down in figures the numbers named in the following exercises:—
*In the foreign system of numeration a thousand millions is called a billion, a thousand billions a trillion, and so on.
