CONVERSATIONAL ARITHMETIC.

CHAT I.

THE UNITS OF ARITHMETIC.

ONE day a bright little boy went to school just to see the crabbed old teacher. Soon after he arrived the teacher, much to the boy's surprise, in a friendly manner said, "Jimmie, we are now about to commence the study of arithmetic, and you may put it down in your noddle as a fact, never to be forgotten, that the first thing to learn in this science of numbers is COUNTING; and, if you ever expect to be wealthy, you must not only learn to count straight, but you must learn to count your pennies.

"Now, here is a handful of pennies, and I ask you to tell me how many are there in all? This would seem to be a very easy question, yet you cannot answer it correctly, because your mind has not been trained to grasp a group of things at once; and this brings us to the first fact in arithmetic, which is that you must begin at the beginning and count one thing at a time. But, you ask, what is this one thing that we must have to count? and the answer is that a single portion into which anything is naturally divided is called one or a unit, as one penny, one horse, etc.,-and if you divide this unit into two equal parts you then have one-half of a unit, as one half-penny, or, as it is more commonly called, 'a ha'-penny.' If you increase this unit by adding another unit of the same kind, you will then have two of a kind, as two pennies.

"Thus, you find that one penny and one more make two pennies; two pennies and one more make three pennies; three pennies and one more make four pennies; four pennies and one more make five pennies, and so on, but when we wish to use a unit without applying it to any particular thing, we simply speak of it as 'one' without stating whether it is a sheep or a goat,-as five and one more make six, six and one more make seven, seven and one more make eight, eight and one more make nine, nine and one more make ten. Units used in this way are called numbers, and simply denote how many; hence, we see that a number may be a unit or a collection of units, as one, two, three, etc.

Right here let me say that we have three methods by which we may express numbers:

"First. As you have seen, by words.

"Second. By letters, or the Roman method.

"Third. By figures, or the Arabic method.

And no matter which of these methods we use the result is the same; and the act of recording or writing numbers is called notation.

"The system of notation used by the Romans consists of seven capital letters, namely, I, V, X, L, C, D, M; the respective value of these letters being one, five, ten, fifty, one hundred, five hundred, and one thousand; and by a uniform system of juggling we find that all other integral numbers may be expressed by these seven letters.

"Now, if you will pay particular attention (please notice, that I do not ask you to pay money), you may ascertain the secret of this great Roman system of counting, and you soon will be able to express numbers as quickly and as correctly as a Roman general. The first thing that you must remember in learning this secret is that every time a letter is repeated the number is increased by the value of the letter,—as I., II., III. Secondly, if a letter denoting a less value be

four

one two three

written on the left of a letter denoting a greater value the number expressed will be the difference between the value of the two letters, as IV. But if the letter denoting the less value be written on the right of a letter denoting a greater value the number expressed will be the sum of the two letters, as VI. Thirdly, a dash (-) placed over a letter increases the value of that letter a thousand times, as I., or one

thousand.

six

"Now, you have the whole secret, and from these three rules the Roman notation table may be written as follows:

I. or one,
II. or two,

III. or three,
IV. or four,

V. or five,

VI. or six,
VII. or seven,
VIII. or eight,

IX. or nine,
X. or ten,

and so on, expressing any number, however great; but it is a poor rule that doesn't work both ways, and there seems to be no way of expressing a number of less than one by the use of this method, and at present it is chiefly used to express dates and for numbering the chapters and pages of books.

"The Arabs seem to have invented a less cumbersome method of expressing numbers by the use of ten characters called figures, and these ten characters are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. naughty one two three four five six seven eight nine The naught, which simply means naughty, or nothing, is used to denote

the absence of one of the nine characters which have, like a great many people, assumed for themselves the title of significant figures.

"You have seen that you can express any number from one to nine by writing one of the significant figures, but when you wish to express a number greater than nine you will have to resort to the burglar-proof combination of locking two or more of these figures together by writing them side by side; and in order to do this properly, you must know the following combination and follow it carefully:

"First. When you write one figure, as, for instance, 1, it means one unit, and when you write another figure, as 9, it means that one unit is counted nine times, and, as you have seen, 9 and one more make ten; but, as there is no single representative character for that number, we have to combine at least two of the given characters, and this we do according to the law of units, which may be stated as follows:

"We pass from a lower to the next higher order by considering how many units of the lower make oné unit of the next higher, and in the Arabic notation the unit of any place is ten times as great as the unit of the next place to the right.

"Now, if you draw a vertical line and put the figure 1 on each side of it, thus 11, you will have two columns of figures, in each of which you have expressed one unit. And if you name these columns you will have one unit of the

first order and second order, thus

26

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1st order

first order and one unit of the second order, but the unit of the second is equal to ten units of the first order; hence, if you remove the separating line the number represented will be ten units and one unit, or eleven, which is one more than you desire.

"Now, by replacing the figure 1 in the first or right-hand column, which is generally called unit column, by the naught, you will have nothing in the first column, but ten units in the second, or one ten. Thus you see

Formerly, in the English notation, six places were given to millions, and they were read: millions, tens of millions, hundreds of millions, thousands of millions, tens of thousands of millions, hundreds of thousands of millions; but the French adopted the method of giving three places to the unit of each period, and this is now the general practice, and the names of the periods extend to the twenty-second place.1

1 Fourth, billions; fifth, trillions; sixth, quadrillions; seventh, quintillions; eighth, sextillions; ninth, septillions; tenth, octillions; eleventh, nonillions;

"In order to make the foregoing more applicable it may be reduced to the four following principles,-viz. :

"1. The same figure expresses different units according to the place which it occupies.

"2. That the figure occupying the place at the right is called units of the first order; the figure occupying the next place to the left of the first is called units of the second order, and so on, the unit of any figure being determined by its place.

"3. Ten units of the first order make one of the second, and ten of the second make one of the third, and so on, for the higher orders.

"4. When figures are written together on the same line, as 67548, etc., ten units in any one place make one unit of the next place on the left. Therefore, in writing numbers, begin at the left hand and write in each order the character representing the number of units required. Thus, in writing six hundred and four, write six units in the third order, no units, or naught, in the second order, and four units in the first order.

"The simple expressing of numbers by characters, however, is not sufficient for practical purposes until you learn to read the numbers thus expressed, and this act or art of reading numbers is called numeration. Now, if you expect ever to be able to read numbers correctly, you must observe the two following principles,—namely:

"1. If the number you desire to read consists of a long row of figures, begin at the right and divide it, either mentally or by means of a comma, into periods of three figures each; but the left-hand period need not always contain three figures.

"2. Begin again at the right hand and name the unit of each figure to the left, then read each period as if it stood alone, as 305,468,three hundred and five thousand four hundred and sixty-eight.

"By bearing in mind what you have been told, so far, you should be able to read any ordinary simple number,—that is, a number having a single unit, whether abstract or denominate, as 9688 or 7 cows; but there is another kind of number, having different units of value, as 1 yard, 2 feet, and called a compound number.

"This brings us to a consideration of units and scales, and we find that the integral units of arithmetic are divided into eight classes:

1. Units of abstract numbers.

2. Units of time.

3. Units of weight.

4. Units of currency.

5. Units of length.

6. Units of surface.

7. Units of volume.
8. Units of angular measure.

twelfth, decillions; thirteenth, undecillions; fourteenth, duodecillions; fifteenth, tredecillions; sixteenth, quatuordecillions; seventeenth, quindecillions; eighteenth, sexdecillions; nineteenth, septendecillions; twentieth, octodecillions; twenty-first, novendecillions; twenty-second, vigintillions.

We have seen that the first unit of arithmetic is the abstract unit, 1; and this is the primary base of all abstract numbers, and it becomes the base of any denomination of a compound number by simply naming the particular denomination to which it is applied. The other units, as units of time, may be separately considered, so we will not trouble about them just now, but we will seek to learn something about a scale. You say, 'a scale? Why, what is a scale, anyhow?' Well, a scale in arithmetic is a series of numbers expressing the law of relation between the different units of any number, and there are two kinds of scales,-uniform and varying.

"A uniform scale is one in which the law of relation is the same between the different steps of the scale, as units, tens, hundreds.

"A varying scale is one in which the law of relation is different between the different steps of the scale, as inches, feet, yards.

"I hope that I have not made these remarks too long, for I intended to say only just enough to make you entertain a strong desire to take the first step towards acquiring a practical knowledge of arithmetic, a desire to reason from cause to effect, a desire to increase your mental power."

And this little boy became so much interested that the next day he said, "Teacher, I wish you would tell me something more about the use of figures, for they are very funny little things, and I want to learn how to use them.”

CHAT II.

ADDITION AND SUBTRACTION.

"BEFORE proceeding directly to the application of numbers," said the teacher, "let me say a few words for your consideration. The rules that are given you by the teachers may pave the way for self-culture, but they cannot take the place of it. As creatures possessing the faculty of reasoning, you must educate yourselves. And let me tell you just one thing,-man's education does not stop where childhood ends. He must seek to increase the power of his mind all the days of his life, and so grow old with ever-increasing efficiency to the end.

"From what has been said about the units of arithmetic, you may have discovered that the study of figures is good training for the mind. Arithmetic teaches us how to make use of numbers, and in applying numbers to every-day questions there are four constantly occurring operations to be performed, namely, addition, subtraction, multiplication, and division; therefore, it is necessary that each of these operations be thoroughly understood before attempting to solve problems combining in an intricate manner all four of them.

"In demonstrating a problem signs or characters are used to denote