THE MEANINGS OF THE SIGNS USED IN THIS WORK.

+plus, between two numbers or expressions, shews that they are to be added.

minus, between two numbers or expressions, shews that the latter is to be subtracted from the former.

X into, between two numbers or expressions, shews that they are to be multiplied together.

divided by, between two numbers or expressions, shews that the former is to be divided by the latter.

V the root of, shews that a root of the number standing under it is to be

found.

= is equal to, shews that the expressions between which it stands are equal.

{}( ), brackets, shew that all quantities placed between them are to

be considered as forming one quantity.

.. means therefore.

... means because.

is greater than, shews that the former of two expressions is greater than the latter.

is less than, shews that the former of two expressions is less than the

latter.

G.C.M. stands for greatest common measure.

L.C.M. stands for least common multiple.

L.C.D. stands for least common denominator.

a stands for varies as.

ARITHMETICAL RULES AND EXAMPLES.

I.

PART I.

NOTATION AND NUMERATION.

Def. 1. Number is that which expresses how many articles there are in a collection.

Def. 2. If we merely speak of the number of articles, without any reference to the articles themselves, the number is called an abstract number. Def. 3. If we not only specify the number, but the nature of the articles, the expression used is called a concrete number.

Def. 4. Numeration is the method of expressing different degrees of nuinber by distinct names.

Def. 5. Notation is the method of representing different degrees of number by distinct signs, or figures.

Def. 6. A single article of any kind is called a unit of that kind.

Def. 7. The unit is expressed by the number one, and is called the unit of magnitude, or the unit of repetition, the concrete or abstract unit, according as we connect or not, the article with the number of times it is repeated.

Thus, one inch, one foot, one yard, are severally units of magnitude, with which other lengths may be compared. But the number one, which merely expresses how often the inch, foot, or yard, is repeated in the unit, is called the unit of repetition, or the abstract unit.

Def. 8. If to one article there be added another, the number of articles is called two and three, four, five, six, seven, eight, nine, ten, express the numbers in collections differing successively by one.

Def. 9. If from a single article there be taken itself, the number remaining (or rather the absence of number) is expressed by the word naught

or zero.

Def. 10. The numbers naught or zero, one, &c. up to nine, are represented by the symbols, or figures, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these is called a cipher; the others have the same names as the numbers which they represent.

These figures are also called digits.

Numbers from ten, differing by one, are named as follows:

Def. 11.

Def. 12.

Def. 13. Numbers from one hundred to two hundred, are named by joining those from one up to one hundred to the number one hundred. Those from two to three hundred; from three to four hundred; from four to five hundred, &c. are named in the same way, as far as ten hundred. As three hundred and eighty-four.

Def. 14. Ten hundred is also called a thousand.

Def. 15. Numbers from one thousand to two thousand are named by joining to the number one thousand those from one to one thousand. In the same way numbers from two to three thousand; from three to four thousand, &c. as far as one thousand thousand are named. Thus one hundred and sixty-three thousand nine hundred and fifty-two.

Def. 16. One thousand thousand is called a million.

Def. 17. Numbers from one million to two millions are named by joining to the number one million, the numbers from one as far as one million. In the same way numbers from two to three millions, from three to four, &c. as far as one million millions are named.

Def. 18. One million millions is called a billion.

Def. 19. One million billions is called a trillion.

Def. 20. One million trillions is a quadrillion; and quintillion, sexillion, septillion, octillion, nonillion, are names given to one million quadrillions, &c.

Def. 21. The figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which alone cannot represent more than the number nine, are by a certain agreement made to represent numbers however large, several of them being used at once.

Def. 22. The agreement is that each of these figures shall have, besides its own primary value, another value, depending on the place it occupies, such that, with every increase of its distance from the first figure on the right hand, its value is increased ten fold. Thus the 1st figure on the right represents so many single things or units; the 2nd figure ten times as many as it would represent if it stood in the first place, or alone, i. e. denotes so many collections of tens; the 3rd figure denotes as many collections of hundreds, &c. &c. as shewn in the subjoined

NUMERATION TABLE.

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Tens.
Hundreds.
Thousands.

Units.

Tens of Thousands.

Hundreds of Thousands.

Millions.

Tens of Millions.

Hundreds of Thousands of Millions

Tens of Thousands of Millions.

Thousands of Millions.

Hundreds of Millions.

Billions.

Tens of Billions.

Tens of Thousands of Billions.

Thousands of Billions.

Hundreds of Thousands of Billions.

Hundreds of Billions.

Trillions.

Obs. The pupil should be required to learn this table as far as billions by heart, so as to be able to assign immediately the proper place to a number of given denomination, and vice versa. This may be readily acquired by learning the names of the first six places, as the others are merely a repetition of these with the addition of millions, billions, &c. Dividing then the figures into periods of six, it is readily seen that, as thousands is the name of the 4th place in the 1st period, thousands of millions is that of the 4th place in the 2nd period, or of the (4+6) th, or 10th place: thousands of billions that of the 4th in the 3rd period, or of the 16th place, &c.

1. TO WRITE IN FIGURES A GIVEN NUMBER.

Rule. Write a series of dots, in a line, in groups of six; under each, beginning from the right, place the number of single units, tens, &c. which occur in the number, taking care when any denomination of figures is wanting to write a cipher in its place.

II. TO READ THE NUMBER REPRESENTED BY A SERIES OF FIGURES. Rule. Divide the figures into periods of six; by which means the highest denomination is readily found; read each period, beginning on the left hand, as though it were the first on the right, adding the proper denomination of the period.

EXAMPLES.

1. Write in figures the number:-eight hundred and forty thousand and ninety-one millions, three hundred and two thousands, and fifty-nine. Here the highest denomination being hundreds of thousands of millions, I want 12 figures. I write the number therefore thus :

840091,3 02059.

2. Write in words the number 1,000101,010010.

Ans. One billion, one hundred and one million, ten thousand and ten.

II. SIMPLE ADDITION.

Def. Addition is the operation by which we find the number of things in a collection, formed by placing several collections together, the numbers in each being known.

TO ADD SEVERAL NUMBERS TOGETHER.

Rule-1. Place the numbers under one another, units under units, tens under tens, &c.

2. Add together the units in all the numbers, and if the sum be greater than 10, divide it into tens and units.

3.

Write down the units of this sum, under the other units, and carry (in your memory) the number of tens.

4. Now add the numbers in the tens' column, taking in the number carried from the units.

5. Divide this sum into hundreds and tens; write down the number of tens, and carry the number of hundreds to the column of hundreds, which is now to be added.

6. Proceed in this manner through all the columns, always dividing the sum of any column into parts, when it is greater than ten, till you come to the last column, whose sum must be written down to the left of the other figures.

EXAMPLE.

Add together the numbers, 41068, 280934, 3008007, 410608, 209, 90183, 708.

41068

280934

3008007

410608

209

90183

708

3831717 Ans.

III. SIMPLE SUBTRACTION.

Def. Subtraction is the operation by which we find what number is left, when from one given number we have taken another :—or, what number must be added to the less of two given numbers, that it may be equal to the greater.

TO SUBTRACT ONE NUMBER FROM ANOTHER.

Rule-1. Write the number to be subtracted, or the subtrahend, under that from which it is to be taken, or the diminuend, placing units under units, tens under tens, &c.

2. Subtract each figure in the lower line, beginning with the units, from that above it, if possible.