The figures which compose a large number are separated into periods and half-periods, for the more readily ascertaining the precise position which each figure occupies. The period consists of six figures, and the first, beginning on the right, is called the period of units, the second the period of millions, the third the period of billions, a contraction for millions of millions or bi-millions, and so on. Thus the number

490,386; 407,135; 017,693; 125,076

is read thus :-Four hundred and ninety thousand three hundred and eighty-six trillions; four hundred and seven thousand one hundred and thirty-five billions; seventeen thousand six hundred and ninety-three millions; one hundred and twenty-five thousand and seventy-six.

10. There does not appear to be any number naturally adapted for constituting a class of the lowest or any higher rank to the exclusion of others; though it is very probable that our present system of numeration had its origin in the practice of reckoning with the ten fingers. The number ten is called the radix or scale of the common system, because in it we ascend by collections of ten in each class to the next higher class, and though almost all nations have adopted this number as the base of their system of numeration, still it is perfectly arbitrary, and convenient as it may be for general use, there may be other scales, such as the duodenary, whose base is 12, which possess superior advantages. But whatever be the scale of notation made use of, the same principle will enable us to write all numbers in that scale. Thus in the quinary scale whose radix is 5, we need only the five characters, 0, 1, 2, 3, 4, and each figure placed on the left of another will have a value five times greater than if it occupied the place of this last. Hence in this scale, 10 means five, 11 six, 12 seven, and so on. In this as well

as in every other scale, except the denary or decimal one, we find a difficulty in enunciating a number, because there is no longer an accordance with the decimal language which pervades the construction not only of our own, but of all civilized languages.

11. Numbers may be viewed in two ways, either by considering them without particularizing the unit to which they refer, or by designating what they are intended to enumerate. Thus two, three, five, seven are abstract numbers, while three men, five days, seven books are concrete numbers. It is evident that the formation of numbers, by the successive re-union of units, does not depend upon the nature of these units, since 5 days and 7 days together make 12 days, 5 acres and 7 acres together make 12 acres, and 5 and 7 together make 12.

12. Since numbers can only be changed by increasing or diminishing them, it follows that the whole art of arithmetic is comprehended in two operations, which are termed Addition and Subtraction. But as it is frequently required to add several equal numbers together, as well as to subtract several equal sums from a greater, till it be exhausted, other methods have been devised for facilitating the operation in these cases, and named respectively Multiplication and Division. These four rules are the foundation of all arithmetical operations whatever.

ADDITION.

13. ADDITION is the collecting together of two or more numbers, and the amount of all of them is termed their sum. The sign +

(plus) is employed to indicate addition, and 7+ 2 signifies that 2 is to be added to 7. Also the sign (equal) signifies that the numbers between which it is placed are equal: thus 8 +1

EXAMPLES.

9.

1. Let it be required to find the sum of the two numbers 1724 and 4638. Take them to pieces, separating them into thousands, hundreds, tens, and units. Thus

1724 4638

=

=

1 thousand, 7 hundreds, 2 tens, and 4;
4 thousands, 6 hundreds, 3 tens, and 8.

To each of the four parts into which the first number is separated add the part of the second which is under it, beginning at the units. Thus 8 units and 4 units are 12 units; that is, 1 ten and 2 units; again, 3 tens and 2 tens are 5 tens; 6 hundreds and 7 hundreds are 13 hundreds, or 1 thousand and 3 hundreds. Lastly, 4 thousands and 1 thousand are 5 thousands; hence the sum is either

=

= 6362.

5 thousands, 13 hundreds, 5 tens and 12 units; or 6 thousands, 3 hundreds, 6 tens and 2 units 2. Let it be required to find the sum of 26389, 38127, 2815, 6817, 490, 25 and 3745.

26389

38127

2815

6817 490

25

Write the numbers, as at the side, so that the figures of the same class shall be in the same vertical columns. Then taking the sum of each class, we find there are 38 units, 27 tens, 31 hundreds, 25 thousands, and 5 tens of thousands. Now 38 units are 3 tens and 8 units; then writing 8 below the units' column, carry the 3 tens to the 27 tens, which together make 30 tens, or 3 hundreds and 0 tens. Write 0 below the column of tens, and reserve the 3 hundreds to be added to the 31 hundreds. This gives 34 hundreds or 3 thousands and 4 hundreds, and writing 4 below the column of hundreds, 78408 carry the 3 thousands to the 25 thousands, and we get 28 thousands, or 2 tens of thousands and 8 thousands. Writing 8 below the column of thousands, carry the 2 tens of thousands to the 5 tens of thousands, and finally write 7 below the column of tens of thousands, making the entire sum = 78408.

14. From these principles the following rule may be drawn :

3745

RULE. Write the numbers to be added together in vertical columns, so that the units of all the numbers may be in one column, the tens in the second, the hundreds in the third, and so on. Draw a line under the last number, and, beginning with the column of units, add successively the numbers contained in each column: if the sum does not exceed nine, write it down under the line; but if it contain tens, reserve them to be added to the next column, writing down only the units of each column; and under the last column put the entire sum whatever it may be. If the sum of any column be an exact number of tens, write 0 for the units and carry the tens to the next column.

15. The results of the partial additions being furnished by the memory, it is desirable to have some plan of testing the accuracy of the final sum, and this may be done in various ways; but we shall only mention the two following methods of the proof of addition:

1. Having found the sum in the usual way, begin at the top and

add the numbers together downward; then if the summation is the same by both methods, it is very probable that the sum is correctly obtained. 2. Separate the numbers into two or three parts; find the sum of each separately, and then add all these partial sums together, which will give the whole amount. It is usual to separate the numbers into two parts only, the uppermost number forming one part.

SUBTRACTION.

16. SUBTRACTION is the taking a less number from a greater, and finding their difference. The process of subtraction involves two principles-the one is the equal augmentation or diminution of each of the numbers. In either way the difference of the two numbers will not be altered; for if the greater number be either increased or diminished by 7, for example, and the less be likewise increased or diminished by 7, the numbers themselves will be altered but not their difference. The other principle is this: since 12 exceeds 7 by 5, and 8 exceeds 6 by 2, then 12 and 8 together, or 20, exceed 7 and 6 together, or 13, by 5 and 2 together, or 7. The sign - (minus) is used to indicate subtraction, and 97 signifies that 7 is to be taken from 9.

EXAMPLES.

1. Let it be required to take 231 from 574.

Write the numbers as in the margin, units under units, tens under tens, and hundreds under hundreds.

Then

3

4

3

5 hundreds exceed 2 hundreds by 3 hundreds. Therefore by the second principle all the first column together exceeds all the second column together by all the third column together, that is, by 3 hundreds 4 tens and 3 units, or 343, which is the difference between 574 and 231.

2. Let it be required to subtract 23957 from 802126. Write the numbers at length; thus

Now here a difficulty immediately arises, since 7 is greater than 6, and cannot be taken from it, neither can 5 be taken from 2; 9 from 1; 3 from 2; nor 2 from 0. To obviate this we must have recourse to the first principle, and add the same number to both of these numbers, which will not alter their difference. Add ten to the first number, making 16 units; and add ten also to the second number, but instead of adding ten to the number of units, add one to the number of tens, making 6 tens. Again add ten tens to the first number and one hundred to the second; then add ten hundreds to the first and one thousand to the second, and so on, adding equal numbers to each. In this way the numbers will be changed into the following:

Hence,

and the difference 778169 is obtained in the usual manner. when the upper figure is the less, we must augment it by ten, and retain one to be added to the lower figure immediately to the left.

The proof of subtraction is deduced from the simple fact, that th difference added to the smaller number is equal to the greater number.

MULTIPLICATION.

17. MULTIPLICATION is the finding the amount of a number repeated any number of times. The number which is repeated is called the multiplicand, the number denoting the repetitions is called the multiplier, and the amount the product. The multiplicand and multiplier are termed the factors of the product, and the sign × (into) denotes multiplication. Thus 12 x 3 signifies that 12 is to be repeated three times, and added together; thus 12 + 12 + 12 = 36.

18. When the multiplicand and multiplier are large numbers, as 1269 and 423, we should have to write 1269, the multiplicand, 423 times, and then to make an addition of enormous length. This operation can be abridged, however, by reducing it into a certain number of partial multiplications which may be easily effected mentally; but previous to explaining a shorter method, the following table must be committed to

memory.

Multiplication Table.

1 2 3 4 5 6 7 8 9 10 11 12

12 24 36 48 60 72 84 96 108 120 132 144

To form this table, write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, in the first horizontal line; then add each of these numbers to itself to form the second line, which is composed of the products of

each of these numbers multiplied by 2. To each number in the second line add the corresponding one in the first, and the third line is formed, containing the several products of the numbers in the first line multiplied by 3. Again adding the numbers in the third line to the corresponding ones in the first, a fourth line is formed, containing the products of each number of the first line by 4; and so on to the last line. The table may be extended if required. If we take any of the numbers in the first line, as 8, and proceed downwards, we shall find the same succession of numbers as if we had taken 8 at the side and proceeded to the right; hence 8 x 5 40 = 5 x 8. This may be shown in the following manner. Place 8 counters in a line, and repeat that line 5 times; then the number of counters in the whole is 5 times 8 if they are counted by rows from the top to the bottom; but if they are counted by vertical columns, we shall find eight rows with five in each row, the whole number of which is 8 times 5. Hence we see that

5 +5 +5 +5 +5 +5 +5 + 5, or 8 x 5 5 x 8.

=

This method of proof may be applied to any numbers beyond the range of the table, and hence in any multiplication the order of the factors may be changed, that is, either of them may be taken as the multiplier.

19. Let it be required to multiply 739 by the single figure 8. Since the product of 739 by 8 is evidently equal to the sum of the products of all its parts, we have the following operation :

739

5912

In practice the partial products 72, 240, and 5600, are not written down, but combined mentally into one sum: thus we say 8 times 9 are 72, write down 2 and reserve the 7 tens; then 8 times 3 are 24, and the reserved 7 added thereto gives 31, write down 1 and carry the 3 to the product of 8 by 7 or to 56 hundreds, and the entire number of hundreds is 59, the whole product being 5912. 20. Find the products of 2376 multiplied by 10 and by 100. To multiply any number by 10, we have only to remove each of the figures of the multiplicand one place to the left and their value will be increased ten times; hence 2376 × 10 23760. In like manner 2376 × 100 = 237600, where the value of each figure is increased a hundred times.

=

Hence any number will be multiplied by 10, 100, 1000, etc., by writing on the right of the multiplicand as many ciphers as there are in the multiplier.

21. When the significant figure of the multiplier is not a unit, as for example 30, 400, or 7000. Since these multipliers are the same as 10 times 3, 100 times 4, or 1000 times 7; the multiplicand is first multiplied by the significant figure 3, 4, or 7, by Art. 19, and afterwards the