Thus: How many are 7 +3 +5, would be a problem in addition, and the answer, 15, would be the sum of 7, 3, and 5.

(c.) In order that numbers may be added, it is necessary that the things they represent shall be of the same name or denomination.

Illustrations.

-

2 books and 3 slates would be neither 5 books nor 5 slates; but since books and slates are both things, we can change the denomination of both by calling them things; when we shall have, 2 things and 3 things are 5 things.

In like manner 2 tens and 3 units would be neither 5 tens nor 5 units; but by reducing the tens to units, calling them 20 units, we shall have 2 tens + 3 units 20 units+3 units =23 units.

2 shillings + 3 pence are neither 5 shillings nor 5 pence; but since 2 shillings =24 pence, 2 shillings +3 pence must equal 24 pence +3 pence, or 27 pence.

(d.) For such reasons it will be found convenient, in writing large numbers for addition, to write those of the same denomination near each other. This can best be done by writing them in vertical columns, so that units shall come under units, tens under tens, &c., and pounds under pounds, shillings under shillings, pence under pence, &c.

(e.) In adding, we can begin with any denomination we choose; but it will usually be more convenient to begin with the lowest, or the one at the right hand, and to reduce the sum of each column to a higher denomination, when it can be done.

NOTE. Addition is the most important of the four numerical operations, both because it is the foundation of all the others, and because it is the one most frequently used in all the departments of practical life. Moreover, it is the one in which there is the greatest liability to error. For these reasons, and many others which might be urged, the student should be very careful to master it fully.

(f) The methods of applying these principles are illustrated in the following examples and solutions.

50. Simple Addition.

(a.) Abstract numbers, or concrete numbers, which represent values in terms of a single denomination, as in pounds, in bushels, or in dollars, are called SIMPLE NUMBERS; but concrete numbers, which represent values in terms of several

different denominations, as in pounds, shillings, and pence, or in bushels, pecks, and quarts, are called COMPOUND NUMBERS. (b.) SIMPLE ADDITION is the addition of simple numbers. What is the sum of 75798+ 24687 +39764 +86328+ 439528386536?

Solution. We first write the numbers, placing units under units, tens under tens, &c., in order that figures expressing the same denomi nations may be near together & Thus :

75798.

24687.

39764.

86328.

4395.

283.

86536.

317,791.

* Beginning at the bottom of the units column, (because the lowest denomination mentioned is units,) and naming only results, we add thus: 6, 9, 14, 22, 26, 33, 41 units, which are equal to 4 tens and 1 unit.

Writing 1 as the units' figure of the amount, we add the 4 tens with the figures of the tens column; thus, 4, 7, 15, 24, 26, 32, 40, 49 tens, which are equal to 4 hundreds and 9 tens.

Writing 9 as the tens' figure of the amount, we add the 4 hundreds with the figures of the hundreds column, thus; 4, 9, 11, 14, 17, 24, 30, 37 hundreds, which are equal to 3 thousands and 7 hundreds.

Writing 7 as the hundreds' figure of the amount, we add the 3 thousands with the figures of the thousands column; thus, 3, 9, 13, 19, 28, 32, 37 thousands, which are equal to 3 ten-thousands and 7 thousands.

Writing 7 as the thousands' figure of the amount, we add 3 ten-thou sands with the figure of the ten-thousands column; thus, 3, 11, 19, 22, 24, 31 ten-thousands, which are equal to 3 hundreds thousands and 1 ten-thousand; and as there are no higher denominations, we write the 3 and 1 in their appropriate places.

Having thus added all the denominations, we must have the sum, or amount of the numbers, which is 317,791.

NOTE. - Many call the names of the separate numbers added, as well as the results of the addition, and would add, thus: 6 and 3

If the learner does not readily understand this method of addition, let him for a time call the separate numbers added, as explained in the note.

are 9, and 5 are 14, and 8 are 22, and 4 are 26, and 7 are 33, and 8 are 41; 41 units are equal, &c. This method is much less expeditious than the first one given, and therefore should not be adopted. Indeed, we may say, that calling the names of the separate numbers comprising a sum is to addition what spelling, i. e., calling the letters composing a word, is to reading.

51. Compound Addition.

(a.) Compound Addition is the addition of compound

numbers.

What is the sum of £8 15 s. 11 d. 2 qr. + £3 12 s. 8 d. 3 qr. £9 19 s. 9 d. 1 qr. + £7 18 s. 10 d. 3 qr. + £5 18 s. 2 qr. +£6 8 d. 1 qr. + £5 13 s. 3 d. + 16 s. 8 d. 1 qr.?

Solution.-We first write the numbers, placing pounds under pounds, shillings under shillings, &e., in order that figures expressing the same denomination may be near each other.

Beginning with the right hand column, as before, and reducing as we add, we proceed thus: 1 qr. and 1 qr. are 2 qr., and 2 qr. are 4 qr. = 1 d., and 3 qr. are 1 d. 3 qr., and 1 qr. are 1 d. 4 qr.= 2 d., and 3 qr. are 2 d. 3 qr., and 2 qr. are 2 d. 5 qr.= = 3 d. 1 qr.

Writing 1 as the farthings' figure of the sum, we add the 3 d. with the numbers in the pence column, thus: 3 d. and 8 d. are 11 d., and 3 d. are 14 d. = 1 s. 2 d., and 8 d. are 1 s. 10 d., and 10 d. are 1 s. 20 d. 2 s. 8 d., and 9 d. are 2 s. 17 d. = = 3 s. 5 d., 4 s. 1 d., and 11 d. are 4 s. 12 d.*= 5 s.

and 8 d. are 3 s. 13 d.

Writing 0 as the pence figure of the sum, we add the 5 s. with the numbers in the shillings column, thus: 5 s. and 16 s. are 21 s. = £1

Since 12 d. = 1s

I s.,* and 13 s. = £1 14 s., and 18 s. are £1 32 s.*= £2 12 s., and 18 s. are £2 30 s.£3 10 s.,* and 19 s. are £3 29 s. £4 9 s.,* and 12 s. are £4 21 s. = £5 1 s.,* and 15 s. are £5 16 s.

Writing 16 as the shillings' figure of the sum, we add £5 with the numbers of the pounds column, thus: 5, 10, 16, 21, 28, 37, 40, 48.

As all the denominations of the given numbers have been added, the amount sought must be £48 16 s. 0 d. 1 qr.

(b.) In the above form of solution, the numbers to be added have been named merely to insure that the explanations should be understood, but in practical work the additior. should be performed by naming only results.

Thus 1 qr., 2 qr., 4 qr.= 1 d.; 1 d. 3 qr., 1 d. 4 qr.=2 d. ; 2 d. 3 qr., 2 d. 5 qr.=3 d. 1 qr. Write 1 qr. 3 d., 11 d., 14 d. = 1 s. 2 d.; 1 s. 10 d., 1 s. 20 d. 2 s. 8 d.; 2 s. 17 d.= = 3 s. 5 d., &c.

=

52. Compound and Simple Addition compared.

(a.) Compound Addition involves precisely the same prin ciples that Simple Addition does. In both, numbers of the same denomination are placed under each other, in order that they may be more readily distinguished. In both, we commence to add at the lowest denomination, in order to avoid the necessity of erasing or altering figures which have been once written; in both, we reduce the sum of each column to units of the next higher denomination, in order that the answer may appear in its simplest form; and in both we add the units thus obtained with those written in the column of the next higher denomination. Moreover, the same methods of proof apply to both.

(b.) The slight differences in the methods of applying these principles result from the fact, that in simple numbers 10 units of one denomination always equal one of the next higher, while in compound numbers there is no uniformity in this respect.

NOTE TO THE TEACHER. The explanations and examples are so arranged that, should the teacher think it inexpedient to teach Comound Addition at the same time that Simple Addition is taught, he can

Since 20 = £1.

defer it till. in his opinion, the class are prepared for it. We would, however, recommend that whenever it is taught, it should be presented as a further application of the principles involved in Simple Addition. The same thing may be said of Simple and Compound Subtraction, Multiplication, and Division.

53. Methods of Proof.

(a.) We can test the correctness of the work in many ways, a few of which we will mention.

First Method.

Go over the work carefully a second time

in the same manner as at first.

i.

Second Method. — Begin to add at a different part of the column from that at which the first addition was commenced; e., if the first addition was commenced at the top, begin the second at the bottom, and vice versa. This, by presenting the figures in a different order, renders it improbable that any mistake which may have been made in the first work will be repeated.

Third Method.

Separate the numbers to be added into two or more parts, add the parts separately, and then add This method is illustrated below.

their sums.

317791 Sum of the two partial sums first a

Thus showing that the work was correct.