Examples in Notation and Numeration.

1. Write two hundred and nine.

2. Write five thousand and five.

3. Write twelve thousand and twelve.

4. Read 1040; 30706; 6606.

5. Read 2001; 35006; 4070070.

6. Write one hundred thousand, one hundred and one.

7. Read 207600042; 1000860005.

8. Read 100000100; 5000000750001.

9. Write forty-seven millions, two hundred and four thousand, eight hundred and fifty-one.

10. Write six quadrillions, forty-nine trillions, seventy-two billions, four hundred and seven thousand, eight hundred and sixty-one.

11. Write eight hundred and ninety-nine quadrillions, four hundred and sixty trillions, eight hundred and fifty billions, two hundred millions, five hundred and six thousand, four hundred and ninety-nine.

12. Write and read, fifty-nine trillions, fifty-nine billions, fifty-nine millions, fifty-nine thousand, nine hundred and fifty

nine.

13. Eleven thousand, eleven hundred and eleven.

14. Nine billions and sixty-five.

15. Write and read, three hundred and four trillions, one million, three hundred and twenty-one thousand, nine hundred and forty-one.

16. Write and read, nine trillions, six hundred and forty billions, with 7 units of the ninth order, 6 of the seventh order, 8 of the fifth, 2 of the third, 1 of the second, and 3 of the first.

17. Write and read, three hundred and five trillions, one hundred and four billions, one million, with 4 units of the fifth order, 5 of the fourth, 7 of the second, and 4 of the first.

18. Write and read, three hundred and one billions, six millions, four thousand, with 8 units of the fourteenth order, 6 of the third, and two of the second.

ADDITION.

24. 1. John has two apples, and Charles has three: how many have both?

ANALYSIS. They have, together, as many apples as are equal to 2 apples counted with 3 apples, which are 5 apples.

2. James had 5 marbles, and William gave him 7 more. how many had he then?

8 and 5?

3. Mary has 6 pins, and Jane 9: how many have both? 4. How many are 5 and 3? 6 and 4? 5. How many are 4 and 9? 6. How many are 3 and 7? 10 and 0? 0 and 10? 7. How many are 1 and 5 and 6? 3 and 4 and 9?

The answer to any of the above questions, is called, the Sum of the numbers, and the operation by which we find it, is called, Addition.

25. The Sum of two or more numbers, is a number which contains as many units as there are in all the numbers added. ADDITION is the operation of finding the sum of two or more numbers.

Of the Signs.

26. The sign+, is called plus, which signifies, more. When placed between two numbers, it denotes that they are to be added together.

The sign, is called, the sign of equality. When placed between two numbers, it denotes that they are equal to each other. Thus, 3+2 = 5, denotes that the sum of 3 and 2 is equal to 5.

24. What is the sum of two or more numbers? What is Addi tion?

26. What is the sign of Addition? What is it called? What does it signify? Express the sign of equality. When placed between two numbers, what does it show?

27. The operation of Addition is governed by four principles, viz. :

1. A single number expresses a collection of like units. 2. Like units alone can be added together; that is, units must be added to units, tens to tens, hundreds to hundreds, &c.

3. Every number, expressed by two or more figures, is the

sum of its units, tens, hundreds, &c.; thus, 279 is the sum of 2 hundreds, 7 tens, and 9 units

4. The sum of several numbers is equal to the sum of all their parts.

1. James has 14 cents, and John gives him 21: how many cents has he then?

OPERATION.

14

21

35 cents.

ANALYSIS. Since units must be added to units, and tens to tens, the numbers are written so that units of the same order may fall in the same column, and a line is drawn beneath them. The column of the lowest order is first added, and contains 5 units, which are written under the column. The tens are next added, and they amount to tens, which are written under

the tens. The sum is 3 tens and 5 units, or thirty-five.

2. A gentleman bought a carriage for 385 dollars, a team of horses for 286 dollars, and two sets of harness for 96 dollars: what did he pay for all?

ANALYSIS.-Write the numbers so that units of the same value shall fall in the same column; then add each order of units separately.

OPERATION.

385

286

96

17

25

5

767

The following, however, is the method in practice:

OPERATION.

385

286

96

ANALYSIS.-Write the numbers as before. The units are added together, and their sum is 17, which is 1 ten and 7 units; the units are placed under the column of units, and the 1 ten is added with the column of tens, which then amounts to 26 tens, equal to 2 hundreds and 6 tens; the 6 tens are placed under the tens, and the hundreds are added with the column of hundreds, which amounts to 7, and is therefore placed under the hundreds.

767 dollars.

27. By how many principles is the operation of adding governed Name them.

When a column amounts to ten, or more than ten, the nnit figure is set down, and the tens' figure is added to the next column, because, 10 units of any order make 1 unit of the next higher order. This process is called, carrying to the next column.

28. Hence, to find the sum of two or more numbers, we have the following

Rule.

I. Write the numbers to be added, so that units of the same order shall fall in the same column.

II. Add the column of units: set down the units of the sum, and carry the tens to the next column.

III. Add the column of tens: set down the tens, and carry the hundreds to the next column; and so on, till all the columns are added, and set down the entire sum of the last column.

Proof.

The PROOF of any operation in Addition, consists in showing that the result, or answer, contains as many units as there are in all the numbers added, and no more. There are two methods of proof, for beginners:

I. Begin at the top of the units column, and add all the columns downward, carrying from one column to the other, as when they were added upward. If the two results agree, the work is supposed to be right.

II. Draw a line, dividing the numbers into parts. Add the parts separately, and then add the sums. If the last sum is the same as the sum first found, the work may be regarded as right.

28. How do you set down numbers for addition? Where do you begin to add? If the sum of any column can be expressed by a single figure, what do you do with it? When it cannot, what do you do? When you add to the next column, what is it called? What do you set down in the last column? What does the proof consist of, in Addition? What is the first method of proof? What is the second method of proof?