The first period, beginning at the right, contains units, tens, hundreds; the second, thousands, ten-thousands, hundred-thousands; the third, millions, ten-millions, hundred-millions; etc.

The number in the table is read, nine hundred eighty-seven trillion, four hundred thirty-two billion, one hundred ninetyeight million, seven hundred sixty-five thousand, four hundred thirty-two.

21. The writing of numbers is called notation, and the reading of numbers is called numeration. It will be noticed that in reading and writing numbers the s at the end of thousands, millions, etc. is omitted.

ROMAN NOTATION.

22. Roman notation is a method of expressing numbers by means of seven capital letters: These letters are I, V, X, L, C, D, and M. Their values, when standing alone, are as follows: I = 1, V = 5, X = 10, L = 50, C = 100, D= 500, M 1,000. By combinations of these letters all other numbers are expressed. Their combinations are in accordance with the following.

=

V, D, and L are never repeated, and only I, X, C, and M are ever used more than once in any combination.

2. If a letter precedes one of greater value, their difference is denoted; if it follows, their sum is denoted.

3. A bar placed over a letter multiplies its value by one

23. The following table illustrates more fully the foregoing principles:

24. The four fundamental processes of arithmetic are addition, subtraction, multiplication, and division. They are called fundamental processes because all operations in arithmetic are based upon them.

ADDITION.

25. Addition is the process of finding a number that is equal to two or more numbers taken together. The sign of addition is +. It is read plus, and means more. Thus, 5 + 6 is read 5 plus 6, and means that 5 and 6 are to be added.

26. The sign of equality is

equal to.

It is read equals, or is

Thus, 56 11 may be read 5 plus 6 equals 11.

27. Only like numbers can be added. Thus, 6 dollars can be added to 7 dollars, and the sum will be 13 dollars, but 6 dollars cannot be added to 7 feet.

28. The following table gives the sum of any two numbers from 1 to 12:

no value, the sum of any number and 0 is the number itself; thus, 17 and 0 is 17.

II and

4 is 15

II and 5 is 16
II and 6 is 17
11 and 7 is 18
11 and 8 is 19
II and 9 is 20
II and 10 is 21
II and II is 22
II and 12 is 23

29. For addition, place the numbers to be added directly under one another, taking care to place units under units, tens under tens, hundreds under hundreds, and so on.

EXAMPLE 1.-What is the sum of 131, 222, 21, 2, and 413?

EXPLANATION.-After placing the numbers in proper order, begin at the bottom of the units column and add, mentally repeating the different sums. Thus, three, five, six, eight, nine, the sum of the numbers in the units column. Place the 9 directly beneath, as the units figure in the sum.

The sum of the numbers in the tens column is 8, which is the tens figure in the sum.

The sum of the numbers in the hundreds column is 7, which is the hundreds figure in the sum.

EXAMPLE 2.-What is the sum of 61,803 + 43,429 + 47,712 + 62,138?

EXPLANATION.-The sum of the units column is 22; of the tens column it is 6 tens, or 60; of the hundreds column it is 20 hundreds, or 2,000; of the thousands column it is 13 thousands, or 13,000; and of the ten-thousands column the sum is 20 ten-thousands, or 200,000. The sum of these numbers

is 215,082. Ordinarily, the work would be performed as follows, the unnecessary ciphers being omitted:

61803

43429

47712

62138

22

6

20

13

20

sum 215082

This method is very convenient to use when adding a long column of figures.

30. In practice, addition is performed as follows:

425

36

9215

4

907

sum 10587

=

=

EXPLANATION.-The sum of the numbers in the units column 27 units, or 2 tens and 7 units. Write the 7 units as the first, or right-hand, figure in the sum. Reserve the 2 tens and add them to the tens column. The sum of the figures in the tens column plus the 2 tens reserved from the units column - 8, which is written as the second figure in the sum. There is nothing to carry to the next column. The sum of the numbers in the next column is 15, or 1 thousand and 5 hundreds. Write the 5 as the third, or hundreds, figure in the sum and carry the 1 to the next column: 1+9 10, which is written at the left of the other figures. The second method saves space and figures, but the first is to be preferred when adding a long column.