ARITHMETIC

DEFINITIONS

1. Arithmetic is the science of numbers and their use. 2. A Unit is a single thing; as, one, one man, one horse. 3. A Number is one or more units; as 1, 3, 9, 6 boys. 4. The Unit of a Number is one of the kind expressed by the number. The unit of 9 is 1, the unit of 20 feet is 1 foot.

5. An Integer is a whole number.

6. Like Numbers are those which are composed of the same kinds of units. Thus 25 and 34; 3 yards and 10 yards.

7. An Abstract Number is one used without reference to any particular thing or quantity. Thus 15; 64; 280.

8. A Concrete Number is one used with reference to some particular thing or quantity. Thus 25 dollars; 14 days; 100 men.

Concrete numbers are called denominate numbers because the denomination or kind is named.

9. A Sign is a character used to indicate an operation, or express the qualities or relations of numbers.

A Solution is a process of computation used to obtain a required result.

10. A Problem is a question for solution.

11. An Example is a problem solved, illustrating a principle or rule.

12. A Principle is a truth upon which the solution is founded. 13. An Analysis is a statement of the successive steps in a solution.

14. An Explanation is a statement of the reasons for the manner of solving a problem.

15. A Rule is a direction for solving problems.

NOTATION AND NUMERATION

16. Notation is the art of writing numbers by means of characters.

17. Numeration is the art of reading numbers written by characters.

Two systems of Notation are in general use: the Roman and the Arabic. The Roman is supposed to have been invented by the Romans and employs seven capital letters to express numbers. The Arabic is said to be derived from the Arabs and employs ten characters called figures.

ARABIC NOTATION

18. Figures are characters used to represent numbers. There are ten figures:

O,

1, 2, 3, 4, 5, 6, 7, 8, 9.

Zero, One, Two, Three, Four, Five, Six, Seven, Eight, Nine.

The Figure 0, Zero or Cipher, expresses no units, or nothing, when standing alone. The other nine figures express the number of units shown by their names. These figures are called digits.

19. To express numbers greater than nine and less than one hundred, two figures are written side by side; as, thirty-six, 36; seventy-two, 72.

20. To express numbers greater than ninety-nine, three or more figures are written side by side; as, one hundred eighty-five, 185; two thousand six hundred twenty-four, 2624.

21. When figures are written side by side, the one at the right expresses units or ones, the next tens, the next hundreds, the next thousands, etc.

22. The Simple Value of a figure is the value it expresses when standing alone, or in unit's place; as 3, 7, 9.

23. The Local Value of a figure is the value it expresses when used with other figures to represent a number.

In the number 345, the figure 5 expresses a simple value, and the figures 3 and 4 express local values.

24. The Order of a Unit takes its name from the place it occupies. A figure in the first place expresses units of the first order; in the second place, units of the second order, etc.

When a figure stands in the second place it represents tens;

in the third place hundreds; in the fourth place thousands, etc. One ten is written..

One ten and four, fourteen.

Two tens, twenty..

Two tens and seven, twenty-seven.

10

14

20

27

30

Three tens, thirty.

One hundred, one ten and seven, one hundred seventeen.... 117 Three hundred, five tens and nine, three hundred fifty-nine.. 359 Six thousand, five hundreds, nine tens and two, is read, six thousand five hundred ninety-two...

.6592

NOTE.—In reading whole numbers the word AND should not be used. Thus, seven hundred fifty-four, not seven hundred and fifty-four.

Copy and read the following numbers:

19.

20.

Six thousand nine hundred twenty-seven.
Four thousand sixty-four.

21. Seven thousand four hundred one.

23. Nine thousand six hundred ninety-six.

24. Eight thousand eight hundred.

25.

26.

Seven units of the first order and two of the second. Nine units of the fourth order, three of the third, two of the second and one of the first.

27. Five units of the third order, one of the first.

28. Six units of the fourth order, seven of the second,

two of the first.

25. For convenience, figures are arranged in periods of three places each; the first three at the right being called units or one's period; the next three the thousand's period; the next three the million's period, etc.

1. Ten units of any order equal one unit of the next higher order... Ten units equal one ten; ten tens equal one hundred; ten hundreds equal one thousand, etc.

2. Removing a figure one place to the left increases its value ten times. Removing a figure one place to the right decreases its value ten times.

To Write Numbers

a. Begin at the left and write the figures belonging to the highest period.

b. Write the hundreds, tens and units of each period in their order, putting a cipher in the place of any order that is omitted.