15. The place of a figure in a number is the position it occupies with reference to other figures; thus, in 436, the 6, counting from the right, is in the first place, 3 is in the second place, and 4 in the third place.

The figure in the first place represents simple units, or units of the first order; the second figure represents tens, or units of the second order; the third, hundreds, or units of the third order; the fourth, thousands, or units of the fourth order, etc.; thus, in the number 3576, the 6 is 6 units of the first order; the 7 tens is 7 units of the second order; the 5 hundreds is 5 units of the third order, etc.

16. Therefore, to determine the value of a number expressed in figures, two things must be considered: first, how many units each figure represents; and, second, the order of the units represented. Thus, in each of the numbers 2, 20, 200, the left-hand figure represents two units, but in the first it represents two units of the first order, in the second, two units of the second order, or two tens, and in the third, two units of the third order, or two hundreds.

17. It is also evident that a figure is made to represent units of tenfold value by removing it one place toward the left; a hundred fold by removing it two places, etc.; that is, ten units of the first order make one ten, ten units of the second order make one hundred, and, in short, ten units of any order make one unit of the next higher order.

18. The cipher, when used with other figures, fills a place that would otherwise be vacant; thus, in 206 the cipher occupies the place of tens, because there are no tens expressed in the given number.

19. As figures are written on the left of the unit figure, so they are written on the right and are subject to the same law; that is, ten units of any order make one unit of the next higher order.

20. Figures on the right of units are called decimals, and are separated from the unit figure by a period, which is called the decimal point. The first place at the right of the point is tenths, the second, hundredths, the third, thousandths, and so on. Thus, 1.1 is one, and one tenth; 1.01 is one, and one hundredth; 1.001 is one, and one thousandth; and so on. 21. The following table shows the method of writing numbers and the names of the places.

22. For convenience in reading, figures are often separated into periods of three figures, as shown in the table. The first period on the right of the decimal point contains tenths, hundredths, and thousandths, and is called the thousandths' period; the first period at the left of the decimal point contains units, tens, and hundreds, and is called the units' period; the second period contains thousands, tens of thousands, and hundreds of thousands, and is called the thousands' period; and so on, as shown in the table.

NOTE. This table can be extended to any number of periods, as shown in the Appendix.

23. The value of the figures in the table, expressed in words, is seven hundred six billion four hundred seventy-six million one thousand eight hundred forty-three, and five hundred forty-eight thousandths.

24. To read figures at the left of the decimal point,

Rule.

1. Beginning at units, numerate and point off the number into periods of three figures each.

2. Beginning at the left, read each period separately, giving the name of each period except that of units.

To read figures at the right of the.decimal point,

Rule.

Read the figures at the right of the point the same as figures at the left, and then add the name of the place of the right-hand figure.

Thus, 0.6 is six tenths; 0.65 is sixty-five hundredths; 0.659 is six hundred fifty-nine thousandths.

NOTE. To avoid ambiguity in writing in words and in reading a decimal we shall use the conjunction and in such cases only between the unit and the decimal. Thus, 0.203 is not two hundred and three thousandths, but two hundred three thousandths; and 200.003 is two hundred, and three thousandths.

Or, the word decimal may be written before the decimal. Thus, 306.205 is three hundred six, and decimal two hundred five thousandths (or decimal two, naught, five).

26. To write numbers,

Rule.

Beginning at the left, write the figures of each successive period in their order, filling each vacant place with a cipher and placing the decimal point between units and tenths.

NOTE. Where there are no decimal figures the decimal point is usually omitted.

Exercises in Notation and Numeration.

27. Write the following numbers in figures:

25. Two hundred five.

Ans. 205.

NOTE. Since no figure of the second order is given, a cipher is written in the second place.

26. Six hundred thirty-eight.

27. Three hundred fifty-six.

28. Six hundred fifty-three.

29. Five hundred sixty-three.

30. Three hundred sixty-five.

31. Six hundred fifty-one.

32. One thousand six hundred fifty-one.

33. Forty-two thousand five hundred fifty-four.

34. Eight hundred sixteen thousand two hundred.

35. Six million one hundred four thousand two hundred seventy-six.

36. Three hundred six thousand five hundred two.

37. Nine hundred forty-six million five hundred fourteen thousand nine hundred twenty-five.

six.

38. Six billion fifteen million seven thousand four hundred.

39. Five million six hundred fifty-one thousand four hundred

40. Seventy-four million.

41. Sixty-three million fourteen thousand seven hundred.

42. Two, and five tenths.

43. Fifty-two hundredths.

44. Sixty, and four hundredths.

45. Two hundred four thousandths.

46. Eight hundred, and fourteen thousandths.

47. Four million sixteen thousand seven, and four hundredths. 48. Seventeen million seventeen thousand seventeen, and seventeen thousandths.

49. One billion one million one thousand one hundred, and one hundredth.

50. Eleven billion eleven thousand, and eleven thousandths.

28. QUESTIONS.

2. A Number?

ART. 1. What is a Unit? 3. What is a Concrete Number? 4. An Abstract Number? 5. What is Arithmetic ? 6. What is Notation? 7. What is Numeration? 8. How many figures are used in Notation? Name them. 9. What is the first figure, 0, called? 10. What is the greatest number expressed by one figure? 11. Ten, how expressed? Twenty? 12. Numbers from ten to twenty, how expressed? 13. One hundred, how expressed? Two hundred ? 14. Other numbers, how expressed? 15. What is the place of a figure? What does the figure in the first place represent? Second place? Third? 16. Explain the method of determining the value of a number. 17. How does moving a figure towards the left affect its value? 18. For what is a cipher used ? 19. Can figures be written on the right of units? What is the law? 20. What are figures on the right of units called? What separates them from units? What is the first place at the right of units called? The second? The third? 21. Repeat the table. 22. How many figures are there in a period? Name the periods. 23. Read the figures in the table. 24. Give the rule for reading figures at the left of the decimal point. Rule for reading figures at the right of the decimal point. Explain the ambiguity referred to in the note, and tell how it may be avoided. 26. Give the rule for writing numbers.