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NOTATION AND NUMERATION.
NOTATION.

25. Notation is the process of representing numbers by letters, figures, or other symbols.

The common methods of expressing numbers are three: by words, written or spoken; by letters, called the Roman method; and by figures, called the Arabic method.

26. In common language, we express numbers by the terms one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty, twenty-one, etc., giving a distinct name to each unit as far as ten, when we begin a second ten, and pass on to twenty; a third ten, and pass on to thirty; and so on to forty, fifty, sixty, seventy, eighty, and ninety. Proceeding thus we reach ten tens, which we call one hundred, when we begin a second hundred, and pass to two hundred; a third hundred, and pass to three hundred; and so on as far as ten hundred, which we call one thousand. A thousand thousand we call one million; a thousand million, one billion; a thousand billion, one trillion; and so on with numbers still higher.

Note 1. — The term eleven is a contraction of one left after ten; and twelve, of two left after ten. Thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, are derived from three and ten, four and ten, five and ten, etc. Twenty, thirty, forty, fifty, sixty, seventy, eighty, and ninety are contractions of two tens, three tens, four tens, etc.

Note 2. — Billion is a contraction of the Latin bis, twice, and million; and trillion, of the Latin tres, three, and million. In like manner from the Latin numerals, quatuor, four; quinque, five; sex, six; septem, seven; octo, eight; novem, nine; decern, ten; undecim, eleven; duodeeim, twelve; tredecim, thirteen; quatuordecim, fourteen; quindecim, fifteen; sexdecim, sixteen; septendecim, seventeen; octodecim, eighteen'; novemdecim, nineteen; viginti, twenty, — are formed quadrillions, quintillions, sextillions, septillions, octillions, nonUlions, decilr Kors, undedUions, duodecillions, etc.

Roman Notation.

27. The Roman Notation, So called from its having originated with the ancient Romans, employs in expressing numbers seven capital letters, viz.:

I, V, X, L, C, D, M.

one, five, ten, fifty, one hundred, five hundred, one thousand.

All intervening and succeeding numbers are expressed by use of these letters, either in repetitions or combinations. By a letter being written after another denoting equal or less value, the sum of their values is represented; as, II represents two; VI, six. By writing a letter denoting a less value before a letter denoting a greater, their difference of value is represented; as, IV represents four; XL, forty.

A dash (—) placed over a letter increases the value denoted by the letter a thousand times; as, V represents five thousand; IV, four thousand.

Table.

[table]

Note 1. — The Roman method of Notation is now but little used, except in numbering sections, chapters, and other divisions of books; and for indicating the hours on the face of clocks, watches, or dials.

Note 2. — Formerly CIO was used to represent one thousand, and the pre fixing of a C and the annexing of a 0 increased the number denoted ten times; thus, CCIOO represented ten thousand, and CCCIOOO, one hundred thousand.

Exercises.
Represent the following numbers by letters:—

1. Forty-nine. Ans. XLIX.

2. Ninety-seven.

3. One hundred and eighty-eight.

4. Two hundred and nineteen.

5. Six hundred and sixty-three.

6. One thousand five hundred and six.

7. One thousand eight hundred and fifty-seven.

8. Four thousand four hundred and forty-four.

9. Eleven thousand nine hundred and eleven.

10. One hundred fifty thousand and fifty.

11. One million twenty thousand and twenty.

12. Three million one hundred thousand.

Arabic Notation.

28. Arabic Notation, so called from its having been made known through the Arabs, employs in expressing numbers ten characters or figures, viz.:

1, 2, 3, 4, 5, 6, 7, 8, 9, 0. one, two, three, four, five, six, seven, eight, nine, cipher.

The first nine are sometimes called digits, and the cipher, naught or zero.

29. The place of a figure is its particular position with regard to other figures; as in 61 (sixty-one) counting from the right, the 1 occupies the first place and the 6 the second place, and so on for any other like arrangement of figures.

30. The digits have been denominated significant figures, because each of itself expresses a positive value, always representing so many units, or ones, as its name indicates. But the size or value of the units represented by a figure differs with the place occupied by the figure. Thus in 235 (two hundred and thirty-five), each of the figures, without regard to its place, expresses units, or ones; but these units or ones differ in value. The 5 occupying the first place represents 5 single units; the 3 occupying the second place represents 3 tens, or 3 units each ten times the size or value of a unit of the first place; and the 2 occupying the third place represents 2 hundreds, or 2 units each one hundred times the size or value of a unit of the first place; the value expressed by any figure being always made tenfold by each removal of it one place to the left.

31 i The cipher becomes significant when connected with other figures, by filling a place that otherwise would be .vacant; as in 10 (ten) where it occupies the vacant place of units, and in 102 (one hundred and two) where it fills the vacant place of tens.

32. The simple value of a unit is the value expressed by a figure standing alone; or, in a collection, when standing in the right-hand place. Thus 2 alone, or in 32 (thirty-two), expresses a simple value of two single units or ones.

33. The local value of a unit is the value expressed by a figure when it is used in combination with another figure or figures, and depends upon the place the figure occupies. Thus, in 44 (forty-four), the 4 in the first place expresses the local value of 4 units, and the 4 in the second place, the local value of 4 tens, or forty.

34. The successive places occupied by figures are often called orders. Thus a figure in the first or units' place is called a figure of the first order, or of the order of units; a figure in the second place is a figure of the second order, or of the order of tens; in the third place, of the third order, or of the order of hundreds; and so on, each figure next to the left belonging to a distinct order, the unit of which is tenfold the size or value of a unit of the order at the right.

Exercises.

1. Write three units of the first order.

2. Write five units of the first order.

3. Write eight units of the second order, with seven of the first.

4. Write two units of the third order, with none of the second, and one of the first.

5. Write seven units of the fourth order, with two of the third, none of the second, and none of the first. .

6. Write one unit of the fifth order, with none of the four lower orders.

7. Write six units of the sixth order, five of the fifth, four of the fourth, three of the third, one of the second, and two of the first.

8. Write one unit of the eighth order, with none of the seven lower orders.

9: Write nine units of the ninth order, with six of each of the eight lower orders.

10. Write two units of the twelfth order, with none of the eleventh, none of the tenth, one of the ninth, five of the eighth, nine of the seventh, none of the sixth, none of the fifth, none of the fourth, three of the third, none of the second, and three of the first.

11. Write three units of the fifteenth order, with none of the fourteenth, none of the thirteenth, none of the twelfth, one of the eleventh, seven of the tenth, five of the ninth, one of the eighth, none of the seventh, none of the sixth, five of the fifth, three of the fourth, two of the third, two of the second, and seven of the first.

12. Write four units of the twenty-fifth order, with three of the twenty-fourth, two of the twenty-third, none of the twentysecond, none of the twenty-first, none of the twentieth, none of the nineteenth, none of the eighteenth, none of the seventeenth, five of the sixteenth, and none of the fifteen lower orders.

NUMERATION.

35. Numeration is the process of reading numbers when expressed by figures.

36. There are two methods of numeration; the French and the English.

French Numeration.

37. The French method of numeration is that in general use on the continent of Europe and in the United States. Beginning at the right, figures occupying more than three places being separated into as many groups as possible oi three figures each, called periods, it gives a distinct name to each period.

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