To express in words the number represented by 36405.4916, we can separate off one complete group in the integral part and one in the decimal part, and 36,405.4,916 is then read thirty-six thousand four hundred five and four thousand nine hundred sixteen ten-thousandths.

To 'read off' decimals, it is, however, the common practice merely to name the digits in order.

Thus .615 is read 'decimal six, one, five'; 15.0524 is read 'fifteen, decimal naught, five, two, four'; and 1567.0082 is read 'one thousand five hundred sixty-seven, decimal naught, naught, eight, two.'

In reading numbers the word 'and' should be used only when we reach a decimal point.

EXAMPLES I.

1. For what does 5 stand in the numbers 15, 1.57, 514, 352167, and 3561234, respectively?

2. For what does 7 stand in the numbers 70, 37123, 125.479, 274126315, and 370001002003, respectively?

3. Name all the figures which represent their digit value of hundreds in 314, 2167, 50412, and 31024.

4. Name all the figures which represent their digit value of thousands in 2314, 56123, 60417, and 3005167.

5. Express in words the separate value of every figure in 3.5, 15.7, 125.34, 12.53, 800.17, 1200.63, .875, 50.037, 5.00107, 560002.19007.

6. Express in words the numbers 27, 349, 560, 3.06, 1204, and 5020.

7. Express in words the numbers 200.9, 6050, 12345, 10.305, 40050, and 1.20463.

8. Express in words the numbers 518618, 602010, 100010, 504075, 420040, and 107.005.

9. Express in words 111111111, 1203405, 2314100, 504.0314, 20050060, and 30300074.

10. Express in words the numbers 3012004, 1101.11011, 201201201, 1000040305101, and 604102000300004.

11. Write in figures the numbers fifty-eight, eightyfive, two hundred eleven, three thousand twelve, six thousand forty, and nine thousand three hundred.

Write in figures the following numbers:

12. Twelve and three-tenths.

13. Three hundred four and nine-tenths.

14. Twenty-five, three-tenths, and four hundredths. 15. Four, six-tenths, and seven-thousandths.

16. One million, four-tenths, and three-millionths.

17. Write in figures the numbers eleven hundred eleven, fourteen hundred sixty, twelve hundred thousand sixteen hundred, six million twelve hundred sixteen, and eleven billion eleven hundred eleven.

18. Write in figures twenty million twenty thousand, seventeen million fifty thousand nineteen, one hundred four million six hundred two thousand eleven, and six thousand three hundred seven million two thousand fifty six.

18. The ordinary system of notation was introduced into Europe by the Arabians, and is still called the Arabic system of Notation although it is now known that the Arabians derived their knowledge from the Hindoos.

ROMAN NUMERALS.

19. Besides the Arabic system of notation some use is still made of the cumbrous system employed by the Romans.

The symbols which were used by the Romans, and which are called Roman Numerals, are the following:

I for 1, V for 5, X for 10, L for 50, C for 100, D for 500, M for 1000.

A horizontal line over any numeral increases its value one thousand fold: thus V stands for 5000, X for 10000, etc.

Roman numerals are arranged in order of magnitude from left to right, and are repeated as often as may be necessary.

Thus, 2 is represented by II, 30 by XXX, 233 by CCXXXIII, and 1887 by MDCCCLXXXVII.

20. To avoid some of the troublesome repetitions which are common to the Roman system of notation, a numeral is in certain cases placed before another of greater value to denote that the value of the larger is to be diminished by the amount of the smaller.

Thus, IV denotes one less than five, that is, 4; IX denotes one less than ten, that is, 9; XL denotes ten less than fifty, that is, 40; and XC denotes ten less than one hundred, that is, 90; so also, CCXC denotes 290.

21. The symbols CIO, CCOO, CCCIDOO, etc., were anciently employed to denote respectively 1000, 10,000, 100,000 etc.; also IƆ, IƆƆ, IƆƆƆ, etc., to denote respectively 500, 5000, 50,000, etc. In fact, M and D are only modified forms of CIO and I respectively.

22. Roman numerals were used only to register numbers, and were never employed in making numerical calculations. The Romans made their calculations by means of counters and a mechanical apparatus called an Abacus. The counters used were often pebbles (Latin, calculus), whence our word calculation.

EXAMPLES II.

1. Express all the numbers from 1 to 20 by means of Roman numerals.

2. Express by means of Roman numerals the numbers 20, 30, 40, 50, 60, 70, 80, 90, 200, 400, 600, 800, and 900.

3. Express by means of Roman numerals the numbers 39, 49, 59, 69, 79, 89, 99, 96, 444, 1294, and 1889.

4. Write the numbers LVIII, XXXIX, XLIV, XCIV, XCIX, CXCIX, and MMDCCXCIX, in the Arabic notation.

CHAPTER II.

ADDITION-SUBTRACTION

DIVISION.

ADDITION.

MULTIPLICATION

23. THE process of finding a single number which contains as many units as there are in two or more given numbers taken together is called Addition; and this single number is called the Sum.

The sum of the numbers of the units in two or more groups would therefore be found by forming a single group containing them all, and then counting the number of the units in this single group.

24. The following fundamental truth is evident :

The number of the things in any group will always be found to be the same in whatever order they may be counted.

From this it follows that the sum of the numbers of the things in any two groups will be found by first counting all the things in the first group and then proceeding to the second; that is, by increasing the number in the first group by as many units as there are in the second. The same sum will also be found by increasing the number in the second group by as many units as there are in the first.

Thus, the sum of 3 and 5 is found by counting five onwards from three, namely four, five, six, seven, eight; or by counting three onwards from five, namely six, seven,