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, CCIO, CCCIOSO, etc., were anciently employed to denote respectively 1000, 10,000, 100,000 etc.; also 10, 100, 1000, 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. 27 97 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, eight. In the first case we are said to add 5 to 3, and in the second case we are said to add 3 to 5; but the results must be the same. 25. Addition is indicated by the sign +, which is read 'plus.' Thus, 5 + 4 is read five plus four, and denotes that 5 is to be increased by 4, that is, that 4 is to be added to 5; also, 5 + 4 + 3 denotes that 4 is to be added to 5, and then 3 added to the result. 26. The sign, which is read 'equals' or 'is equal to,' is used to denote the equality of two numbers. Thus, 5+ 4 = 9 denotes that the sum of 5 and 4 is 9. 27. When children first begin to add they make use of their fingers, but all counting on the fingers, or with any other real objects, should be discontinued as soon as possible, and the results of adding numbers not greater than nine should be given instantaneously. Tables of the results of the addition of any two numbers each not greater than 10 might at first be made by the pupil, arranged in lines; as for example, 8 and 1 are 9, 8 and 2 are 10, 8 and 3 are 11, etc. EXAMPLES III. Oral Exercises. These examples should be practised until great rapidity is attained. 1. Add 1 and 9, 3 and 8, 2 and 6, 4 and 7, 6 and 3, 4 and 4. 2. Add 7 and 8, 7 and 6, 3 and 9, 5 and 4, 3 and 5, 9 and 8. 3. Add 4 and 3, 9 and 9, 8 and 8, 6 and 9, 7 and 2, 3 and 3. 4. Add 5 and 9, 9 and 4, 6 and 8, 5 and 7, 2 and 9, 8 and 5. 5. Add 7 and 7, 5 and 5, 6 and 6, 8 and 4, 6 and 4, 9 and 7. 6. Add 8 to 15, to 25, to 35, to 45, to 65, and to 95. 7. Add 13 and 7, 23 and 7, 43 and 7, 63 and 7, 83 and 7, 93 and 7. 8. Add 9 to 17, to 27, to 57, to 67, to 87, and to 97. 9. Begin with 7 and add 2 again and again up to 27. Do not say 7 and 2 are 9 and 2 are 11 and 2 are 13, etc., but state results; thus, 7, 9, 11, 13, etc. 10. Begin with 2 and add 3 again and again up to 35. 11. Begin with 85 and add 4 again and again up to 101. 12. Begin with 50 and keep on adding sevens until the sum exceeds 100. 13. Begin with 15 and keep on adding nines until the sum exceeds 100. 14. Add the following numbers in order, first beginning at the right and then at the left: (1) 2, 7, 4, 0, 6, 9, 5, 2, 6, 5, 9, 3, 4, 8. State results only; thus, 2, 9, 13, 13, 19, 28, etc. (2) 7, 9, 5, 4, 0, 8, 6, 7, 3, 5, 9, 8, 2, 6. 28. The sum of any two numbers may be found by counting onwards from the first as many units as there are in the second, but this method would obviously be very troublesome except when the second number is very small. |