22 



THE POPULAR EDUCATOE. 



LESSONS IN ARITHMETIC. II. 



THE ROMAN METHOD OF NOTATION. 

 THE symbols by which the Romans expressed all numbers 

 wore : 



denoting 



one 

 five 

 ten 



C 



D or IQ 



M or CI 



denoting 



a hundred 

 five hundred 

 a thousand 



L fifty 



By combining these symbols according to the following rules all 



numbers can be represented : 



When two symbols are placed together, if the one denoting 

 the less value is on the left of the other, then the less number 

 is to be subtracted from the greater ; if on the right hand, it 

 is to be added to it. Thus IX denotes ten with one subtracted, 

 or nine ; XI denotes eleven ; XL denotes forty ; LX, sixty. If 

 the symbols are of equal value, then they are simply to be 

 added. Thus XX denotes twenty ; CO, two hundred, etc. The 

 value represented by IQ is increased tenfold by every additional 



placed on the right. Thus 5,000 is denoted by IQO, and 

 50,000 by IQQQ. The value of the symbol ClQ becomes in- 

 creased tenfold by the addition of C and Q, one on each side of 

 the line I. Thus 100,000 is denoted by CCClQQQ, 1,000,000 by 

 CCCClQQQQ, and so on. A straight line placed over any one 

 of these symbols increases its value a thousand-fold. Thus 



1 denotes 1,000; V, 5,000; L, 50,000; C, 100,000. 2,000 was 

 usually denoted by ClQClQ, but sometimes by IIClQ, or IIM, 

 or MM. Similarly, 4,000 was denoted by IVClQ, etc. 



The above remarks will sufficiently explain the following 

 Table of Roman Numerals : 



EXEKCISE 3. 



1. Write out the names of all the numbers from one to a 

 hundred, and express them in figures. 



2. Write out the names of the numbers which immediately 

 follow : 



1. One hundred. 



2. One hundred and ninety-nine. 



3. Pour hundred and ninety- nine. 



3. Express, in figures, the numbers named in the preceding 

 example, and those which immediately follow them. 



4. Write the names of the numbers which are next to the 

 following numbers, and express both sets in figures : 



4. Nine thousand nine hundred 



and ninety-nine. 



5. One million. 



1. One million and ninety-nine. 

 -. One million five thousand nine 

 hundred and ninety-nine. 



3. Nine millions nine hundred 

 and ninety - nine thousand 

 nine hundredaud ninety-nine. 



5. Read or express the following numbers in words : 



1. 202 7. 20030208 13. 100010001000 



2. 1001 8. 1010101 14. 3000000000000 



3. 15608 9. 9999999 15. 777666555444 



4. 306042 10. 347125783 



5. 567S914 11. 202021010 



6. 26312478 12. 9090909090 



16. 123456789123 



17. 48484848484848 



18. 10210230430400 



6. Write or express the following numbers in figures : 



1. Four hundred and four. 



2. Three thousand and thirty-two. 



3. Twenty -four thousand and 



eighty-six. 



4. Six hundred and five thousand 



and nineteen. 



5. Eleven thousand eleven hun- 



dred and eleven. 



6. Three hundred and forty-one 

 thousand seven hundred and 

 eighty- two. 



7. One million. 



8. Nine thousand nine hundred 



and ninety-nine millions, 

 nine hundred and ninety- 

 nine thousand nine hundred 

 and ninety-nine. 



9. Write the number which 



follows this last ono in 

 order. 

 10. One trillion and three. 



11. Eighty millions two hundred 

 and three thousand and two. 



12. Two hundred and two millions 



twenty thousand two hun- 

 dred and two. 



13. Twenty thousand millions. 



14. Two hundred thousand and 



twenty millions two thou- 

 sand. 



15. The next number to thirty 



thousand billions nine hun- 

 dred and ninety-nine thou- 

 sand. 



ADDITION. 



1. The process of uniting two or more numbers together, so 

 as to form a single number, is called Addition. The number 

 thus formed is called the sum of the separate numbers. 



2. The sign -f- placed between two numbers indicates that 

 they are to be added together. This symbol is called plus. The 

 sign = placed between two numbers denotes that they are 

 equal. Thus 2+3 = 5, expresses that 2 and 3 added together 

 are equal to 5. 



3. Suppose that it be required to add the two numbers 3452 

 and 4327 together. 



These are respectively 



3 thousands, 4 hundreds, 5 tens, and 2 units, 



4 thousands, 3 hundreds, 2 tens, and 7 units, 



which, added together, are equal to 



7 thousands, 7 hundreds, 7 tens, and 9 units. 



The sum, therefore, of 3452 and 4327 is 7 thousands, 7 hun- 

 dreds, 7 tens, and 9 units, which, according to our system of 

 notation, will be written 7779. 



This is got by putting down the two numbers one under the 

 other, the units under the units, the tens under the tens, and so 

 on ; and then adding up the lower to the upper figure in each 

 place, thus : 



3452 



4327 



7779 



4. In the example we have taken, the sum of the numbers 

 of the thousands amounts only to a number expressed by one 

 figure, namely, 7 ; and similarly for the hundreds, the tens, 

 and units. 



Suppose, however, that we have a case in which this is not 

 so ; for instance, to add 



8976 and 4368. 

 These are respectively equal to 



8 thousands, 9 hundreds, 7 tens, and 6 units, 

 4 thousands, 3 hundreds, 6 tens, and 8 units; 



or, added together, to 



12 thousands, 12 hundreds, 13 tons, and 14 units. 



This, however, is not at present in a form which can be at once 

 written down according to our system of notation. We must, 

 therefore, alter its form. 



Now 14 units are the same as 1 ten and 4 units ; therefore 



13 tens and 14 units are the same as 14 tens and 4 units. 



But 14 tens are the same as 1 hundred and 4 tens ; therefore 

 12 hundreds and 14 tens are the same as 13 hundreds and 

 4 tens. 



But 13 hundreds are the same as 1 thousand and 3 hundreds; 

 therefore 12 thousands and 13 hundreds are the same as 13 

 thousands and 3 hundreds. 



Hence we see that 12 thousands, 12 hundreds, 13 tens, and 



14 units, are the same as 13 thousands, 3 hundreds, 4 tens, and 

 4 units, which, by our notation, is written 13344. 



5. The preceding process will sufficiently explain the following 1 

 Rule far Addition : 



Write down the numbers under each other, so that units may 

 stand under units, tens under tens, etc., and draw a line beneath 

 them. Then, beginning with the units, add tha columns sepa- 

 rately. Whenever the sum of the figures in a column is a number 

 expressed by more than one figure, write down the right-hand 

 figure of such number under the column, and add the other 

 figure or figures into the next column. Proceed in this way 



