4th. A letter with a line over it represents a number one thousand times as great as the same letter without the line; thus X stands for ten, but X stands for one thousand times ten, i. e. ten thousand; M stands for one thousand, but M for one thousand times one thousand. TABLE OF ROMAN NUMERALS. EXERCISES IN ROMAN NOTATION. 36. Express the following numbers by letters: 1. Twelve. 2. Eighteen. Ans. XII. Ans. XVIII. 3. Twenty-nine. 4. Ninety-nine. 5. Two hundred and eighty-four. 6. One thousand four hundred and forty-six. 7. One thousand six hundred and forty-four. 8. The present year, A. D. NOTE. The Roman notation is very inconvenient for Arithmetical oper ations, and the Roman Numerals are now seldom used, except for number ing the pages of a preface, the divisions of a discourse, and the sections, chapters, and other divisions of a book. 35. What is the fourth principle in Roman Notation? 36. Are Roman numerals much used in arithmetical operations? Why? For what are they used? 37. Besides the Arabic and the Roman figures, there are various marks used to indicate that certain operations are to be performed, such, c. g., as the sign of addition,+; the sign of subtraction, -; etc. These signs will be given, and their uses explained when their aid is needed. ADDITION. 38. ADDITION is the process of putting together two or more numbers of the same kind, to find their sum or amount. The sum or amount of two or more numbers is a number which contains the same number of units as the two or more numbers put together; thus, 7 is the sum of 3 and 4, because there are just as many units in 7 as in 3 and 4 put together; for a like reason 11 days is the sum of 2 days, 4 days, and 5 days. Ex. 1. James has 4 marbles, John has 5, and Henry has 3; how many marbles have they all? To solve this example, add the numbers 4, 5, and 3: thus, 4 and 5 are 9, and 3 are 12; therefore James, John, and Henry have 12 marbles, Ans. 2. How many are 3 and 6? 6 and 3? 2 and 5 and 7? How many are 5 and 6? 4 and 7? 9 and 3 and 8 ? How many are 3 and 6 and 7 and 8? 8 and 9 and 7 and 4? 39. A SIGN is a mark which indicates an operation to be performed, or which is used to shorten some expression. 40. The sign of dollars is written thus, $; e. g. $2 represents two dollars; $10, ten dollars, etc. 41. The sign of equality,=, signifies that the quantities between which it stands are equal to each other; thus, $1=100 cents, i. e. one dollar equals one hundred cents. 37. What characters are used in Arithmetic be ides the Arabic and Roman figures? For what? 38. What is Addition? Sum or amount? 39. A sign? 40. Make the sign of dollars on the blackboard. 41. Make the sign of equality? What does it mean? 42. The sign of addition, +, called plus, denotes that the quantities between which it stands are to be added together; thus, 325, i. e. three plus two equals five, or three and two are five. 43. Three dots, thus, .., are the symbol for therefore, hence, or consequently; thus, 2+3= 5, and 3 + 2 = 5, .. 2+3=3+2, i. e. therefore the sum of 2 and 3 is equal to the sum of 3 and 2. Ex. 3. William paid $4 for a pair of skates, $3 for a sled, and $1 for a knife; what did he pay for all? $4+$3+$1 = $8, Ans. 4. What is the sum of $6+$3? $5+$2+$8? 5. What is the sum of 4+6+2+3? 3+5+8+2? 44. To add when the numbers are large and the amount of each column is less than 10. 6. A manufacturer sold 125 yards of cloth to one merchant, 342 to another, and 231 to another; how many yards did he sell in all ? Ans. 698. OPERATION. 125 342 Sum, 698 Having arranged the numbers so that units. stand under units, tens under tens, etc., add the units; thus, 1 and 2 are 3, and 5 are 8, and set the result under the column of units. Then add the tens; thus, 3 and 4 are 7, and 2 are 9, set down the result, and so proceed till all the columns are added. 42. Make the sign of addition. 43. Sign for therefore. 44. How are numbers arranged for addition? Which column is added first? Its sum, where placed? 15. What is the sum of 1243, 2112, and 1313? Ans. 4668. 16. What is the sum of 2013, 1421, 2132, and 1231? 17. A gentleman paid $125 for a horse, $231 for a chaise, and $32 for a harness; what did he pay for all? Ans. $388. 45. To add when the amount of any column is 10 or more. 18. Add together 27, 93, and 145. OPERATION. 27 93 145 Ans. 265 Ans. 265. Having arranged the numbers, add the column of units; thus, 5 and 3 are 8, and 7 are 15 units (=1 ten and 5 units). The 5 units are placed under the column of units, and the 1 ten is added to the column of tens; thus, 1 and 4 are 5, and 9 are 14, and 2 are 16 tens (=1 hundred and 6 tens). The 6 tens are set under the tens, and the 1 hundred is added to the 1 hundred in the third column, making 2 hundreds to be set under the third column. 31. Add 3467, 82, 946, 13845, and 426. Ans. 18766. 46. The examples already given embrace all the principles in addition. Hence, to add numbers, RULE. Write the numbers in order, units under units, tens under tens, etc. Draw a line beneath, add together the figures in the units' column, and, if the sum be less than ten, place it under that column; but, if the sum be ten or more, write the units as before, and add the tens to the next column. Thus proceed till all the columns are added. 47. PROOF. The usual mode of proof is to begin at the top and add downward. If the work is right, the two sums will be alike. NOTE 1. By this process, we combine the figures differently, and hence shall probably detect any mistake which may have been made in adding upward. ILLUSTRATION. Ex. 35. 37684 48297 68746 94852 Sum, 249579 Proof, 249579 In adding upward we say, 2 and 6 are 8, and 7 are 15, and 4 are 19, etc.; but in adding downward, we say, 4 and 7 are 11, and 6 are 17, and 2 are 19, etc.; thus obtaining the same result, but by different combinations. If we do not obtain the same result by the two methods, one operation or the other is wrong, perhaps both, and the work must be carefully performed again. NOTE 2. In adding it is not desirable to name the figures that we add; thus, in example 35, instead of saying 2 and 6 are 8, and 7 are 15, and 4 are 19, it is shorter, and therefore better, to say 2, 8, 15, 19; setting down the 9, say 1, 6, 10, 19, 27, etc. 36. What is the sum of 8432, 42698, 34, 1892, 70068, 5142, and 68742? Ans. 197008. 37. What is the sum of 2468, 13579, 276, and 42? 38. What is the sum of 3406, 872, 6541, 2, and 17? 39. What is the sum of 3910, 4, 876, 27, and 89462? 46. If the amount of any column is ten or more, where is the right-hand figure of the amount. written? What is done with the left-hand figure? Repeat the rule for Addition. 47. How is Addition proved? Why not add upward a second time? Is it desirable to name the figures as we add them? |