lionths, hundred millionths, billionths, &c.; which are placed in divisions more and more removed from the right of the separatrix or comma. 24. The parts of an unit, which have been described, are called decimals. 25. The manner of expressing them is the same, as for whole numbers. After having expressed the figures, which are to the left of the separatrix, the decimals, (which are written in the same manner as whole numbers,) are expressed; but at the end, the name of the decimal unit of the last kind is added. Thus, to express the number 34,572, we say thirty four units and five hundred and seventy two thousandths. If the number were leagues, it would be read thirty-four leagues and five hundred and seventy-two thousandths of a league. The reason for this is easily perceived, if we observe, that in the number 34,572, the figure 5 can be indifferently expressed, either by calling it five tenths, or five hundred thousandths; since the tenth (22) has the value of 10 hundredths, and the hundredth (23) the value of 10 thousandths, the tenth will contain ten times the 10 thousandth, or 100 thousandths; five tenths then have the value of 500 thousandths. For the same reason, the figure may be expressed by saying seventy thousandths; sincer (23) each hundredth has the value of 10 thousandths. 26. The kind of units of which the last figure is com-. posed, may be easily determined by counting each figure successively, from the separatrix towards the right, by the following names; tenths, hundredths, thousandths, ten thousandths, &c. 27. If in the sum there are no entire units, but only parts of an unite, a cipher is put in the place of units; thus to express 125 thousandths, we write 0,125. If 25 thousandths are to be expressed, we write 0,025, putting a cipher between the separatrix and the other figures, as much, to denote a deficiency of tenths, as to give to the following parts their true value. For the same reason, to express 6 thousandths, we write 0,006. 28. We will now examine the changes, that may be made in the value of numbers by changing the place of the separatrix. Since the separatrix determines the places of units and of all other figures, whose value de: pends upon their distance from it; if the separatrix is advanced one, two, or three places towards the left, the number is rendered 10,100,1000, &c. times smaller; and on the contrary, if the separatrix is removed one, two, or three places towards the right, it becomes 10,100, or 1000 times greater. Thus, having the sum 4327,5264, if we advance the separatrix towards the left and write 432,75264, it is evident, that the thousands of the first number have become the hundreds of the second or new.. number; the hundreds are the tens, and the tens are the units; the units are the tenths, the tenths are the hundredths, and so on. Each part then has become ten times smaller by the change. On the contrary, by removing the separatrix one place towards the right, the sum is expressed 48275,264; and the thousands of the first number are changed into tens of thousands, the hundreds into thousands, the tens into hundreds, the units into tens, the tenths into units, and so on. And this new number has become ten times greater than the first. 29. This reasoning shows, that by removing the separatrix two, or three places, towards the left, the number is made 100, or 1000 times smaller; on the contrary, 100 or 1000 times greater by removing it two or three places towards the right. 30. The last observation, which we shall make upon the notation of decimals, is, that their value is not altered by adding any number of ciphers after the last decimal figure. Thus 43,25 is the same as 43,250, or as 43,2500, or 43,25000, &c. Because each hundred has the value of 10 thousandths or 100 ten-thousandths, &c.; and the twenty-five hundredths have the value of 250 thousandths, or 2500 ten-thousandths; and in short, it is the same thing as when we say one hundred cents instead of one dollar. Let the pupil express the following numbers by figures. 1. Seventy-nine. 2. One hundred and one. 3. Five hundred and twenty-four. 4. Eight hundred and seven. 5. Five tenths. 6. One and eight tenths. 8. Seven hundred whole numbers and seven hun dredths. 9. Nine thousand seven hundred and ten. 10. Ten thousand and four. 11. Fifteen thousand, and fifteen thousandths. 12. One million and one. 13. One hundred millions and one hundred thousand. 14. Five millions, and five hundredths. 15. Two billions, two millions, and two. ARITHMETICAL OPERATIONS. 1. Addition, Subtraction, Multiplication, and Division, are called the four fundamental operations of Arithmetic. All questions, which depend for their solution upon numbers, are reduced to the practice of some of these rules. It is therefore important they should be thoroughly and familiarly understood. 32. The design of Arithmetic, as has already been explained, is to furnish the most ready method for calculating numbers. The means used for this purpose, consists in reducing the calculation of complicated numbers, to that of numbers more simple, or expressed by the smallest possible number of figures. The following treatise has this object. ADDITION OF WHOLE NUMBERS AND DECIMAL PARTS. 33. Addition teaches to express the total value of many numbers by a single number. When the numbers, which are proposed to be added, are expressed by one figure, there is no need of a rule; but when they consist of many figures, their total value is found by observing the following method. Write under each other all the proposed numbers, in such order, that all the unit figures may be in the same vertical column. In like manner place all the tens in the same vertical column; and also the hundreds, thousands, &c. in their respective columns; and draw a line under the whole. Then add all the numbers which are in the lumn of units. If the sum does not exceed 9, write it underneath. If it exceed nine, it belongs to the tens. Write underneath the excess of simple units over tens, and count these tens as so many units to be added with the units of the next column. Proceed with the second column as with the first, and continue in the same manner from column to column to the last, under which write the whole amount that is found in it. This rule is explained by the following examples. EXAMPLE I. If 54925 is to be added to 2023; the numbers are written as below. 54925 56948 sum. To add these numbers, begin with the units by saying 5 and 3 make 8, which write under the unit column. Then pass to the column of tens and say 2 and 2 are 4; and write 4 under that column. In like manner, at the column of hundreds, 9 and cipher make 9, which is put under its column. And at the column of thousands, 4 and 2 make 6, which is put under its column. And lastly in the column for tens of thousands, we say 5 and cipher make 5, and write it under the same column. The number 56948, found by this operation, is the sum of the two proposed numbers; since it contains the units, tens, hundreds, thousands, and tens of thousands of both successively collected together. EXAMPLE II. Ruired the sum of the four following numbers; 6903, 7854, 953, 7327. They are to be thus written. 6903 7854 953 7327 23037 sunt. Commencing, as in the first Example, at the right hand column, we say 3 and 4 make 7, and 3 are 10, and 7 make 17; 7 units are written under the first column, and the ten is retained to be joined as an unit to the next column, which is the column of tens. Passing to the second column, we say 1, which was retained from the first column, and cipher make 1, and 5 make 6, and 5 are 11, and 2 make 13; 3 is written under this column; and for the ten, an unit is retained and added to the following column; where we say 1 and 9 inake 10, and 8 make 18, and 9 are 27, and 3 make 30; a cipher is written under this column, and for the three tens of it, three units are retained, and added to the next column, by saying as before, 3 and 6 make 9, and 7 are 16, and 7 make 23; 3 is written under this column, and as there is not another, to which the two remaining tens might be added, they are written in the place of such column. The number 23037 is the sum of the four proposed numbers. 34. When decimals are to be added; since they are reckoned like other numbers, by tens, in passing from right to left, the same rule is observed as for whole numbers; units of the same order being placed in the same column. Thus, if it is proposed to add these three numbers, 72,957, 12,8, 124,03; we write them in this man ner. 72,957 124,03 209,787 sum. This sum is obtained by following the same method as in the preceding cases. EXAMPLES FOR PRACTICE. 1. What is the sum total of the following numbers; 845674, 98736, 456752, and 7654 ? Ans. 903816. 2. What is the sum of ,014+,9816+,32+,159144 72918+,0047 ? Ans. 2,20852. 3. I lent a friend 942 dollars, at one time; at another, 741; at another 91 dollars; what is the whole suin lent ? Ans. 1774 dollars. 4. What is the sum of one millionth part of an unit added to 9,999999? Ans. 10. |