6903 7854 953 7327 23037 sum. And commencing as before at the top of the right hand column, I say, 3 and 4 are 7, and 3 are 10, and 7 are 17; I write the 7 units under the first column, and the ten I retain to add as a unit to the numbers in the next column, the units of which are also tens. Proceeding to the second column I say 1 which I carry and 0 are 1, and 5 are 6, and 5 are 11, and 2 are 13; I write 3 under the same column, and I carry for the ten a unit to the next column, saying: 1 and 9 are 10, and 8 are 18, and 9 are 27, and 3 are 30; I place 0 under that column, and I carry for the three tens, three units which I add to the next column, saying: 3 and 6 are 9, and 7 are 16, and 7 are 23; I write 3 under that column, and as there are no more columns, I advance one place farther to the left the two tens which would belong to the next column if there were another. 34. If there be decimal parts, since these are counted like the other numbers, by tens, as we advance from the right hand towards the left, the rule for adding them is absolutely the same, observing always to place the units of the same order under each other in the same column, in which case the commas or decimal points will also be under each other. Thus, if we propose to add the three numbers 72,957... 12,8...124,03, I shall write 72,957 12,8 124,03 209,787 And in following the above rule, I shall have 209,787 for the sum. EXAMPLES. 1. Required the sum of twenty-nine; six hundred and fifty-eight; two hundred and thirty-four; and seven thousand seven hundred and eighty-five. six. Answer. The sum is eight thousand, seven hundred and 2. Required the sum of two and nine tenths; sixty-five, and eight tenths; twenty-three, and four tenths; seven hundred and seventy-eight, and five tenths. Answer. Eight hundred and seventy, and six tenths. 3. Required the sum of the four numbers six, and nine hundred and three thousandths; seven, and eight hundred and fifty-four thousandths; nine hundred and fifty-three thousandths; seven, and three hundred aud twenty-seven thousandths. Answer. Twenty-three, and thirty-seven thousandths. 4. Required the sum of the three numbers, seven, and two thousand nine hundred and fifty-seven ten-thousandths; one, and twenty-eight hundredths; twelve, and four hundred and three thousandths. Answer. Twenty, and nine thousand seven hundred and eighty-seven ten-thousandths. 5. Required the sum of the three following numbers, three hundred, and three ten-thousandths; seventy, and eighty. four thousandths; nine, and three tenths. Answer. Three hundred and seventy-nine, and three thousand eight hundred and forty-three ten-thousandths. 6. Required the sum of the answers to the five preceding examples. Answer. Ten thousand. As there are certain signs used by mathematicians to denote the four fundamental operations of arithmetic, we shall consider each of these together with the rules to which it applies. There is also an auxiliary sign, viz. two equal and horizontal straight lines thus, which signifies that the numbers be tween which it is placed are equal to one another. For example 4=4 is read four equal to four. A straight cross +, which is called plus, or more, is the sign of addition, and signifies that the numbers between which it is placed are to be added together. Thus 4+3 signifies that 4 is to be added to 3, and is read four plus three, also 4+3=7, is read four plus three equal to seven, and the expression 9+5+8=22, is read nine plus five plus eight equal to twenty-two, in adding the numbers 9, 5, and 8 together. Note. It is immaterial in what order they are added as 9+5+8 is the same as 9+8+5 or 8+5+9, etc. . EXAMPLES. 1. 19+9+9+9+9+9+9+9+9+9=100. It will be seen in this example, that (because 9+1=10) in adding 9 to any significant figure in a number, that figure will be diminished and the next figure increased, each by a unit: Thus 19+9=28; 28+9=37; 37+9=46; 46+9== 55; 55+9=64, etc. where in each new addition the unit figure is one less and the figure in the place of tens one greater. 2. 27+36+107+9=179. 3. 93,7+54,36+0,007=148,067 When there are only two or three numbers, the addition may easily be performed without placing the numbers under each other, by commencing with the lowest units and being careful to add only those of the same order to cach other, carrying for the tens as usual. 4. 75+62,3+58,9=196,2. 5. 556+829,43+66,078+3,008+5,833=1460,349. 6. 725+596 +274+592+673+889+438+299=4486. In order to prove the work, when the addition is somewhat long, as in the present example, it is usual to add both ways; that is, after having placed the numbers, to add from the top downward and from the bottom upward, if the sum be the same both ways, the work can scarcely be wrong, 3 ANOTHER METHOD OF PROVING ADDITION. This proof of addition is made in adding anew, by parts, but in beginning at the left, the sums which we have already added. We take the sum of the first column, from that part of the whole sum which answers to it: we write the remainder underneath, which we consider as so many tens, to join it to the following figure of that sum, and from the whole we take the sum of the column above it, we continue thus to the column of units, the sum of which being subtracted there should be no remainder. Thus, having found that the four numbers 6903 7854 953 7327 give the sum 23037 3110 To verify this result, I add the same numbers in commencing at the left, and I say, 6 and 7 are 13, and 7 are 20, this taken from 23, leaves 3 or three tens, which with the cipher following makes 30. I pass to the next column, and I say, 3 and 8 are 17, and 9 are 26, and 3 are 29, which I take from 30; there remains 1 or ten, which being added to the next figure makes 13. I add all the figures in the third column saying, 5 and 5 are 10, and 2 are 12, this taken from 13, leaves 1 or ten, which added to the figure 7, makes 17. I add likewise all the figures in the column of units, saying, S and 4 are 7, and 3 are 10, and 7 are 17, which taken from 17 leaves nothing; whence I conclude that the first operation is correct. We have reason to conclude that the first operation was correctly performed, since after this proof there remains nothing, because having taken away successively all the thousands, all the hundreds, all the tens, and all the units, of which we had composed the sum, there should nothing remain. 7. 7+8+6+5+9+8+7+8+2+4+6+8+9+7+3+ €+9+8+2+6=128. In adding any question, whenever two figures succeed each other, the sum of which is ten, it will facilitate the operation to comprehend them both together; thus, in the present example I say, 7 and 8 are 15, and 6 are 21, and 5 are 26, and 9 are 35, and 8 are 43, and 7 are 50, and 10 (comprehending the 8+2) are 60, and 10 (comprehending 4+6) are 70, and 8 are 78, and 9 are 87, and 10 (comprehending 7+3) are 97, and 6 are 103, and 9 are 112, and 10 (comprehending 8+2) are 122, and 6 are 128. OF THE SUBTRACTION OF WHOLE NUMBERS AND DECIMAL PARTS. 35. Subtraction is the operation by which we take one number from another. The result of this operation is called remainder, excess, or difference. To perform this operation, we write the number which we would take away under the other, in the same manner as in addition; and having drawn a line underneath the whole, we take away, in going from the right hand towards the left, each lower figure from its correspondent upper one; that is to say, the units from units, the tens from tens, etc.; we write each remainder underneath, and a cipher when nothing remains. When the lower figure is greater than the corresponding upper one, we add ten units to the upper one in borrowing by the thought a unit from the next figure on the left, which should be regarded as one unit less in the next operation. Instead of diminishing by a unit the figure from which we have borrowed, we can, if we please, leave it as it is, and increase by a unit that which is to be taken from it: the remainder will always be the same. |