CHAPTER II. ADDITION AND SUBTRACTION. 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. = Thus 2+3=5 2. The sign 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. 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, 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 hundreds, 7 tens, and 9 units, which, according to the 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. Or, added together, to 12 thousands, 12 hundreds, 13 tens, 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 I 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 i 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 I 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 Rule for Addition: Write down the numbers under each other, so that units may stand under units, tens under tens, &c., and draw a line beneath them. Then, beginning with the units, add the columns separately. 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 throughout all the columns, and set down the whole sum of the last or left hand column. Thus: 8976 4368 13344 Adding the units, 8 and 6 are 14. Therefore write down 4 and add 1 to the tens column. Adding the tens, 1 and 6 and 7 are 14. Therefore write down 4 and add 1 to the hundreds column. Adding the hundred, 1 and 3 and 9 are 13. Therefore write down 3 and add 1 to the thousands column. Adding the thousands, 1 and 4 and 8 are 13. N.B. The same rule evidently applies if there are more than two lines of figures to be added together. 6. Test of Correctness.—There are various methods by which the correctness of the process of addition may be tested. If Perhaps the most convenient test is to add the numbers together in the reverse order; that is, to commence with the top line instead of the bottom. the second result be the same as the first, the work may be presumed to be right; for it is highly improbable that the same error will have been made in performing the operation in two different orders. EXERCISE III. (1.) Find the sum of 75234 + 41015 + 19075 + 176. (2.) 85064 +9035 + 72358 + 919. (3.) 1500267 + 45085 + 4652 + 4780400 + 90276 + 89760841. (4.) 40702135+ 67070420 + 670856 + 4230825 + 750642 + 8790845. (5.) 756 +849 + 934 +680 + 720 + 843 + 657689 + 989876498 +8045685 + 807266780. (6.) 16075 + 250763 + 7561 + 830654 + 293106 + 2537104 +316725. (7.) 493742 + 56710607 + 23461 + 400072 + 6811004 + 8999003 + 26501. (8.) 432678902 + 310046734 + 2167005 + 327861 + 293000428. (9.) Add together the following numbers :-Twenty-three thousand three hundred and forty-nine; seven thousand two hundred and seven; three hundred and twentyfive; five millions two hundred and fifty-three; fiftysix billions three hundred and nine millions five hundred and thirty-one thousand six hundred and nine; four thousand and seventeen millions; four thousand and four. 7. SUBTRACTION.-If a less number be taken away from a greater, or, as it is called, subtracted from it, the number left behind is called the difference of the two numbers, or the remainder. The sign (called minus) placed between two numbers indicates that the one before which it stands is to be subtracted from the other. 8. When the individual figures composing the larger number are respectively larger than the corresponding figures of the smaller number, the process is evident. We have only to take the differences of the numbers of units, tens, hundreds, &c., respectively, and the resulting number can be at once written down. Thus, for instance, suppose it be required to find the difference between 9876 and 7653. Write down the numbers one under the other, the units under the units, the tens under the tens, the hundreds under the hundreds, and so on, thus : 9876 2223 number 3 units in the less taken from 6 units {in the larger leaves 2 tens. 6 hundreds 8 hundreds,, 2 hundreds. 7 thousands 9 thousands 2 thousands. Thus, the difference is 2 thousands, 2 hundreds, 2 tens, and 3 units, or, as it is written, according to the rules of our notation, 2223. 9. But suppose that the figures in the less number are not respectively less than the corresponding figure in the other number; we must then proceed somewhat differently. The method we employ depends upon the following self-evident proposition, or Axiom. If two numbers be increased by the same quantity, their difference will not be altered. 10. Suppose that it be required to subtract 4789 from 5231. Place the numbers, one under the other, as before- 5231 4789 442 9 units in the less cannot be taken from I unit of the greater; add, however, 10 units to the I unit in the upper, and add 10 to the lower number by changing the 8 in the tens place into a nine. The numbers are now 5 thousands, 2 hundreds, 3 tens, and II units; and 4 thousands, 7 hundreds, 9 tens, and 9 units. Now, 9 units from 11 units leave 2 units. Again, tens cannot be taken from 3 tens, but if we increase the 3 in the tens place of the upper number by ten, and the 7 in the hundreds place in the lower by one, we shall be adding the same quantity (a hundred) to each number, since any figure indicates a number 10 times as great as the same figure in a place immediately on its right. Then 9 tens from 13 tens leave 4 tens. Again, 8 hundreds cannot be taken from 2 hundreds, but if we increase the 2 in the hundreds place of the upper number by 10, and the 4 in the thousands place |