56. The simplest test of the correctness of an addition is to add a second time, beginning at the top, instead of the bottom of the columns. Another test is found in adding two columns at once. .61803 .43429 .47712 .62138 2.15082 To illustrate these checks or tests, take the example in the margin. Adding from the bottom, the computer says to himself, 8, 10, 19, 22; 5, 6, 8; 1, 8, 12, 20; 4, 11, 14, 15; and 7, 11, 15, 21. Next, adding from the top, he begins, 3, 12, 14, 22; 4, 5, 8; 8, 12, 19, 20; 3, 6, 13, 15; 7, 11, 15, 21. But in the two-column mode, he begins with the tens, and says to himself, 38, 48, 50, 79, 82, which is written. down. Then he continues, 21, 91, 98, 128, 132, 142, 150; the 50 is written, and he goes on, 7, 11, 15, 21. Many bookkeepers and merchants strongly recommend this addition of two columns at once, as the most expeditious and the least liable to error.* 57. Add the following by double columns, and test by adding with single columns: * Many computers find it convenient, when adding, to write on a piece of waste paper the full sum of each column or double column. Thus, in the example above, they would write, Then they copy the last number and the right-hand figure (or figures) of the others in succession upward, and thus obtain 2.15082, as above. An expert computer, when adding, catches at a glance the couples which make ten, and considers them as one number. Thus, he would, when adding the above by single columns, say to himself, 10, 19, 22; 5, 6, 8; 8, 12, 20; 4, 14, 15; 11, 21. CHAPTER IV. SUBTRACTION. 58. SUBTRACTION means taking away. The sign (-) is read minus, and means that the number before which it is placed is to be subtracted. For example, 8-5=3 is read, 8 minus 5 equals 3; 17-8=9. 59. The number to be subtracted is called the subtrahend; the number from which it is taken is the minuend; the number left is the difference, or remainder. 60. Count fifty backward. Name the even numbers from fifty down to naught. Name the odd numbers from 51 down to 1. 61. Begin 60, 57, and name every third number down to naught. Do the same, beginning 61, 58, and beginning 59, 56. 62. Subtract 4 from every number between 8 and 14; between 38 and 44. Subtract 5 from every number between 5 and 15; between 85 and 95. 63. Subtract 6 from each number between 6 and 16; between 46 and 56. Subtract 7 from each number between 17 and 27. 64. Subtract 8 from each number between 18 and 28; 9 from each number between 59 and 69. изве 65. A number with the sign minus before it is called a minus number. When the subtrahend is larger than the minuend, the remainder obtained by subtracting the minuend from the subtrahend is a minus number. Such results are common in Algebra, but are avoided in Arithmetic. The meaning of a minus number is generally manifest; as in the example, The thermometer was at 17° and fell 25°, how high was it then? This would be written, 17° 25° -8°, and shows that the mercury fell to 8° below zero. 66. An expression containing the sign of equality (=) is called an equation. It is like a balanced scale-beam : the plus numbers represent weights; the minus numbers represent balloons lifting up the beam of the scale. The one rule in working with equations is, Keep the balance true. In other words, do to each side whatever you do to either. 5+3-17. 67. When numbers are connected by the signs + or −, part of them are also sometimes joined by parentheses, brackets, or bars; as, 7+(5-8)=4; 7-(5-8)=10; 7-(8-5)= 4. 7+5-8=4; 7-5-8=10; 7 8 = 4. The operations on the joined numbers must be performed first, and the result treated as a single number, plus or minus. The sign outside the parenthesis must also be prefixed; and thus arise four cases of a double sign: ++, +−, − +, and and − + is taking The++ is equivalent to a single +; the + are each equivalent to a single ; while the -away a minus; it is the removal of a balloon, which is equivalent to adding the weight which the balloon was lifting; that is, it is equivalent to +. 68. The parenthesis, therefore, may be, according to the signs, useless or needful; it may or may not affect the result, and must be handled carefully. To illustrate : 8-35 (8-3)+5=10; but 8-(3+5)=0. 12 (63) 12-3=9. = 70. The following questions will illustrate the meaning of minus numbers: Starting 90 miles south of Chicago, I go 50 miles due north; and the next day, 80 miles still north. How far from Chicago am I now? With only 67 dollars I undertake to pay three bills, of $47, of $13, and of $11. Can I pay the bills? How much shall I lack? 71. If we add the same number to a minuend and to a subtrahend, we do not alter the remainder. For example, 27-15= 12; and if we add any number whatever, both |