56. A simple test of the correctness of an addition is to add a second time, beginning at the top instead of the bottom of the columns, or to add two columns at once.

It is of great advantage to educate the eye to take in at a glance digits enough to make 10 or more, and then these sums can be added instead of the separate digits.

61803

To illustrate, take the example in the margin. Adding from the bottom, the computer says to himself (seeing 8+2=10, and 9+3=12), 10, 12, 22; 43429 8, 8, 12, 20; 11, 415; 11, 10, 21. 47712 In the two-column mode he says, 50, 32, 82;

62138

215082

and writes down the 82. Then he continues, 98, 42, 150, and writes the 50 at the left of 82; then he proceeds, 11, 10, 21, and writes the 21.

Many bookkeepers and merchants strongly recommend the addition of two columns at once, as the most expeditious and the least liable to error.

Many computers begin at the bottom of the right-hand column in adding, and write on a piece of waste-paper the full sum of each column or double column; then they begin at the top of the left-hand column, and add each column or double column, also writing the full sum; finally, they add the sums obtained in the first addition, and the sums obtained in the second addition, and compare the results. Thus, in the example above,

57. Add the following by double columns, and test by

adding with single columns:

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, the minuend; the resulting number is called the difference or remainder.

60. Count 50 backward.

Name the even numbers from

50 down to 0. Name the odd numbers from 51 down to 1.

61. Subtract by threes: beginning 60, 57; beginning 61, 58; 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.

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 may be represented by weights, the minus numbers by 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 one side whatever you do to the other.

5+3-17.

67. When numbers are connected by the signs + or −, part of them are also sometimes joined by parentheses,

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

The++ is equivalent to a single +; the+- and the − + are each equivalent to a single ; while the means the taking away of a minus; it is equivalent to the removal of a balloon, and this is the same as adding the weight which the balloon was lifting; that is, 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: 8-3+5=(8-3)+5=10; but 8-(3+5)=0.

12-(6-3)=12-3=9.
12-(3-6)=12-(-3)=15.

69. Write the second members to the following equa

11

tar

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, 27-1512; and if we add any number whatever, both