DEFINITIONS. 70. Subtraction is the process of finding the difference between two numbers. 71. The Minuend is the greater of two numbers whose difference is to be found. 72. The Subtrahend is the smaller of two numbers whose difference is to be found. 73. The Difference or Remainder is the result obtained by subtraction. 74. The Process of Subtraction consists in comparing two numbers, and resolving the greater into two parts, one of which is equal to the less and the other to the difference of the numbers. 75. The Sign of Subtraction is, and is called minus. When placed between two numbers it indicates that their difference is to be found; thus, 14-6 is read, 14 minus 6, and means that the difference between 14 and 6 is to be found. 76. Parentheses () denote that the numbers inclosed between them are to be considered as one number. 77. A Vinculum affects numbers in the same manner as parentheses. Thus, 19 + (13 — 5), or 19 + 13 5 signifies that the difference between 13 and 5 is to be added to 19. 78. PRINCIPLES.-I. Only like numbers and units of the same order can be subtracted. II. The minuend is the sum of the subtrahend and difference, or the minuend is the whole of which the subtrahend and difference are the parts. III. An equal increase or decrease of the minuend and subtrahend does not change the difference. REVIEW AND TEST QUESTIONS. 79. 1. Define the process of subtraction. Illustrate each step by an example. 2. Explain how subtraction should be performed when an order in the subtrahend is greater than the corresponding order in the minuend. Illustrate by an example. 3. Indicate the difference between the subtraction of numbers and the subtraction of objects. 4. When is the result in subtraction a remainder, and when a difference? 5. Show that so far as the process with numbers is concerned the result is always a difference. 6. Prepare four original examples under each of the following problems and explain the method of solution: PROB. I. Given the whole and one of the parts to find the other part. PROB. II.-Given the sum of four numbers and three of them to find the fourth. 7. Construct a Subtraction Table. 8. Define counting by subtraction. 9. Show that counting by addition, when we add a number larger than one, necessarily involves counting by subtraction. 10. What is the difference between the meaning of denomination and orders of units? 11. State Principle III and illustrate its meaning by an example. 12. Show that the difference between 63 and 9 is the same as the difference between (63 + 10) and (9 + 10). 13. Show that 28 can be subtracted from 92, without analyzing the minuend as in (64), by adding 10 to each number. 14. What must be added to each number, to subtract 275 from 829 without analyzing the minuend as in (64)? 15. What is meant by borrowing and carrying in subtraction? 80. STEP I.-To find the sum of 2 times, 3 times, and so on to 12 times, any number expressed by one figure. 1. The sum of any number of times a given number is found by addition. Thus, 3 times 9 or three nines equals 9+9+9= 27. 2. The sign x stands for the word times. Thus, 4 x 5 is read either 4 times 5 which means 5 + 5 + 5 + 5, or 5 times 4 which means 4 + 4 + 4 + 4 + 4. 3. Be particular to notice that the sum of 2 threes and 3 twos is the same, and so with 3 fours and 4 threes, 4 fives and 5 fours, and so on. This may be shown with marks on your slate, thus: 4. The sum of any number of times a given number is called a Product. Thus, 3 times 77+7+7=21, hence 21 is the product of 3 and 7; and 3 and 7 are called Factors of 21. 5. The Multiplication Table consists of the products of numbers from 2 to 12 inclusive. These products are found by addition, and then memorized so that they can be given at sight of their factors. EXERCISES FOR PRACTICE. 81. Copy on your slate and find by addition the product of each of the following examples, and write it in place of the 6. Find by addition each of the products in the following Multiplication Table. 82. STEP II. To memorize the Multiplication Table. Pursue the following course: 1. Write on your slate in two sets and in irregular order thus: (1.) 2 times 2 are 2 times 2 are, 3 times 2 are, and so on, up to 12 times 2 are, (2.) 7 times 2 are 2. Find, by adding, the product of each example and write it after the word "are." 3. Read very carefully the first set several times, then erase the products and write them again from memory as you read the example. Continue to erase and write the products in this way until they are firmly fixed in your memory. In every case where you feel the least uncertainty about the correctness of a product, find it again by addition. 4. Practice on the second set in the same way. When you have its products fixed in your memory, then practice on writing at once the products of both sets. To vary the exercises, erase all the products, and give them orally as you repeat the example mentally; thus, Two times two are four; Five times two are ten; three times two are six, and so on. 5. Write on your slate a series of twos, and write under them in irregular order the numbers from 2 to 12 inclusive; thus, 2 2 2 2 2 2 2 2 2 2 2 Write the product under each example as you repeat mentally the number of twos. Thus, as you say mentally two twos, write 4 under the first example; as you say, six twos, write 12 under the second example, and so on with each of the other examples. Erase the products and write them again and |