GENERAL PRINCIPLES OF THE FUNDAMENTAL RULES PRINCIPLES OF ADDITION. 1. The sum of all the parts equals the whole. 2. The whole diminished by one or more parts equals the sum of the other parts PRINCIPLES OF BUBTRACTION. 1. The Remainder equals the Minuend minus the Subtrahend. 2. The Minuend equals the Subtrahend plus the Remainder. 8. The Subtrahend equals the Minuend minus the Renainder. PRINCIPLES OF MULTIPLICATION. 1. The Product cquals the Multiplicand into the Multiplier. 2. The Multiplicand equals the Product divided by the Multiplier. 3. The Multiplier equals the Product divided by the Multiplicand. PRINCIPLES OF DIVISION. 1. The Quotient equals the Dividend divided bathu Divisor. 2. The Dividend equals the Divisor multiplied vy the Quotient. 8. The Divisor equals the Dividend divided by the Quotient. 4. The Dividend equals the Divisor multiplied by the Quotient plus the Remainder. 8. The Divisor equals the Dividend minus the Remainder, divided by the Quotient. OTHER PRINCIPLES OF DIVISION. 1. Multiplying the Dividend or dividing the Divisor by any num. ber, multiplies the Quotient by that number. 2. Dividing the Dividend or multipiying the Divisor by any number, divides the Quotient by that number. 3. Multiplying or dividing both Dividend and Divisor by the same number, does not change the Quotient. 4. A General Laro.-A change in the Dividend by multiplication or division produces a similar change in the Quotient; but such a change in the Divisor produces an opposite change in the Quotient. NOTE TO TEACHER.—Let the pupils be required to show the reason for the above principles, and give illustrations of them. No demonstrations are given, since it is better for the pupil to learn to depend somewhat upon himself, that he may become, not a mere imitator, but an original thinkcor. INTRODUCTION TO SECONDARY OPERATIONS. MENTAL EXERCISES. 1. What numbers multiplied together will produce 47.69 8? 10? 197 149 16? 18? 20? 247 267 289 309 2. What numbers can be produced out of the numbers 2 and 88 3 and 5? 2, 3, and 5? 3, 4, and 5? 2, 3, 4, and 5? 3. Will the product of any two numbers, each greater than a unit produce 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 81, and 37 ? 4. What may we call a number which is composed by multiplying several numbers together? Ans. A Composite Number. 5. What shall we call numbers that cannot be produced by multiplying several numbers together? Ans. Prime Numbers. 6. Which are prime and which composite numbers in the follow ing list : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15? 7. What may we call the numbers whose product makes a composite number? Ans. Makers of the numbers. 8. If the word Factor means the same as maker, what may we call the makers of a composite number? Ans. Hactors. 9. Form composite numbers out of the factors 3 and 4; 8, 4, and 6; 4, 5, and 6. What are the factors of 12? 15? 18? 20? 217 24? 10. Form a composite number by using 2 twice as a factor; 3 twice as a factor; 2 three times as a factor; 8 four times as a factor. 11. Required one of the tro equal factors of 9; of 16; of 25; of 36 one of the three equal factors of 8; of 27; of 64; of 125. 12. A number composed of two equal factors is called the second poroer of that factor ; of three equal factors, the third poroer, etc. 13. Required the second power of 3; of 4; of 6; of 7; of 8: the third power of 2; of 3; of 4; of 6. 14. One of the troo equal factors of a number is the second root of a numler; one of the three equal factors is the third root, etc. 15. What is the second root of 16? of 257 of 88 ? of 49? What is the third root of 87 of 277 of 647 of 125 ? 18. What would it seem natural to call the process of making composite numbers? Ans. Composition. 17. What would it seem natural to call the process of inding the factors of a number? Ans. Factoring. 18. What are the first four operations of arithmetic called ? Ane. The Fundamental or Primary Operations of arithmetic. 19. What would it be natural to call these operations which are derived from the fundamental operations? Ans The Derioative or Secondary Operations. SECTION III. SECONDARY OPERATIONS. 100. The Primary Operations of arithmetic are those of synthesis and analysis, including the four fundamental mules. 101. The Secondary, or Derivative Operations, are those which arise from or grow out of the primary operations of synthesis and analysis. 102. The Secondary Operations are Composition, Factoring, Greatest Common Divisor, Least Common Multiple, Involution, and Evolution. COMPOSITION. 103. Composition is the process of forming composite numbers when their factors are given. 104. A Composite Number is a number which can be produced by multiplying together two or more numbers, each greater than a unit; as 8, 12, 15, etc. 105. The Factors of a composite number are the numbers, which, when multiplied together, will produce it; thus 4 and 2 are the factors of 8. 106. A Prime Number is one that cannot be produced by multiplying together two or more numbers, each greater than a unit; as, 2, 5, 7, 11, etc. 107. A composite number consisting of two equal faciors is said to be the 2d power of that factor; of three equal factors, the 3d power, etc.; thus, 9 is the 2d power of 3, and 64 is the 3d power of 4. NOT.-An even number 18 one that is exactly divisible by 2; an odd number is one that is not exactly divisible by 2. HENTAL AND WRITTEN EXERCISES. 1. Tell which of the following numbers are prime or tomposite: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. 2. Name the prime numbers from 1 to 53. Name the prime numbers from 53 to 101. 3. Write the numbers from 1 to 100, and cut out all the composite numbers, leaving the primes. 4. What is the second power of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20? 5. What is the 3d power of each of the above? The 4th power? The 5th power? The 6th power ? PRINCIPLES. 1. Every composite number is equal to the product of its factors. 2. A factor of a number is a factor of any number of times that number, 108. To form composite numbers out of any factors 1. Form a composite number out of 4, 9, and 5. SOLUTION.—A composite number formed out of the factors, 4, 9, and 5, is equal to their product, which is 4X9Xbee 180 180, OPERATION. WBITTEN EXERCISES, Form composite numbers out 2. Of 5, 6, 7, and 8. Ans. 1680. 3. Of two 2’8, 3, and 7. Ans. 84. 4. Of tbree 3's, four 2's, and two 5'8. Ans. 10800. 5. Find a number consisting of four 5's. Ans. 625 6. Find the fifth power of 3, of 4, of 7. Ans. 243; 1024; 16807. 7. Form a composite number out of the first four primo Dumbers after unity. Ans. 210. 8. Form a composite number out of all the prime numbers between 11 and 29. Ans. 96577. 9. Form all the composite numbers you can out of 2, 3, 5, and 7. Ans. 6; 10; 14; 15; 21; etc. 10. Form all the composite numbers you can out of 2, 3, 5, 7, and 11. Ans. 6; 10; 14; 22; 15; 21; etc. 11. Find a composite number consisting of three factors, the first being 2, the second being twice as great, and the third three times as great. Ans 48. DIVISIBILITY OF COMPOSITE NUMBERS. 109. Composite Numbers can be divided by the factors which produce them. 110. The Factors of many composite numbers may be seen by inspection from the following principles : PRINCIPLES. 1. A number is divisible by 2 when the right hand term is zero or an even digit. For, the number is evidently an even number, and all even numbers are divisible by 2. 2. A number is divisible by 3 when the sum of the digits is divisible by 3. This may be shown by trying several numbers, and, seeing that it is true with these, we infer that it is true with all. A rigid demonstration is too difficu for this place. 3. A number is divisible by 4 when the two right hand terms are ciphers, or when the number they express is divisible by 4. If the two right hand terms are ciphers, the number equals a number of hundreds, and since 100 is divisible by 4, any number of hundreds is divisible by 4. If the number expressed by the two right band digits is divisible by 4, the number will consist of a number of hundreds plus the number expressed by the two right hand digits (thus 1232=1200+32); and since both of these are divisible by 4, their sum, which is the number itself, is divisible by 4. 4. A number is divisible by 5 when its right hand term is 0 or 5. When the unit figure is 0, the last partial dividend must be 0, 10, 20, 30, or 40, each of which is divisible by 5. When the unit figure is 5, the last partial dividend must be 15, 25, 35, or 45, each of which is divisible by 5. Therefore, etc. 5. A number is divisible by 6 when it is even, and the sum of the digits is divisible by 3. Since the number is even it is divisible by 2, and since the sum of the digits is divisible by 3 the number is divisible by 3, and since it contains both 2 and 3, it will contain their product, 3x2, or 6. fo A number is divisible by 8 when the three right hana terms are ciphers, or when the number expressed by them is divisible by 8. |