GENERAL PRINCIPLES OF FUNDAMENTAL

OPERATIONS.

135. The Fundamental Operations of Arithmetic are Addition, Subtraction, Multiplication, and Division. They are fundamental because others depend upon them.

136. The Principles of the fundamental operations are the general truths which relate to them.

137. The Problems of the fundamental operations are the different classes of questions which arise under them. NOTE. The pupils will illustrate the following principles and problems.

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.

PROBLEMS.

1. Given, the parts to find the whole.

2. Given, the whole and all the parts but one, to find that part.

3. When several numbers are given, how do you find their sum?

4. When the sum of several numbers and all of them but one are given, how is that one found?

PRINCIPLES OF SUBTRACTION.

1. The Remainder equals the Minuend minus the Subtrahend.
2. The Minuend equals the Subtrahend plus the Remainder.
3. The Subtrahend equals the Minuend minus the Remainder.

PROBLEMS.

1. Given, the minuend and subtrahend, to find the remainder. 2. Given, the minuend and remainder, find the subtrahend. 3. Given, the subtrahend and remainder, to find the minuend. 4. When two numbers are given, how is their difference found? 5. When the greater of two numbers and the difference between them are given, how is the less found?

6. When the less of two numbers and the difference between them are given, how is the greater found?

PRINCIPLES OF MULTIPLICATION.

1. The Product equals the Multiplicand into the Mltiplier.

2. The Multiplicand equals the Product divided by the Multiplier. 3. The Multiplier equals the Product divided by the Multiplicand.

PROBLEMS.

1. Given, the multiplicand and multiplier, to find the product. 2. Given, the product and multiplier, to find the multiplicand. 3. Given, the product and multiplicand, to find the multiplier.

4. When two or more numbers are given, how is their product found? 5. When the product and one of two factors are given, how is the other found?

6. When the several factors of a number are given, how is the number found?

7. When the continued product of several factors and all the factors but one are given, how is that one found?

PRINCIPLES OF DIVISION.

1. The Quotient equals the Dividend divided by the Divisor.
2. The Dividend equals the Divisor multiplied by the Quotient.

3. The Divisor equals the Dividend divided by the Quotient.

4. The Dividend equals the Divisor multiplied by the Quotient, plus the Remainder.

5. The Divisor equals the Dividend minus the Remainder, divided by the Quotient.

PROBLEMS.

1. Given, the divisor and dividend, to find the quotient. 2. Given, the divisor and quotient, to find the dividend. 3. Given, the dividend and quotient, to find the divisor.

4. Given, the divisor, quotient, and remainder, to find the dividend. 5. Given, the dividend, quotient, and remainder, to find the divisor. 6. Given, the final quotient of a continued division and the several divisors, to find the dividend.

7. Given, the final quotient of a successive division, the first dividend, and all the divisors but one, to find that divisor.

8. Given, the dividend and several divisors of a successive division, to find the quotient.

PRINCIPLES OF CHANGES OF TERMS.

OF ADDITION.

1. Increasing or diminishing any term by any number, increases or diminishes the sum by that number.

OF SUBTRACTION.

1. Increasing or diminishing the minuend and subtrahend by the same number does not change the remainder.

2. Increasing or diminishing the minuend by any number, increases or diminishes the remainder by that number.

3. Increasing or diminishing the subtrahend by any number, diminishes or increases the remainder by that number.

OF MULTIPLICATION.

1. Multiplying either the multiplicand or multiplier by any number, multiplies the product by that number.

2. Dividing either the multiplicand or multiplier by any number, divides the product by that number.

3. Multiplying both multiplicand and multiplier by a number, multiplies the product by both numbers.

4. Dividing both multiplicand and multiplier by a number, divides the product by both numbers.

5. Multiplying one factor and dividing the other by the same number, does not alter the product.

6. Adding a number to either factor increases the product by as many times the other factor as there are units in the number added.

7. Subtracting a number from either factor diminishes the product by as many times the other factor as there are units in the number subtracted.

OF DIVISION.

1. Multiplying the dividend multiplies the quotient, and dividing the dividend divides the quotient.

2. Multiplying the divisor divides the quotient, and dividing the divisor multiplies the quotient.

3. Multiplying or dividing both dividend and divisor by the same number does not alter the quotient.

4. Adding a number to the dividend increases the remainder by this number, if the quotient remains the same.

5. Subtracting a number from the dividend diminishes the remainder by this number, if the quotient remains the same.

6. Adding any number to the divisor diminishes the quotient by as many units as the new divisor is contained times in the product of the quotient by the number added.

7. Subtracting any number from the divisor increases the quotient by as many units as the new divisor is contained times in the product of the quotient by the number subtracted.

GENERAL LAWS.

1. A change, by addition or subtraction, of any term in addition, produces a similar change in the sum.

2. A change in the minuend by addition or subtraction, produces a similar change in the difference; but such a change in the subtrahend produces an opposite change in the difference.

3. A change in either factor in multiplication, by multiplication or division, produces a similar change in the product.

4. 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.

SECTION III.

SECONDARY OPERATIONS.

138. The Primary Operations of Arithmetic are those. of synthesis and analysis, including the four fundamental rules.

139. The Secondary, or Derivative Operations are those which arise from or grow out of the primary operations of synthesis and analysis.

140. The Secondary Operations are Composition, Factoring, Greatest Common Divisor, Least Common Multiple, Involution, and Evolution.

COMPOSITION.

141. Composition is the process of forming composite numbers when their factors are given.

142. 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.

143. 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.

144. 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.

145. A Power of a number is a number formed by taking the given number several times as a factor; thus 64 is the third power of 4.

146. The Second Power of the number is the composite number formed by using the number twice as a factor.

9 is the second power of 3.

Thus

147. The Third Power of a number is the composite number formed by using the number three times as a factor. Thus 27 is the third power of 3.

148. The Symbol for the power of a number is a small figure, called an exponent, placed a little above it at the right; thus, 53 denotes the third power of 5, etc.

NOTE. In the fundamental operations, each synthetic process has its corresponding analytic process; it follows, therefore, that there should be a synthetic process corresponding to the analytic process of Factoring. This process I have called Composition. This new generalization, given in my Algebra, has already been approved by several mathematicians.

149. Cases.-The subject is treated under six cases. The development of these cases is based upon the following principles:

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.

CASE I.

150. To form a composite number out of any factors.

1. Form a composite number out of 3, 4, and 5.

SOLUTION.-A composite number formed out of

the factors 3, 4, and 5, is equal to their product, which is 60.

OPERATION.

3X4X5=60

Rule. Take the product of the factors as indicated by

151. To form a composite number out of equal factors.

OPERATION.

1. Find the composite number consisting of eight 2's. SOLUTION.-Multiplying 2 by 2 we have 4, which consists of two 2's; multiplying 4 by 4 we have 16, which consists of two+two, or four 2's; multiplying 16 by 16 we have 256, which consists of four+four, or eight 2's.

2x2=4 4X4 16 16×16=256