188. QUESTIONS FOR EXAMINATION

IN MULTIPLICATION AND DIVISION OF LARGE NUMBERS.

CASE 1.

1. How are large numbers multiplied?

2. How are they divided?

3. Let the student define the terms multiplicand, multiplier, dividend, divisor, and quotient. When numbers have to be multiplied together, is the result the same whether these numbers be multiplied as wholes, or be divided into parts, and these several parts multiplied together, and their products added?

4. It has been shown that figures increase in a ten-fold ratio as they advance from right to left; explain this principle as connected with Multiplication and Division.

5. What effect is produced on a number by annexing a cypher to the right of it? What by annexing two, three, four, &c. cyphers?

189. QUESTIONS FOR EXAMINATION

UNDER CASES 2, 3, 4.

1. When the multiplier consists of two figures, how should questions be arranged for multiplying by long Multiplication? 2. When the divisor consists of two figures, how should questions be arranged for dividing by long Division ?

3. What advantage is derived from performing operations by different ways, and placing them side by side ?

4. In multiplying a number by a multiplier, consisting of several figures, is great care necessary in arranging the products of the several figures of which the multiplier is composed, according to the relative value of the figures which compose them?

5. In multiplying by a multiplier consisting of two or more figures, we place the right hand figure of the second product under the second figure from the right hand of the first product, and so on, as the figures of the multiplier advance towards the left hand, their products are placed a space towards the left. Why is this done?

6. In dividing one number by another, 3786480 by 432, for instance, the first figure we place in the quotient is 8. Now how does the student know the value of this figure before the division is completed; whether it be 8 thousands, 8 hundreds, 8 tens, or 8 units?

7. In dividing one number by another, if the divisor be small, there is no difficulty in deciding how many times it is contained in the portion of the dividend which we select; but

when the divisor is large this is not so easy; in such cases what is the best method of ascertaining the proper figure to place in the quotient?

190. QUESTIONS FOR EXAMINATION UNDER

CASE 6.

1. In what cases can numbers be divided by short Division? 2. What is a prime number? Are all numbers either prime or reducible into a certain number of primes?

3. By what different means may it be ascertained, whether a number be prime or not?

4. If the sum of the figures which compose a number be divisible by 9 or by 3, by what number is the number itsel divisible?

5. With what figures do prime numbers end?

6. What numbers are divisible by 5? What numbers by 4 ? What numbers by 8? What numbers by 2?

7. Define the terms multiple, and sub-multiple, and the difference between multiple and product, and sub-multiple and factor.

8. How is short Division performed?

9. Is short Division preferable to long Division?

10. In what cases is it preferable?

11. When the divisor is large, and the question is to be performed by short Division, what is the best means of finding the most convenient sub-multiples of the divisor to work with?

12. When we have one number to multiply by another, can we use with the same effect such sub-multiples of the multiplier, as will, by their multiplication into themselves, produce the multiplier, as if we multiply by the whole multiplier in the manner shown in paragraphs previous to the 138th ?

13. Let the student explain, by proper examples, the method of treating remainders in short Division.

191. QUESTIONS FOR EXAMINATION

CONCERNING WHOLE NUMBERS AND FRACTIONS UNDER

CASE 7.

1. What is an integer? From what Latin word is the term derived? Give examples of integers.

2. What is a Fraction? From what Latin word is the term derived? Give examples of fractions also of mixed; and compound numbers.

3. Which is the numerator of a fraction? Which the denominator? What are their offices?

4. Let the student describe, as far as his present limited knowledge of the nature of fractions will admit, how they are added, subtracted, multiplied, and divided.

5. When a whole number is the product of other whole numbers, can it be divided by any of its factors without a remainder?

6. What does the student understand by the term measure? 7. Do sub-multiples measure all their multiples? Are multiples measured by all their sub-multiples? If a number be measured by another number, and the quotient be measured by a third, can the original number be measured by the product of the two former measures?

8. If a number be the product of two prime numbers, can it be measured by any numbers other than its two factors?

9. Of what advantage is a thorough knowledge of the properties and relationships of multiple and sub-multiple? Give an instance of the application of this knowledge in shortening ope

rations.

192. QUESTIONS FOR EXAMINATION CONCERNING COMMON MEASURE, &C. UNDER CASE 8.

1. Define the meaning of the term common measure? 2. When two numbers have not a common measure, what are they said to be?

3. Can numbers, which are not prime themselves, be prime to each other? Give an instance.

4. In what particular circumstances are two numbers said to have a common measure?

5. What are the common sub-multiples of two numbers? 6. Is the product of one set of prime numbers, prime to the product of another set?

7. What is the rule for finding the prime sub-multiples of numbers?

8. What is the greatest common measure of two numbers? 9. Is it a general principle in numbers, that when one number will measure two others, it will also measure their sum, their difference, and their product?

10. In dividing one number by another, can the remainder, if any there be, be divided by any number that will measure the divisor and dividend?

11. If one number be divided by another, and there be no remainder, can the dividend be restored by multiplying the divisor and quotient together? If there be a remainder, how is the dividend restored?

12. In cases, then, where one number is divided by another, without leaving a remainder, is the product of the divisor and quotient equal to the dividend?

13. In cases where one number is divided by another, and there be a remainder, is the product of the divisor and quo tient equal to the dividend, less the remainder? Give an example.

14. What are the rules for finding the greatest common measures of two numbers ?

15. How do we find the greatest common measure of three and of four numbers?

193. QUESTIONS FOR EXAMINATION

CONCERNING COMMON MULTIPLES, &C. CASE 9.

1. What are common multiples?

2. What is understood by the least common multiple ? 3. What rules are there for finding the least common multiples of two numbers ?

4. What rule is there for finding the least common multiple of three or more numbers ?

CHAPTER V.

194. COMPOUND ARITHMETIC

Is the art of computing* numbers of different denominations. 195. At paragraph 9 I have slightly touched upon compound numbers; but there the term compound has but a very limited signification. It is applied to express the different denominations of the same thing into which numbers are necessarily divided, owing to the few symbols+ we possess.

196. A number above 10 may be considered either as simple or compound; for example, if we wish to express the number 364 of men, or any thing else; and we consider this number as a whole, as so many units, then it is a simple number; but if we consider it as so many hundreds, and parts of a hundred, or so many tens and parts of ten, then it is a compound number, but in a limited sense. Every number, however, used for expressing the numbers or magnitude of the same thing is, generally considered, a simple number; but when we have to express different denominations of number, different denominations of magnitude, or different denominations of weights of the same thing, the symbols used for expressing these different denominations are called compound numbers; as, for instance, money is divided into pounds, shilling, and pence; time into years, months, days, &c.; weight into tons, hundred weights, &c. To express these different divisions of

Computing, calculating, reckoning, a casting together of several sums or particulars so as to ascertain the amount aggregate, from Com, with or together, and Rut-o, to lop, to think, to adjust accounts. + Symbols, types, representatives.

money, time, weight, &c., we use compound numbers, the computation of which is called Compound Arithmetic.

197. The same principles which apply to simple numbers, apply also to compound numbers. They are capable of being added, subtracted, multiplied, and divided in the same manner as simple numbers. The only particular in which compound and simple numbers differ is their notation; and here let me observe that the notation of Compound Arithmetic is various; money, time, the different weights, measures, &c. have all a different notation; it is, therefore, necessary that the student make himself acquainted with the different tables. The tables are scales of the different denominations into which any thing is divided, showing how many of one denomination is contained in a higher one, and also the contrary.

198. Numbers in one denomination are changed into another by Reduction, and as this rule is in continual practice in working compound numbers, we will commence with it first.

REDUCTION,

ASCENDING AND DESCENDING.

199. REDUCTION from Re, back again, and Duc-o, to lead or bring. The literal signification of the term then is a bringing back again. In this sense it may be said to be applied where, in the different questions performed in the foregoing rules, they are brought back by a contrary process for the purpose of proof. In common parlance (speech), to reduce means to make less or diminish, to degrade, to subdue, but as applied in this rule, the term is not intended to convey any idea of diminution; it is applied to signify that process of calculation, by which quantities of one denomination are brought into quantities of another denomination, as when, for example, pounds are reduced to shillings, or by a contrary process, shillings are brought back again to pounds. In these reductions from one denomination to another, there is no decrease of quantity, weight, or value, but only an expression of the same quantity, weight, or value in other terms. Thus if we reduce £20 into 400 shillings, or 4800 pence, in each denomination we express precisely the same sum of money. I have been thus particular in defining the term Reduction, lest the student should carry with him into this rule the common idea attached to the term, and thereby commit error.

200. By REDUCTION we bring numbers of one denomination into their equivalent value in another denomination.

201. REDUCTION ASCENDING is bringing from a lower to a higher denomination.