633) 79125 (125 633 1582 1266 3165 3165 0 DIVISION. The student should accustom himself to work questions in different systems of numeration, which will give him a clearer insight into the nature of arithmetical processes than he could obtain by any other method. When he uses a system in which numbers are counted by a number greater than ten, he will want some new symbols for figures. For example, in the duodecimal system, where twelve is the number of figures supposed, twelve will be represented by 1'0; there must, therefore, be a distinct sign for ten and eleven, a nine and six reversed, thus, and d, might be used for these. CHAPTER III. Elementary Rules of Arithmetic. As soon as the beginner has mastered the notation of arithmetic, he may be made acquainted with the meaning of the algebraical signs +, X,=, and also with that for division, or the common way of representing a fraction. There is no difficulty in these signs or in their use. Five minutes' consideration will make the symbol 5+3 present as clear an idea as the words "5 added to 3." The reason why they usually cause so much embarrassment is, that they are generally deferred until the student commences algebra, when he is often introduced at the same time to the representation of numbers by letters, the distinction of known and unknown quantities, the signs of which we have been speaking, and the use of figures as exponents of letters. Either of these four things is quite sufficient at a time, and there is no time more favourable for beginning to make use of the signs of operation than when the habit of performing the operations commences. The beginner should exercise himself in * To avoid too great a number of accents, Roman numerals are put instead of them; also, to avoid confusion, the accents are omitted after the first line. 7" 7' 3) 1 3iv 0 4 76* (1" 4' 8 7 7 3 4 2 1.7 3 4 2 3 68 46 6 8 4 6 0 putting the simplest truths of arithmetic in this new shape, and should write such sentences as the following frequently :— 2+7=9, 6-4=2, 1+8+46=4+2+1, These will accustom him to the meaning of the signs, just as he was accustomed to the formation of letters by writing rules of arithmetic he should take care copies. As he proceeds through the never to omit connecting each operation with its sign, and should avoid confounding operations together, and considering them as the same, because they produce the same result. Thus, 4×7 does not denote the same operation as 7x4, though the result of both is 28. The first is four multiplied by seven, four taken seven times; the second is four times; and that 4x7=7x4 is a seven multiplied by four, seven taken for granted. Again, × 4 and are proposition to be proved, not to be taken marks of distinct operations, though their result is the same, as we shall show in treating of fractions. The examples which a beginner should choose for practice should be simple, and should not contain very large numbers. The powers of the mind cannot be directed to two things at once: if the complexity of the numbers used requires all the student's attention, he cannot observe the principle of the rule which he is following. Now, at the commencement of his career, a principle is not received and understood by the student structor. He does not, and cannot, geas quickly as it is explained by the inneralize at all; he must be taught to do so; and he cannot learn that a particular fact holds good for all numbers unless by having it shown that it holds good for some numbers, and that for those some numbers he may substitute others, and use the same demonstration. Until he can do this himself he does not understand the principle, and he can never do this except by seeing the rule explained, and trying it himself on small numbers. He may, indeed, and will, believe it on the word of his instructor, but this disposition is to be checked. He must be told that, whatever is not gained by his own thought is not gained to any purpose; that the mathematics are put in his way purposely because they are the only sciences in which he must not trust the authority of any one. The superintendence of these efforts is the real business of an instructor in arithmetic. The merely showing the student a rule by which he is to work, and comparing his answer with a key to the book, printed for the preceptor's private use, to save the trouble which he ought to bestow upon his pupil, is not teaching arithmetic any more than senting him with a grammar and dictionary is teaching him Latin. When the principle of each rule has been well established by showing its application to some simple examples (and the number of these requisite will vary with the intellect of the student), he may then proceed to more complicated cases, in order to acquire facility in computation. The four first rules may be studied in this way, and these will throw the greatest light on those which succeed. pre The student must observe that all operations in arithmetic may be resolved into addition and subtraction; that these additions and subtractions might be made with counters; so that the whole of the rules consist of processes intended to shorten and simplify that which would otherwise be long and complex. For example, multiplication is continued addition of the same number to itself twelve times seven is twelve sevens added together. Division is a continued subtraction of one number from another; the division of 129 by 3 is a continued subtraction of 3 from 129, in order to see how many threes it contains. All other operations are composed of these four, and are, therefore, the result of additions and subtractions only. The following principles, which occur so continually in mathematical operations that we are, at length, hardly sensible of their presence, are the foundation of the arithmetical rules : I. We do not alter the sum of two numbers by taking away any part of the first, if we annex that part to the second. This may be expressed by signs, in a particular instance, thus: (206) + (32+6) = 20+32. II. We do not alter the difference of two numbers by increasing or diminishing one of them, provided we increase or diminish the other as much. This may be expressed thus, in one instance :- 22. (45+7) (22+7)=45 (458) (22-8)=45 — 22. III. If we wish to multiply one number by another, for example 156 by 29, we may break up 156 into any number of parts, multiply each of these parts by 29, and add the results. For example, 156 is made up of 100, 50, and 6. Then 156 × 29=100 × 29 +50×29 +6 × 29. IV. The same thing may be done with the multiplier instead of the multiplicand. Thus, 29 is made up of 18, 6, and 5. Then 156×29 156 x 18+ 156×6+156×5. V. If any two or more numbers be multiplied together, it is indifferent in what order they are multiplied, the result is the same. Thus, The student should discover the reason for himself. A prime number is one which is not divisible by any other number except 1. When the process of division can be performed, it can be ascertained whether a given number is divisible by any other number, that is, whether it is prime or not. This can be done by dividing it by all the numbers which are less than its half, since it is evident that it cannot be divided into a number of parts, each of which is greater than its half. This process would be laborious when the given number is large; still it may be done, and by this means the number itself may be reduced to its prime fac tors, as it is called, that is, it may either be shewn to be a prime number itself or made up by multiplying several prime numbers together. Thus, 306 is 34 × 9, or 2x 17x9, or 2×17x3x3, and has for its prime factors 2, 17, and 3, the latter of which is repeated twice in its formation. When this has been done with two numbers, we can then see whether they have any factors in common, and, if that be the case, we can then find what is called their greatest common measure or divisor, that is, the number made by multiplying all their common factors. It is an evident truth that, if a number can be divided by the product of two others, it can be divided by each of them. If a number can be parted into an exact number of twelves, it can be parted also into a number of sixes, twos, or fours. It is also true that, if a number can be divided by any other number, and the quotient can then be divided by a third number, the original number can be divided by the product of the other two. Thus, 144 is divisible by 2; the quotient, 72, is divisible by 6; and the original number is divisible by 6×2 or 12. It is also true that, if two numbers are prime, their product is divisible by no numbers except themselves. Thus, 17 x 11 is divisible by no numbers except 17 and 11. Though this is a simple proposition, its proof is not so, and cannot be given to the beginner. From these things it follows that the greatest common measure of two numbers (measure being an old word for divisor) is the product of all the prime factors which the two possess in common. For example, the numbers 90 and 100, which, when reduced to their prime factors, are 2 x 5 x 3 x3 and 2 x 2 x 5 x 5, have the common factors 2 and 5, and are divisible by 2 x 5, or 10. The quotients are 3 × 3 and 2 × 5, or 9 and 10, which have no common factor remaining, and 2 x 5, or 10, is the greatest common measure of 90 and 100. The same may be shewn in the case of any other numbers. But the method we have mentioned of resolving numbers into their prime factors, being troublesome to apply when the numbers are large, is usually abandoned for another. It happens frequently that a method simple in principle is laborious in practice, and the contrary. When one number is divided by another, and its quotient and remainder *The factors of a number are those numbers by the multiplication of which it is made. obtained, the dividend may be recovered again by multiplying the quotient and divisor together, and adding the remainder to the product. Thus 171 divided by 27 gives a quotient 6 and a remainder 9, and 171 is made by multiplying 27 by 6, and adding 9 to the product. That is, 171 27 x 69. Now, from this equation it is easy to shew that every number which divides 171 and 27 also divides 9, that is, every common measure of 171 and 27 is also a common measure of 27 and 9. We can also shew that 27 and 9 have no common measures which are not common to 171 and 27. Therefore, the common measures of 171 and 27 are those, and no others, which are common to 27 and 9; the greatest common measure of each pair must, therefore, be the same, that is, the greatest common measure of a divisor and dividend is also the greatest common measure of the remainder and divisor. Now take the common process for finding the greatest common measure of two numbers; for example, 360 and 420, which is as follows, and abbreviate the words greatest common measure into their initials g. c. m.: 360) 420 (1 60) 360 ( 6 0 From the theorem above enunciated it g. c. m. of 60 and 360 is 60; Every number which can be divided by another without remainder is called a multiple of it. Thus, 12, 18, and 42 are multiples of 6, and the last is a common multiple of 6 and 7, because it is divisible both by 6 and 7. The only things which it is necessary to observe on this subject are, 1st, that the product of two numbers is a common multiple of both; 2d, that when the two numbers have a common measure greater than 1, there is a common multiple less have no common measure except 1, the than their product; 3d, that when they least common multiple is their product. The first of these is evident; the second will appear from an example. Take 10 and 8, which have the common measure 2, since the first is 2 x 5 and the second 2×4. The product is 2× 2 × 4×5, but 2×4×5 is also a common multiple, since it is divisible by 2 × 4, or 8, and by 2 × 5, or 10. To find this common multiple we must, therefore, divide the product by the greatest common meaThe third principle cannot be proved in an elementary way, but the student may convince himself of it by any number of examples. He will not, for instance, be able to find a common multiple of 8 and 7 less than 8 x 7, or 56. sure. CHAPTER IV. Arithmetical Fractions. WHEN the student has perfected himself in the four rules, together with that for finding the greatest common measure, he should proceed at once to the subject of fractions. This part of arithmetic is usually supposed to present extraordinary difficulties; whereas, the fact is that there is nothing in fractions so difficult, either in principle or practice, as the rule for finding the greatest common measure. We would recommend the student not to attend to the distinctions of proper and improper, pure or mixed fractions, &c. as there is no distinction whatever in the rules, which are common to all these fractions. When one number, as 56, is to be divided by another, as 8, the process is written thus:-56. By this we mean that 56 is to be divided into 8 equal parts, and one of these parts is called the quotient. In this case the quotient is 7. But it is equally possible to divide 57 into 8 equal parts; for example, we can divide 57 feet into 8 equal parts, but the eighth part of 57 feet will not be an exact number of feet, since 57 does not contain an exact number of eights; a part of a foot will be contained in the quotient 57, and this quotient is therefore called a fraction, or broken number. If we divide 57 into 56 and 1, and take the eighth part of each of these, whose sum will give the eighth part of the whole, the eighth of 56 feet is 7 feet; the eighth of 1 foot is a fraction, which we write, and 57 is 7+1, which is usually written 7. Both of these quantities 57, and 71, are called fractions; the only difference is that, in the second, that part of the quotient which is a whole number is separated from the part which is less than any whole number. There are two ways in which a fraction may be considered. Let us take, for example, g. This means that 5 is to be divided into 8 parts, and stands for one of these parts. The same length will be obtained if we divide 1 into 8 parts, and take 5 of them, or find 1 × 5. To prove this let each of the lines drawn below represent of an inch; repeat five times, and repeat the same line eight times. In each column is th of an inch repeated 8 times; that is one inch. There are, then, 5 inches in all, since there are five columns. But since there are 8 lines, each line is the eighth of 5 inches, or, but each line is also th of an inch repeated 5 times, or x 5. Therefore, × 5; that is, in order to find inches, we may either divide five inches into 8 parts, and take one of them, or divide one inch into 8 parts, and take five of them. The symbol is made to stand for both these operations, since they lead to the same result. The most important property of a fraction is, that if both its numerator and denominator are multiplied by the same number, the value of the fraction is not altered; that is, is the same as 1%, or each part is the same when we divide 12 inches into 20 parts, as when we divide 3 inches into 5 parts. Again, we get the same length by dividing 1 inch into 20 parts, and taking 12 of them, which we get by dividing 1 inch into 5 parts and taking 3 of them. This hardly needs demonstration. Taking 12 out of 20 is taking 3 out of 5, since for every 3 which 12 contains, there is a 5 contained in 20. Every fraction, therefore, admits of innumerable alterations in its form, without any alteration in its value. Thus, = 2 = j = 4 = 15%, &c.; } = &&c. On the same principle it is shewn that the terms of a fraction may be divided by any number without any alteration of its value. There will now be no difficulty in reducing fractions to a common denominator, in reducing a fraction to its lowest terms; neither in adding nor subtracting fractions, for all of which the rules are given in every book of arith metic. We now come to a rule which presents more peculiar difficulties in point of principle than any at which we have yet arrived. If we could at once take the most general view of numbers, and give the beginner the extended notions which he may afterwards attain, the mathema tics would present comparatively few impediments. But the constitution of our minds will not permit this. It is by collecting facts and principles, one by one, and thus only, that we arrive at what are called general notions; and we afterwards make comparisons of the facts which we have acquired, and discover analogies and resemblances which, while they bind together the fabric of our knowledge, point out methods of increasing its extent and beauty. In the limited view which we first take of the operations which we are performing, the names which we give are necessarily confined and partial; but when, after additional study and reflection, we recur to our former notions, we soon discover processes so resembling one another, and different rules so linked together, that we feel it would destroy the symmetry of our language if we were to call them by different names. We are then induced to extend the meaning of our terms, so as to make two rules into one. Also, suppose that when we have discovered and applied a rule, and given the process which it teaches a particular name, we find that this process is only a part of one more general, which applies to all cases contained under the first, and to others besides. We have only the alternative of inventing a new name, or of extending the meaning of the former one so as to merge the particular process in the more general one of which it is a part. Of this we can give an instance. We began with reasoning upon simple numbers, such as 1, 2, 3, 20, &c. We afterwards divided these into parts, of which we took some number, and which we called fractions, such as,,, &c. Now there is no number which may not be considered as a fraction in as many different ways as we please. Thus 7 is 14 or 21, &c.; 12 is 144, 13, &c. 12 Our new notion of fraction is, then, one which includes all our former ideas of number, and others besides. It is then customary to represent by the word number, not only our first notion of it, but also the extended one, to which our first notions applied we call of which the first is only a part. Those whole numbers, the others fractional numbers, but still the name number is applied both to 2 and, to 3 and . The rules of which we have spoken is another instance. It is called the multiplication of fractional numbers. Now, if we return to our meaning of the word multiplication, we shall find that the multiplication of one fraction by another appears an absurdity. We mul tiply a number by taking it several times and adding these together. What, then, is meant by multiplying by a fraction? Still, a rule has been found which, in applying mathematics, it is necessary to use for fractions, in all cases where multiplication would have been used had they been whole numbers. Of this we shall now give a simple example. Take an oblong figure (which is called a rectangle in geometry), such as ABCD, and find the magnitudes of the sides AB and B C in inches. Draw the line |