189 



FRACTION'S, COMMON AND DECIMAL. 



FRACTIONS, CONTINUED. 



190 



The preceding fraction may be considered in several different ways. 

 It is 1st, the 6th part of a unit repeated a times ; or, in common lan- 

 guage, a-iths of a unit ; 2nd, the number of times, or parts of a time, 

 or both, which a contains b ; 3rd, the proportion which a is of b ; 4th, 

 the expression which ought to be written for a, on the supposition of 

 that which was 6 units being made the unit. Thus J expresses two- 

 fifths of a unit, the part of a time which 2 contains 5, the proportion 

 which 2 is of 5, and the expression which must be written for what is 

 now 2, when that which is now 5 is made the unit. All these mean- 

 ings, except tfte first, are perfectly intelligible when we write a fraction 

 in which the terms are both fractional. Thus 



11 6 1 31 & 



^ IX< 



3? 1J I I 



may be thus explained. We can readily imagine the part of a time 

 which 1 4 is of 3f, the proportion which the first is of the second, and 

 the expression which must be substituted for 1 4 when a larger unit is 

 used, amounting to 3^ of the present unit. But though we see clearly 

 what is meant by dividing 1 into 3 equal parts and into 4 equal parts, 

 what idea are we to attach to the division of 1 into 3f equal parts ? 



The generality of mathematical conceptions is frequently destroyed 

 by the peculiar idiom of a language. The science of arithmetic re- 

 quires the abolition of all those distinctions which depend on singular 

 and plural, noun and pronoun, Ac. Thus, when we speak of the 

 answer to a problem being a number of feet (unknown), it is better to 

 allow the word to imply a part of a foot, a foot itself, or a number of 

 feet together with a part of a foot, than to repeat all those possible 

 cases every time a number is to be mentioned. Again, when one par- 

 ticular phrase seems absurd, but another which is synonymous appears 

 clear, we must either reject the former altogether, or attribute to it 

 the meaning of the latter, and the second course is generally the more 

 convenient. We now observe that the direction to " divide one into 

 10 equal parts " is the same as " find a part such, that ten of them 

 shall make a unit." Now there is no absurdity in requiring to " find 

 a part such that 3f of them shall make a unit," though it is incon- 

 sistent with our idiom to speak of " dividing 1 into 3^ equal parts." 

 The meaning of the phrase which is intelligible should then be ex- 

 tended to that which is not, or " to divide 1 into 3$ equal parts " should 

 mean that the part is to be found which repeated 3 times and f of a 

 time shall give the unit. And this must be extended even to the case 

 in which the number or fraction thus obtained is greater than a unit. 

 Thus in the fourth of the preceding fractions such a number or frac- 

 tion must be found, that |th of it .shall be a unit ; that is, 



stands for the number 7 ; 



and this must be repeated 3| times. The preceding considerations 

 how that fractions with fractional denominators may be explained 

 (without reference to any rule of reduction) by an extension of the 

 definition which applies to integer denominators. The use of such an 

 extension is as follows : at present, algebraical students learn results 

 which are perfectly intelligible with regard to whole numbers, or to 

 fractions with integer terms, but of which they do not see the meaning 

 when fractional or mixed terms are employed. In the latter case they 

 trust to what they see in the former that their results mil remain 

 tnif ; but they can have no distinct perception on this point until they 



have learnt to include every possible form of -^- under one definition. 



6 



The fundamental property of fractions on which all others depend 

 is this that no fraction is changed in value by multiplying or dividing 

 both ita terms by the same number or fraction, that is, 



ma 

 mb 



whatever may be the values of a, 4, and m. This result should be 

 studied in all the variety of ita cases, from such as 



2 _ 3x10 ^^ 2_! = 24xU 



5-6x10 y yxit 



There is another theorem which is much neglected in elementary 

 works, but which is of considerable importance, namely, that if the 

 numerators of two fractions be added for a numerator, and their de- 

 nominators for a denominator, the resulting fraction must lie between 

 the two from which it was derived. Thus of the three fractions, 



u , 



. and 



2+ 6 

 7 + 11 



the third is greater than the first, but less than the second. 



In practice it is convenient to employ fractions having either the 

 same denominators, or which may easily be reduced to others of 

 equal value having the name denominators. The numbers 10, 100, 

 1000, 4c., suggest themselves for this purpose : indeed it may imme- 

 diately be seen that the ordinary system of decimal numeration may 

 be extended so as to allow of a representation of such fractions. If 

 we consider the number 1 1 1 1 1 , we see that for every step which we 

 make to the right, we find a unit which is only the tenth part of the 



preceding unit. Place a point on * the unit's place (to mark its posi- 

 tion), and let the same method of valuation be carried further. Then 

 in 11111-1111, the first 1 after the point should stand for one-tenth of 

 the preceding, or one-tenth of a unit ; the second for one-tenth of a 

 tenth, or one-hundredth, and so on. The fundamental theorem of 

 decimal fractions, in this view of the subject, is that which shows, for 

 example, that 12-2345 (defined to mean 1 ten, 2 units, 2 tenths, 3 

 hundredths, 4 thousandths, and 5 ten-thousandths) is the same as 

 122345 ten thoxisandths ; or that all the number, such as it would be 

 if the units' column were on the right, may be taken as a numerator, 

 and the denomination of the right hand figure as a denominator 

 Thus 



65-483 or 60 + 5 + + JL+ 3 

 10 100 1000 



60000 + 5000 400 



1000 ~ r 1000 1000 



65483 

 1000 



80 

 1000 



3 



1000 



No fraction can be reduced to an equivalent decimal fraction, if its 

 denominator contain any prime factor except 5 or 2 (the divisors of 

 ten). But this is of no consequence in practice, since it may easily be 

 shown that for any fraction can be found a decimal fraction which 

 shall be as near to it as we please. For instance, suppose it required 



Q 



to find a decimal fraction which shall not differ from TT by so much 

 as the hundred thousandth part of a unit. Then 



300000 

 _3_ = 300000 



ilooooo 



41 



41 

 100000 



= 7817ft 



100000 



or 731 7 hundred thousandths of a unit differs from -rr bv onlv 



41 ' '41 



of the hundred thousandth of a unit, or by less than the hundred 

 thousandth part. It is from such a transformation that the common 

 rule is derived. 



It is common to say that a result is true to a certain number of 

 places of decimals when any alteration of any place would make it 

 further from the truth. Thus, the diameter of a circle being unity, 

 the circumference lies between 3-1415 and 3'1416, but nearer to the 

 latter ; whence the same circumference, true to four places of decimals, 

 is 3-1416. Similarly, 62'13299, taken true to two places, is 62'13; to 

 three, 62-133 ; to four, 62-1330. Again, -625, taken true to two places, 

 might be either -62, or -63 ; but the latter -is generally taken. When 

 a decimal fraction cannot be found exactly equal to a given common 

 fraction, the division by which the numerator is found, leads to what 

 is called a CIRCULATING DECIMAL. 



For subjects closely connected with the theory of fractions, see 

 RATIO ; PROPORTION ; INCOMMENSURABLE. 



FRACTIONS, CONTINUED. A continued fraction is one which 

 has a fraction in its denominator, which again has a fraction in its 

 denominator, and so on : such as 



1 

 2 + 



7 + 6 



A more convenient way of writing such fractions is desirable ; in the 

 present article we shall adopt the following : 



4 3 6 2 



Thus 



6 + c 



d + e 

 f 



is written 



* + 



The use of continued fractions is as follows : by converting a 

 common fraction with a large numerator and denominator, into a 

 continued fraction, we are able to find a succession of more simple 

 fractions which are alternately greater and less than the given fraction, 



and approach to it with great rapidity. Let T be the given frac- 

 tion, a being less than l> ; proceed as in the rule for finding the 

 greatest common measure of o and 6, and let q, r, s, t, &c., be the 

 quotient* obtained in the process ; then 



&c. 



1 + 



1 



t + 



&c. 



* It will b very useful to the student to remember that this decimal point 

 belongs to the unit's place, and is not an introduction between the integers and 

 the fractions. 



