Fraction (mathematics)

“ To Simon Stevin of Bruges in Belgium (1548–1620) , a man who did a great deal of work in most diverse fields of science, we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes in very express terms the advantages, not only of decimal fractions, but also of the decimal division in systems of weights and measures. Stevin applied the new fractions "to all the operations of ordinary arithmetic." [25] What he lacked was a suitable notation. In place of our decimal point, he used a cipher; to each place in the fraction was attached the corresponding index. ”

“ The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un = anun-1 + bnun-2. When we can solve this equation we have an expression for the nth convergent to the fraction, generally in the form of the quotient of two series, each of n terms. ”

“ If we write 5·4⁄7·4 in the form 4·5⁄4·7 we may say that the value of a fraction is not altered by multiplying or dividing the numerator and denominator by any number.
52. Fraction of a Fraction.—To find 11⁄4 of 5⁄7 of A we must convert 5⁄7 of A into 4 times some unit. This is done by the preceding paragraph. For 5⁄7 of A = 5·4⁄7·4 of A = 4·5⁄7·4 of A; i.e. it is 4 times a unit which is itself 5 times another unit, 7·4 times, which is A. Hence, taking the former unit 11 times instead of 4 times,
A fraction of a fraction is sometimes called a compound fraction. ”

“ Then, if there be any figures before the point, they make the whole number which the fraction contains. The first figure after the point with the denominator 10, the second with the denominator 100, and so on, are the fractions of which the first fraction is composed.
135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. ”

“ The upper number is called the numerator, the lower number is called the denominator, and both of these are called terms of the fraction. As long as the numerator is less than the denominator, the fraction is less than a unit: thus, 6/17 is less than a unit; for 6 divided into 6 parts gives 1 for each part, and must give less when divided into 17 parts. Similarly, the fraction is equal to a unit when the numerator and denominator are equal, and greater than a unit when the numerator is greater than the denominator. ”

“ The true definition of a fraction may be thus summed up: It is the division of its numerator by its denominator; as seven-eighths are equivalent to the whole number 7 divided by 8—whence a fraction is obtained. Decimal fractions simplify calculations greatly, as they are constructed on the principle of having one common denominator—a multiple of 10; and thus fractions can be added, subtracted, and divided without repeating over the tedious process of bringing down to a common denominator. See Arithmetic; Decimal Fraction. ”

“ Case I. If we perform many additions of two or more fractions the denominators of which contain only terms in , and no terms in , , or any other powers of , we always find that the denominator of the final resulting fraction is the product of the denominators of the fractions which were added to form the result. It follows that by factorizing the denominator of this final fraction, we can find every one of the denominators of the partial fractions of which we are in search. ”

“ The fifth view is nothing more than a change of words: A number reduced to 7/9 of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation 7/9 of a = 7 times a/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion.
When the student comes to algebra, he will find that, in all the applications of that science, fractions such as a/b most frequently require that a and b should be themselves supposed to be fractions. ”

“ Case III. When, among the factors of the denominator there are some which are raised to some power, one must allow for the possible existence of partial fractions having for denominator the several powers of that factor up to the highest. For instance, in splitting the fraction we must allow for the possible existence of a denominator as well as and . ”

“ In teaching arithmetic there does not exist any greater difficulty in getting small children to grasp the nature of the fraction as such than in getting them to grasp the idea of the simpler whole numbers. It is true that the fractions 1⁄2, 1⁄3, 1⁄4, etc., as symbols, are a little more complex than are the single digits; but as to the real meaning, when once the fractional idea has been properly developed by the teacher and the significance of the idea apprehended by the pupil, it is as easily understood as any other simple truth. ”

“ RULE VIII.
FRACTIONS.
The word Fractions is from the Latin "Fractus," broken. A Fraction is therefore a part or broken piece. A broken head is a fraction; a broken heart is a fraction; a bankrupt is a fraction—he is [60] broken up; yet a horse is not a fraction, although he may be broken in—but his rider may have a broken neck, which is called an irreducible fraction. Speaking generally, therefore, a fraction may be considered as a "Tarnation Smashification."
FRACTIONAL SIGNS.
Fractions are of two kinds, Vulgar and Decimal. ”

“ Thus, if the ratio of the less to the greater period is expressed by the fraction 1⁄2, where the difference between the numerator and the denominator is one, the commensurability is of the first order ”

“ When the same thing is done with two fractions, the result is still called their product, and the process is still called multiplication. There is this difference, that whereas a whole number is made by adding 1 to itself a number of times, a fraction is made by dividing 1 into a number of equal parts, and adding one of these parts to itself a number of times. ”

“ Lemma 2.—The constant term in any proper algebraical fraction developed in ascending powers of its variable is the same as the residue, with changed sign, of the sum of the fractions obtained by substituting in the given fraction, in lieu of the variable, its exponential multiplied in succession by each of its values (zero excepted, if there be such) , which makes the given fraction infinite. ”

“ Though not elementary algebra, it is within the reach of a student of ordinary books. [674]
Let a continued fraction, such as
a —— b + c —— d + e - f + etc., be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity ”

“ Every convergent is nearer to the value of the whole fraction than any preceding convergent.
Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent. ”

“ Now, when I have gone through such various states, passionate, wise, foolish, æsthetic, stoic; now that I see that the most perfect of them crooks and cripples either my earthly roots in the earth, or my twigs in the ether, and that, even if it did not do that, still it could not last over an hour, to say nothing of a life;—as, therefore, I clearly see that we are a fraction only and not a unit, and that all reckoning and reduction of the fraction is only an approximation between numerator and denominator, a change of the 1000/1001 into 10000/10001, I say: 'Well, for all me—be it so! ”

“ The equation must be true for all values of ; therefore it must be true for such values of as will cause and to become zero, that is for and for respectively. If we make , we get , so that ; and if we make , we get , so that . Replacing the and of the partial fractions by these new values, we find them to become and ; and the thing is done.
As a farther example, let us take the fraction . The denominator becomes zero when is given the value ”

“ If we precede the series of convergents by 0/1 and , then the numerator (or denominator) of each term of the series 0/1, 1/0, a/1, ab+1/b . . ., after the first two, is found by multiplying the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that. If a is zero, we may regard 1/b as the first convergent, and precede the series by 1/0 and 1/1.
(ii) Each convergent is a fraction in its lowest terms.
(iii) The convergents are alternately less and greater than the true value. ”

“ As there is 2⁄5 in the denominator, we must take the part of which 2⁄5 will be a unit. That part is larger than a unit; it is 21⁄2 units; 21⁄2 is that of which 2⁄5 is 1. The above fraction then directs us to repeat 21⁄2 units 31⁄4 times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 21⁄3 yards cost 31⁄2 shillings, how much does one yard cost? ”

“ If one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable, in a series prescribed by the ratio to the powers of this variable, it is easy to see that in supposing the variable equal to unity the fraction becomes 1⁄2, and the series becomes plus one, minus one, plus one, minus one, etc. ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ In transforming a vulgar fraction, say 1/17, to a decimal [Pg 229] fraction, we proceed as below, adding as many noughts to the dividend as we like until there is no remainder, or until we get a recurring series of figures, or until we have carried it as far as we require, since every additional figure in a never-ending decimal carries us nearer and nearer to exactitude.
Now, since all powers of 10 can only contain factors of the powers of 2 and 5, it clearly follows that your decimal never will come to an end if any other factor than these occurs in the denominator of your vulgar fraction. ”

“ It appears at first paradoxical, but it is well known to all mathematicians, that, through all eternity, fractions may be taken from a weight without the weight ever being annihilated. It is sufficient that each successive fraction be less than the preceding one, in a determined and regular proportion. ”

“ Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. ”

“ Devices and illustrations of a material kind are necessary to build up in the pupil’s mind at the beginning a clear concept of a tenth, etc., etc., and then to show that one-tenth written as a decimal is only a shorthand way of writing 1/10 as a common fraction, and so on. He sees very soon that the decimal is only a shorthand common fraction, and this notion he must hold to. This is the vital point in decimals. The idea that they can be changed into common fractions and the reverse at will establishes the fact in the pupil’s mind that they are common fractions and not uncommon ones. ”

“ The quantities a1 ..., b2 ... may follow any law whatsoever. If the continued fraction terminates, it is said to be a terminating continued fraction; if the number of the quantities a1 ..., b2 ... is infinite it is said to be a non-terminating or infinite continued fraction. If b2/a2, b3/a3 ..., the component fractions, as they are called, recur, either from the commencement or from some fixed term, the continued fraction is said to be recurring or periodic. It is obvious that every terminating continued fraction reduces to a commensurable number. ”

“ Take a mathematical line of one foot in length, divide it into ten parts, each of those parts into ten parts, and so on continually. If a point A be taken at hazard in the line, it does not appear self-evident that if the decimal division be continued ever so far, one of the points of division must at last fall exactly on A: neither would the same appear necessarily true if the division were made into sevenths, or elevenths, or in any other way. There may then possibly be a part of a foot which is no exact numerical fraction whatever of the foot ”

“ The Conversion of Series and Products into Continued Fractions. 1. A continued fraction may always be found whose nth convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. In fact, a continued fraction
can be constructed having for the numerators of its successive convergents any assigned quantities p1, p2, p3, . . ., pn, and for their denominators any assigned quantities q1, q2, q3, . . ”

“ If a or b is unity, a/b cannot be converted into a continued fraction with unit numerators, and the above method fails. In this case the solutions can be derived directly, for if b is unity, the equation may be written y = ax − c, and solutions are obtained by giving x positive integral values greater than c/a. ”