FRACTION

, or Broken Number, in Arithmetic and Algebra, is a part, or some parts, of another number or quantity considered as a whole, but divided into a certain number of parts; as 3-4ths, which denotes 3 parts out of 4, of any quantity.

Fractions are usually divided into Vulgar, Decimal, Duodecimal, and Sexagesimal. For the last three sorts, see the respective words.

Vulgar Fractions, called also simple Fractions, are usually denoted by two numbers, the one set under the other, with a small line between them: thus 3/4 denotes the Fraction three-fourths, or 3 parts out of 4, of some| whole quantity considered as divided into 4 equal parts.

The lower number 4, is called the Denominator of the Fraction, shewing into how many parts the whole or integer is divided; and the upper number 3, is called the Numerator, and shews how many of those equal parts are contained in the Fraction. Hence it follows, that as the numerator is to the denominator, so is the Fraction itself, to the whole of which it is a Fraction; or as the denominator is to the numerator, so is the whole or integer, to the Fraction: thus, the integer being denoted by 1, as the Fraction.— And hence there may be innumerable Fractions all of the same value, as there may be innumerable quantities all in the same ratio, viz, of 4 to 3; such as 8 to 6, or 12 to 9, &c. So that if the two terms of any Fraction i. e. the numerator and denominator, be either both multiplied or both divided by any number, the resulting Fraction will still be of the same value: thus, 3/4 or 6/8 or 9/12 or 12/16 &c, are all of the same value with each other.

Fractional expressions are usually distinguished into Proper and Improper, Simple and Compound, and Mixt Numbers.

A Proper Fraction, is that whose numerator is less than the denominator; and consequently the Fraction is less than the whole or integer; as 3/4.

Improper Fraction, is when the numerator is either equal to, or greater than, the denominator; and consequently the Fraction either equal to, or greater than, the whole integer, as 4/4, which is equal to the whole; or 5/4, which is greater than the whole.

Simple Fractions, or Single Fractions, are such as consist of only one numerator, and one denominator; as 3/4, or 5/4, or 12/25.

Compound Fractions are Fractions of Fractions, and consist of several Fractions, connected together by the word of: as 2/3 of 3/4, or 1/2 of 2/3 of 3/4.

A Mixt Number consists of an integer and a Fraction joined together: as 1 3/4, or 12 2/3.

The arithmetic of Fractions consists in the Reduction, Addition, Subtraction, Multiplication, and Division of them.

Reduction of Fractions is of several sorts; as 1. To reduce a given whole number into a Fraction of any given denominator. Multiply the given integer by the proposed denominator, and the product will be the numerator. Thus, it is found that 3 = 6/2, and 5 = 20/4, or 7 = 35/5.

If no denominator be given, or it be only proposed to express the integer Fraction-wise, or like a Fraction; set 1 beneath it, for its denominator. So 3 = 3/1, and 5 = 5/1, and 7 = 7/1.

2. To reduce a given Fraction to another Fraction equal to it, that shall have a given denominator. Multiply the numerator by the proposed denominator, and divide the product by the former denominator, then the quotient set over the proposed denominator will form the Fraction required. Thus, if it be proposed to reduce 3/4 to an equal Fraction whose denominator shall be 8; then , and the numerator, so that 6/8 is the Fraction sought, being = 3/4, and having 8 for its denominator.

3. To Abbreviate, or reduce Fractions to lower terms. Divide their terms, i. e. numerator and denominator, by any number that will dividē them both without a remainder, so shall the quotients be the corresponding terms of a new Fraction, equal to the former, but in smaller numbers. In like manner abbreviate these new terms again, and so on till there be no number greater than 1 that will divide them without a remainder, and then the Fraction is said to be in its least terms. Thus, to abbreviate 15/60; first divide both terms by 5, and the Fraction becomes 3/12; next divide these by 3, and it becomes 1/4: so that 15/60 = <*>/12 = 1/4, which is in its least terms.

4. To reduce Fractions to other equivalent ones of the same denominator. Multiply each numerator, separately taken, by all the denominators except its own, and the products will be the new numerators; then multiply all the denominators continually together, for the common denominator, to these numerators. Thus, 2/3 and 4/5 reduce to 10/15 and 12/15; and 2/3, 3/4, and 4/5 reduce to 40/60, 45/60, and 48/60.

5. To find the value of a Fraction, in the known parts of its integer. Multiply always the numerator by the number of parts of the next inferior denomination, and divide the products by the denominator. So, to find the value of 9/16 of a pound sterling; multiply 9 by 20 for shillings, and dividing by 16, gives 11 for the shillings; then multiply the remainder 4 by 12 pence, and dividing by 16 gives 3 for pence: so that 11s. 3d. is the value of 9/16l. as required.

6. To reduce a mixt number to an equivalent improper Fraction. Multiply the integer by the denominator, and to the product add the numerator, for the new numerator, to be set over the same denominator as before. Thus 3 5/8 becomes 29/8.

7. To reduce an improper Fraction to its equivalent whole or mixt number. Divide the numerator by the denominator; so shall the quotient be the integral part, and the remainder set over the denominator will form the fractional part of the equivalent mixt number. Thus 29/8 reduces to 3 5/8, and 32/4 = 8.

8. To reduce a compound Fraction to a simple one. Multiply all the numerators together for the numerator, and all the denominators together for the denominator, of the simple Fraction sought. Thus, 1/2 of 3/4 = 3/8, and 2/3 of 4/5 of 7/9 = 56/135.

To reduce a Vulgar Fraction to a decimal. See Decimals. And for several other particulars concerning Reduction, as well as the other operations in Fractions; see my Arithmetic.

Addition of Fractions. First reduce the Fractions to their simplest form, and reduce them also to a common denominator, if their denominators are different; then add all the numerators together, and set the sum over the common denominator, for the sum of all the Fractions as required. Thus, ; And .

Subtraction of Fractions. Reduce the Fractions the same as for addition; then subtract the one numerator| from the other, and set the difference over the common denominator. So ; And .

To Multiply Fractions together. Reduce them all to the form of simple Fractions, if they are not so; then multiply all the numerators together for the numerator, and all the denominators together for the denominator of the product sought. Thus ; And .

To Divide Fractions. Divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide. Thus, .

But if they will not divide without a remainder, then multiply the dividend by the reciprocal of the divisor, that is, by the Fraction obtained by inverting or changing its terms. Thus, .

Algebraic Fractions, or Fractions in Species, are exactly similar to vulgar Fractions, in numbers, and all the operations are performed exactly in the same way; therefore the rules need not be repeated, and it may be sufficient here to set down a few examples to the foregoing rules. Thus,

1. The Fraction aab/bc abbreviates to aa/c.

2. , by dividing by 3a.

3. , by dividing by a - x. See Common Measure.

4. a/b and c/d become ad/bd and bc/bd, when reduced to a common denominator.

5. .

6. .

7. .

8. .

Continued Fraction, is used for a Fraction whose denominator is an integer with a Fraction, which latter Fraction has for its denominator an integer and a Fraction, and the same for this last Fraction again, and so on, to any extent, whether supposed to be infinitely continued, or broken off after any number of terms. Euler, Analys. Inf. vol. 1, p. 295. As , or , or . Or, using letters instead of numbers, . or .

When these series are not far extended, it is not difficult to collect them by common arithmetic.

Lord Brounker, it seems, was the first who considered Continued Fractions, or at least, who applied them to the quadrature of curves, in Wallis's Arith. Infin. prop. 191, vol. 1, p. 469 &c, where this author explains the manner of forming them, giving several numeral examples, in approximating ratios, as well as the geneneral series &c, as he denotes it. Huygens also used it for the like purpose, viz, to approximate the ratios of large numbers, in his Descrip. Autom. Planet. in Oper. Relig. p. 173 &c, edit. Amst. 1728. And a special treatise on Continued Fractions was given by Euler, in his Analys. Infin. vol. 1, pa. 295 &c.

This subject is perhaps capable of much improvement, though it has been rather neglected, as very little use has been made of it, except, by those authors, in approximating to the value of Fractions, and ratios, that are expressed in large numbers; besides a method of Goniometry by De Lagny, explained in the Introduction to my Logarithms, pa. 78; as also some use I have made of it in summing very slowly converging series, in my Tracts, p. 38 & seq.

As to the reducing of common Fractions, and ratios, that are expressed in large numbers, to Continued Fractions, it is no more than the common method of finding the greatest common measure of those two numbers, by dividing the greater by the less, and the last divisor always by the last remainder; for then the several quotients are the denominators of the Fractions, the numerators being always 1 or unity. Thus, to find approximating values of the Fraction 31415926535/10000000000, or to the ratio of 31415926535 to 10000000000, being the ratio of the circumference of a circle to its diameter, by means of a Continued Fraction; or, to change the said Common Fraction to a Continued Fraction: Dividing the greater term always by the less, the same as to find the greatest common measure of the said numbers or terms, the several quotients will be 3, 7, 15, 1, 292, 1, 1, &c, which, after the first, will be the denominators, to the common numerator 1; and therefore the said Fraction will be changed into this Continued Fraction, . Hence, stopping at any part of these single Fractions, one after another, will give several values of the proposed ratio, all successively nearer and nearer the truth, but alternately too great and too little. So, stopping| at 1/7, it is 3 1/7 = 22/7 = 3.142857 too great, or 22 to 7, the ratio of the circumference to the diameter as given by Archimedes. Again, stopping at 1/15, it is 3 1/(7 1/15) = 3 15/106 = 333/106 = 3.141509 &c, too little. But stopping at 1/1, it is (the ratio of Metius) = 3.1415929 &c, which is rather too great. And so on, always nearer and nearer, but alternately too great and too little.

And, in like manner is any algebraic Fraction thrown into a Continued Fraction. As the Fraction , which being in like manner divided, the quotients are a/a, b/b, g/c, d/d; which single Fractions being considered as denominators to other Fractions whose common numerator is 1, these will be the reciprocals of the former, and so will become a/a, b/b, c/g, d/d; and hence the proposed common Fraction is equal to this terminate Continued Fraction, .

On the other hand, any Continued Fraction being given, its equivalent common Fraction will be found, by beginning at the last denominator, or lowest end of the given Continued Fraction, and gradually collecting the Fractions backwards, till we arrive at the first, when the whole will thus be collected together into one common Fraction; as was done above in collecting the Fractions And in like manner the Continued Fraction collects into the Fraction .

When the given Continued Fraction is an infinite one, collect it successively, first one term, then two together, three together, &c, till the sum is sufficiently exact. Or, if these collected sums converge too slowly to the true value, having collected a few of the terms into successive sums, these being alternately too great and too little, the true value will be found as near as you please by the method of arithmetical means, explained in my Tracts, vol. 1, Tract 2, pa. 11.

Vanishing Fractions. Such Fractions as have both their numerator and denominator vanish, or equal to 0, at the same time, may be called Vanishing Fractions. We are not to conclude that such Fractions are equal to nothing, or have no value; for that they have a certain determinate value, has been shewn by the best ma- thematicians. The idea of such Fractions as these, first originated in a very severe contest among some French mathematicians, in which Varignon and Rolle were the two chief opposite combatants, concerning the then new or differential calculus, of which the latter gentleman was a strenuous opponent. Among other arguments against it, he proposed an example of drawing a tangent to certain curves at the point where the two parts cross each other; and as the fractional expression for the subtangent, by that method, had both its numerator and denominator equal to 0 at the point proposed, Rolle looked upon it as an absurd expression, and as an argument against the method of solution itself. The seeming mystery however was soon explained, and first of all by John Bernoulli. See an account of this affair in Montucla, Hist. Math. vol. 2, pa. 366.

Since that time, such kind of fractions have often been contemplated by mathematicians. As, by Maclaurin, in his Fluxions, vol. 2, pa. 698: Saunderson, in his Algebra, vol. 2, art. 469: De Moivre, in Miscel. Anal. pa. 165: Emerson, in his Algebra, pa. 212: and by many others. The same fractions have also proved a stumbling-block to more mathematicians than one, and the cause of more violent controversies: witness that between Powell and Waring, when they were competitors for the professorship at Cambridge. In the specimen of a work published on occasion of that competition, by Waring, was the fraction , which he said became 4 when p was = 1. This was struck at by Powell, as absurd, because when p = 1, then the fraction , which was one chief cause of his not succeeding to the professorship. Waring replied that (by common division) , when p is = 1. See the controversial pamphlets that passed between those two gentlemen at that time.

There are two modes of finding the value of such fractions, that have been given by the gentlemen above quoted. The one is by considering the terms of the fraction as two variable quantities, continually decreasing, till they both vanish together; or finding the ultimate value of the ratio denoted by the fraction. In this way of considering the matter, it appears that, as the terms of the fraction are supposed to decrease till they vanish, or become only equal to their fluxions or their increments, the value of the fraction at that state, will be equal to the fluxion or increment of the numerator divided by that of the denominator. Hence then, taking the example when x = 1; the fluxion of the numerator is , and of the denominator - x^.; therefore , the value of the fraction when x = 1.——Or, thus, because x = 1, therefore ; then the fluxion of the numerator, - 4x³x^., divided by| the fluxion of the denominator, or - x^., gives 4x³ or 4, the same as before.

The other method is by reducing the given expression to another, or simple form, and then substituting the values of the letters. So in the above example , or , when x = 1; divide the numerator by the denominator, and it becomes , which when x = 1, becomes 4, for the given fraction, the same as before.—Again, to find the value of when x is = a, in which case both the numerator and denominator become = 0. Divide the numerator by the denominator, and the quotient is ; which when x = a, becomes , for the value of the fraction in that state of it.