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152. A fraction is a number which denotes one or more parts of unity; the number 1 being supposed divisible into any number of equal parts at pleasure: whatever expresses any assigned number of those parts is called a fraction.

Fractions are usually divided into Vulgar, Decimal, Duodecimal, and Sexagesimal'.


153. Vulgar Fractions are those which express the parts of unity, into whatever number of equal parts it may be supposed to be divided. 154. A Vulgar Fraction is denoted by two numbers placed one over the other, with a small line between them, thus, 3. 155. The number below the line is called the denominator; it shews how many parts the unit is supposed to be divided into. 156. The number above the line is called the numerator; it shews how many of the aforesaid parts are to be understood by the fraction. - Thus in the above fraction 3, the number 7 is the denominator, and shews that l is supposed to be divided into 7 equal parts; 4 is the numerator, and shews that 4 of those parts are to be understood by the fraction; that is, the value of the fraction is four sevenths, or four of such equal parts of which seven

just make the number 1.

f The word fraction is derived from the Latin frango, to break, and is a name descriptive of the numbers included under it. Decimals and Duodecimals will be distinctly treated of; Sexagesimals (or Sixtieths) will likewise be explained when we treat of Practical Geometry, Trigonometry, &c. The term vulgar comes from vulgus, the common people. Vulgar Fractions mean frac

tions that admit of any denomination whatever.

157. When the numerator is less than the denominator, it is evident that the fraction expresses fewer parts than 1 is supposed to be divided into; consequently the value of the fraction is less than 1 : such a fraction is called a proper fraction ; thus, + (one half), 3 (two thirds), or (three elevenths), &c. are proper fractions.

158. When the numerator is equal to the denominator, the fraction expresses just so many parts as 1 is supposed to be divided into; consequently its value will be equal to 1. In like manner, when the numerator is greater than the denominator, the fraction expresses more parts than 1 is divided into, and its value is greater than 1. In either case the fraction is called an improper fraction ; thus, 4 (four fourths), # (five fifths), 3 (seven thirds), *** (twenty-one ninths), &c. are improper fractions.

159. But this division is not confined to unity; we may conceive the parts themselves susceptible of a similar division; every fraction may be subdivided into other parts or fractions, and these into others, and so on without end : the expression denoting a fraction arising from such a division and subdivision of unity is called a compound fraction; thus, + of + (one half of one

2 half), 3 of 3 (two thirds of three fifths), , of 3 of 3 (one fourth of two sevenths

of three eighths); such expressions as these consisting of two or more fractions, with the word of interposed between them, are, as is evident, meant to express a part of a part, or parts of parts, and are called compound fractions. 160. A simple fraction is that which is expressed by one mumerator and one denominator, and both whole numbers. 161. When a whole number and a fraction are connected, so that both together form but one number, such an expression is called a mired number; thus, 13 (one and two sevenths), 53 (five and eight ninths), 14 (fourteen and one sixth), &c. are mixed numbers. 162. When either the numerator or the denominator is a mixed number, or when both are mixed, the fraction is called a

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2 ; : five sixths), go (two and two thirds, three and four fifths 5), &c. are complex Ts


s When both terms are complex, this reading will scarcely be intelligible; perhaps it will be better to read these fractions in the following manner; viz.

l - c) 2 # three and one half by four ; + two by three and five sixths; ă two and 3# +

two thirds by three and four fifths; and the like of others.

163. A common measure of two or more numbers is such a number as will divide each without leaving any remainder; and the greatest common measure is the greatest number possible that will divide each without remainder. , 164. A common multiple of two or more numbers is a number which each of them will divide without remainder; and the least common multiple is the least number that each of them will so divide.


165. Reduction of Fractions is the changing them from one form to another without altering their values. By the operations of reduction, fractions are expressed in the most eonvenient form for the readily understanding of their values, and likewise prepared for adding, subtracting, multiplying, and dividing".

166. To find the greatest common measure of two numbers.

Rule I. Divide the greater number by the less, and divide the divisor by the remainder.

II. Proceed in this manner, dividing continually the last divisor by the last remainder, until nothing remains: the last di

There are some other denominations of fractions, as continued fractions, used for approximating to indeterminate ratios in small numbers; vanishing fractions, the properties of which are best explained by Fluxions, &c.

h Previous to entering on the Reduction of Fractions it will be proper to remark, that one fraction is always equal to another when the numerator of the first is to its denominator as the numerator of the second is to its denominator; thus,” is equal to 4 or to 3 or to # or to #3 or to or or to +}}, &c. for

2 : 4
3 : 6
... o . . 4 : 8
1 + 2 + i < 10 20
50 : 100

100 : 200, &c. Hence it follows, that since numbers have the same ratio to one another that their like multiples or like parts have respectively, both terms of any fraction may be either multiplied or divided by any (the same) number without altering the original value of the fraction; thus both terms of 3-3 may be multiplied by 7, 9, &c. or divided by 10, 5, &c. and the results #3, ###, 3, 4, &c. will be equal to each other and to the given fraction #3.

visor is the greatest common measure of the two given numbers as was required'. ExAMples.

1. Required the greatest common measure of 72 and 120.

OPERATION. Earplanation - a planation. 72)120(1 First I divide the greater by the 72 less, viz. 120 by 72, and the remain48)72(1 der is 48. Next 1 divide 72 by 48, 48 and the remainder is 24. Lastly, I - divide 48 by 24, and there is no re24)48(2 mainder, wherefore the last divisor 24 48 is the greatest common measure re


2. Required the greatest common measure of 536 and 792. 536)792(1 536 256)536(2 51.2 24)256(10 24 16)24(l 16 8) 16(2 16

Answer 8 the common measure required.

3. What is the greatest common measure of 12 and 15 Ans. 3.

* If any number measures two other numbers, it will measure both their sum and difference; this being premised, the truth of the rule may be shewn from the first Example; thus, 24 measures 48, as appears by there being no remainder; wherefore it measures 48+ 24, or 72 ; and since it measures both 48 and 72, it likewise must measure 48+ 72, or 120; wherefore 24 is a common measure of 72 and 120.

It is likewise the greatest common measure; for if not, let there if possible be a greater; then, since this greater measures 72 and 120, it will measure their difference, (viz. 120–72 or): 48; and since it measures 72 and 48, it will likewise measure their difference, (or 72–48) = 24; wherefore a number greater than 24 will measure 24, which is absurd; wherefore 24 is the greatest Common measure,

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