ART. 178. When a number or thing is divided into two equal parts, one of those parts is called one half. If the number or thing is divided into three equal parts, one of the parts is called one third; if it is divided into four equal parts, one of the parts is called one fourth, or one quarter; and, universally, When a number or thing is divided into equal parts, the parts take their name from the number of parts into which the thing or number is divided. 179. The value of one of these equal parts manifestly depends upon the number of parts into which the given number or thing is divided. Thus, if an orange is successively divided into 2, 3, 4, 5, 6, &c., equal parts, the thirds will be less than the halves; the fourths, than the thirds; the fifths, than the fourths, &c. OBS. A half of any number is equal to as many units, as 2 is contained times in that number; a third of a number is equal to as many, as 3 is contained times in the given number; a fourth is equal to as many, as 4 is contained in the number, &c. 180. When a number or thing is divided into equal parts, these parts are called FRACTIONS. OBS. Fractions are used to express parts of a collection of things, as well as of a single thing; or parts of any number of units, as well as of one unit. Thus, we speak of of six oranges; of 75, &c. In this case the collection, or number to be divided into equal parts, is regarded as a whole. 181. Fractions are divided into two classes, Common and Decimal. For the illustration of Decimal Fractions, see Section IX. QUEST.-178. What is meant by one half? What is meant by one third? What is meant by a fourth? What is meant by fifths? By sixths? How many sevenths make a whole one? How many tenths? What is meant by twentieths? By hundredths? When a number or thing is divided into equal parts, from what do the parts take their name? 79. Upon what does the value of one of these equal parts depend? 180. What are frac tions? 181. Into how many classes are fractions divided? 182. Common Fractions are expressed by two numbers, one placed over the other, with a line between them. One half is written thus; one third, ; one fourth, 1; nine tenths, f; thirteen forty-fifths, 1, &c. The number below the line is called the denominator, and shows into how many parts the number or thing is divided. The number above the line is called the numerator, and shows how many parts are expressed by the fraction. Thus, in the frac tion, the denominator 3, shows that the number is divided into three equal parts; the numerator 2, shows that two of those parts are expressed by the fraction. The denominator and numerator together are called the terms of the fraction. OBS. 1. The term fraction, is of Latin origin, and signifies broken, or separated into parts. Hence, fractions are sometimes called broken numbers. 2. Common fractions are often called vulgar fractions. This term, however, is very properly falling into disuse. 3. The number below the line is called the denominator, because it gives the name or denomination to the fraction; as, halves, thirds, fifths, &c. The number above the line is called the numerator, because it numbers the parts, or shows how many parts are expressed by the fraction. 183. A proper fraction is a fraction whose numerator is less than its denominator; as, , †, †. An improper fraction is one whose numerator is equal to, or greater than its denominator; as, 3, 景 A mixed number is a whole number and a fraction expressed together; as, 4, 25H. A simple fraction is a fraction which has but one numerator and one denominator, and may be proper, or improper; as, 3, . A compound fraction is a fraction of a fraction; as, % of † of 7, of off of 3. QUEST.-182. How are common fractions expressed? What is the number below the line called? What does it show? What is the number above the line called? What does it show? What are the denominator and numerator, taken together, called? Obs. What is the meaning of the term fraction? What are common fractions sometimes called? Why is the lower number called the denominator? Why is the upper one called the numerator? 183. What is a proper fraction? An improper fraction? A mixed number? A simple fraction? A compound fraction ? A complex fraction is one which has a fraction in its numerator or denominator, or in both; as, 24 21 184. Fractions, it will be seen both from the definition and the mode of expressing them, arise from division, and may be regarded as expressions of unexecuted division. The numerator answers to the dividend, and the denominator to the divisor. (Arts. 25, 182.) Hence, 185. The value of a fraction is the quotient of the numerator divided by the denominator. Thus, the value of is two; of is one; ofis one third, &c. Hence, 186. If the denominator remains the same, multiplying the numerator by any number, multiplies the value of the fraction by that number. For, since the numerator and denominator answer to the dividend and divisor, multiplying the numerator is the same as multiplying the dividend. But multiplying the dividend, we have seen, multiplies the quotient, (Art. 141,) which is the same as the value of the fraction. (Art. 185.) Thus, the value of §=2; now, multiplying the numerator by 3, the fraction becomes 18, whose value is 6, and is the same as 2X3. 187. Dividing the numerator by any number, divides the value of the fraction by that number. For, dividing the dividend, divides the quotient. (Art. 142.) Thus, §=2; now dividing the numerator by 2, the fraction becomes, whose value is 1, and is the same as 22. Hence, OBS. With a given denominator, the greater the numerator, the greater will be the value of the fraction. 188. If the numerator remains the same, multiplying the denominator by any number, divides the value of the fraction by that number. For, multiplying the divisor, we have seen, divides the QUEST.-What is a complex fraction? 184. From what do fractions arise? 185. What is the value of a fraction? 186. What is the effect of multiplying the numerator, while the denominator remains the same? Explain the reason. 187. What is the effect of dividing the numerator? Obs. With a given denominator, what is the effect of increasing the numerator? 188. What is the effect of multiplying the denominator? quotient. (Art. 143.) Thus, 244; now multiplying the denom inator by 2, the fraction becomes 24, whose value is 2, and is the same as 42. 189. Dividing the denominator by any number, multiplies the value of the fraction by that number. For, dividing the divisor multiplies the quotient. (Art. 144.) Thus, 2-4; now dividing the denominator by 2, the fraction becomes 24, whose value is 8, and is the same as 4×2. Hence, OBS. With a given numerator, the greater the denominator, the less will be the value of the fraction. 190. It is evident from the preceding articles, that multiplying the numerator by any number, has the same effect on the value of the fraction, as dividing the denominator by that number. (Arts. 186, 189.) And, Dividing the numerator has the same effect, as multiplying the denominator. (Arts. 187, 188.) OBS. It will be observed, that multiplying or dividing the numerator of a fraction, has the same effect upon its value, as the same operation has upon a whole number; but, the effect of multiplying or dividing the denominator is exactly contrary to that of the same operation upon a whole number. 191. If the numerator and denominator are both multiplied or both divided by the same number, the value of the fraction will not be altered. (Art. 146.) Thus, 12=3; now if the numerator and denominator are both multiplied by 2, the fraction becomes 24, whose value is 3. If both terms are divided by 2, the fraction becomes, whose value is 3; that is, 12=24=&=3. 192. Since the value of a fraction is the quotient of the numerator divided by the denominator, it follows, If the numerator and denominator are equal, the value is a unit Thus, 1, 7=1, &c. or one. QUEST.-189. What is the effect of dividing the denominator? Why? Obs. With a given numerator, what is the effect of increasing the denominator? 190. What may be done to the denominator to produce the same effect on the value of the fraction, as mul tiplying the numerator by any given number? What, to produce the same effect as divid ng the numerator by any given number? 191. What is the effect if the numerator and denominator are both multiplied, or both divided by the same number? 192. When the numerator and denominator are equal, what is the value of the fraction? If the numerator is greater than the denominator, the value is greater than one. Thus, =2, =1. If the numerator is less than the denominator, the value is less than one. Thus, 3=1 third of 1, ‡=4 fifths of 1. 193. Fractions may be added, subtracted, multiplied, and divided, as well as whole numbers. But, in order to perform these operations, it is often necessary to make certain changes in the terms of the fractions. OBs. It is evident that any changes may be made in the terms of a fraction, which do not alter the quotient of the numerator divided by the denominator; for, if the quotient is not altered, the value remains the same. Thus, the terms of the fraction may be changed into 4, 4, 16, &c., without altering its value ; for in each case the quotient of the numerator divided by the denominator is 2. Hence, for any given fraction, we may substitute any other fraction, which will give the same quotient. REDUCTION OF FRACTIONS. 194. The process of changing the terms of a fraction into others, without altering its value, is called REDUCTION OF FRAC obtain, whose terms are the lowest to which the given fraction can be reduced. Second Operation. 10)18 Ans. If we divide both terms by 10, their greatest common divisor, (Art. 170,) the given fraction will be reduced to its lowest terms by a single division. Hence, QUEST.-When the numerator is larger than the denominator, what? When smaller, 194. What is what? Obs. What changes may be made in the terms of a fraction? meant by reduction of fractions? 195. How is a fraction reduced to its lowest terms? |