CHA P IV. Sect. 1. Of Notation. Fraction, or Broken Number, is that which represents a Part ng or Parts of any thing proposed, (vide Page 3.) and is express sed by two Numbers placed one above the other with a Line drawn betwixt them: ... S 3 Numerator, Thus, Denominator. The Denominator, or Number placed underneath the Line, denotes how many equal Parts the thing is supposed to be divided into (being only the Divisor in Division). And the Numerator, or Number placed above the Line, Thews how many of those Parts are contained in the Fraction (it being the Remainder after Division). (See Page 29.) And these admit of three Distinctions: Proper or Simple ( Compound ) As A proper, pure, or Simple Fraction, is that which is less than an Unit. That is, it represents the immediate Part or Parts of any thing less than the whole, and therefore it's Numerator is al. ways less than the Denominator. Anse is one Fourth Part And Si is one Half. An Improper Fraction is that which is greater than an Unit. That is, it represents some Number of Parts greater than the whole thing; and it's Numerator is always greater than the De nominator. As for or *} &c. A Compound Fration is a part of a Part, confifting of several Numerators and Denominators connected together with the Word [of]. As şof of}, &c. and are thus read, The one Third of the three Fourths of the two Fifths of an Unit. That is, when an Unit (or whole thing) is first divided into any Number of equal Parts, and each of chose Parts are subdivided fubdivided into other Parts, and so on: Then those last Parts are called Compound Fractions, or Fractions of Fractions, As for instance, suppose a Pound Sterling (or 20 s.) be the Unit or Whole; then is 8 s. the į of it, and 6 s. the of those two Fifths, and 2 s. is the } of those three Fourths; viz. 25. = ; of of į of one Pound Sterling. All Compound Fractions are reduced into single ones, Thus, RU L E. Multiply all the Numerators into one another for a Numerator, and all the Denominators into one another for the Denominator. Thus the į of $ of } will become a. Or is. For Ix3x2=6 the Numerator, and 3 * 4*5=60 the Denominator, but oor o of a l. Sterl. is 2 s. As above. Sect. 2. To alter or Change different Fractions into one Denomination retaining the same Value. IN order to gain a clear Understanding of this Section, it will I be convenient to premise this Proposition, viz. If a Number multiplying two Numbers produce other Numbers, the Numbers produced of them shall be in the same Proportion that the Numbers multiplied are, 17 Euclid 7. That is to say, If both the Numerator and Denominator of any Fraction be equally multiplied into any Number, their Products will retain the fame Value with that Fraction. As in these, 2x2–4. 0:2*3=9. Or2x52", &c. 3x2 6. 3x3 9 3x5 15 That is, į and . Or į and . Or į and is are of the same Value, in respect to the Whole or Unit. • From hence it will be easy to conceive, how two or more Fractions that are of different Denominations, may be altered or changed into others that shall have one common Denominator, and ftill retain the same Value. Example. Let it be required to change and into two other Fractions that shall have one common Denominator, and yet retain the same Value. According to the foregoing Proposition, if } be equally multiplied with 7, it will become št, viz.* =* Again, if I be equally multiplied with 3, it will become ), viz. 3*3=. 7*3 And And by this means I have obtained two new Fractions, it and a., . that are of one Denomination, and of the same Value with the two first proposed, vizit= and {=}. And from hence doth arise the general Rule for bringing all " Fractions into one Denomination. RU L E. Multiply all the Denominators into each other for a new (and common) Denominator. And each Nunerator into all the Denominators but it's own, for new Numerators. Example. Let the proposed Fractions be , , , and y. Then, by the Rule, A new Denominator And the new Numerators will will be thus found. be thus found. . 2. 3. 420 140 . 168 . 315. 360 Hence 420 is the common Denominator; and 140 . 168. 315. 360, are the new Numerators, which being placed Fraction-wise are 12.193.14. the New Fractions required. 140i 168 _ 2 315 – 3. and 30 420 Sect. 3. To bring mixed Dumbers into fraxions, and the contrary. M IX'D Numbers are brought into improper fractions by the following Rule. RUL E. Multiply the Integers, or whole Numbers, with the Denominator of the given Fraction, and to their Product add the Numerator, the Sum will be the Numerator of the Fraction required. . . Example, 9 by the Rule will become . For 9 *5=4. And, + = the improper Fraction required. To find the true Value of any improper Fraction given, is only the Converse of this Rule. For if =9*, as before is evident: Then Then it follows that if 49 be divided by 5, the Quotient will give 9 And if 206 be divided by 15, it will give 131, &. consequently it follows, that If the Numerator of any improper Fraction be divided by it's Denominator, the Quotient will discover the true Value of that. Fraction. E X A MPLES: v=s. And 4 =45 And 2=6. Or =3, &c. When whole Numbers are to be expressed Fraction-wise, it is but giving them an Unit for a Denominator. Thus 45 is 4 9?, and 25 is , &c. Sect. 4. To abbreviate or Reduce Fracions into their Lowest or Leaf Denominalion. |