EXAMPLES.-XXXVIII.

Find the Highest Common Factor of

1. x2+5x+6, x2+7x+10, and x2+12x+20.
2. x3+4x2-5, x3-3x+2, and x3+4x3-8x+3.
3. 2x2+x-1, x2+5x+4, and æ3+1.

4. y3—y2−y+1, 3y2—2y−1, and y3—y2+y−1.
5. x3-4x2+9x-10, x3 + 2x2 - 3x+20, and

x3+5x2-9x+35.

6. x3-7x2+16x–12, 3x3 — 14x2 + 16o, and

5x3-10x2+7x-14.

7. y3-5y+11y-15, y3-y2+3y+5, and

2y3-7y2+16y-15.

NOTE. We use the term Highest Common Factor instead of Greatest Common Measure or Highest Common Divisor for the following reasons:

(1). We have used the word "Measure" in Art. 33 in a different sense, that is, to denote the number of times any quantity contains the unit of measurement.

(2) Divisor does not necessarily imply a quantity which is contained in another an exact number of times. Thus in performing the operation of dividing 333 by 13, we call 13 divisor, but we do not mean that 333 contains 13 an exact number of times.

IX. FRACTIONS.

135. A QUANTITY a is called an EXACT DIVISOR of a quantity b, when b contains a an exact number of times.

A quantity a is called a MULTIPLE of a quantity b, when a contains b an exact number of times.

136. Hitherto we have treated of quantities which contain the unit of measurement in each case an exact number of times.

We have now to treat of quantities which contain some exact divisor of a primary unit an exact number of times.

137. We must first explain what we mean by a primary unit.

We said in Art. 33 that to measure any quantity we take a known standard or unit of the same kind. Our choice as to the quantity to be taken as the unit is at first unrestricted, but when once made we must adhere to it, or at least we must give distinct notice of any change which we make with respect to it. To such a unit we give the name of Primary Unit.

138. Next, to explain what we mean by an exact divisor of a primary unit.

Keeping our Primary Unit as our main standard of measurement, we may conceive it to be divided into a number of parts of equal magnitude, any one of which we may take as a Subordinate Unit.

Thus we may take a pound as the unit by which we measure sums of money, and retaining this steadily as the primary unit, we may still conceive it to be subdivided into 20 equal parts. We call each of the subordinate units in this case a shilling, and we say that one of these equal subordinate units is one-twentieth part of the primary unit, that is, of a pound. These subordinate units, then, are exact divisors of the primary unit.

139. Keeping the primary unit still clearly in view, we represent one of the subordinate units by the following

notation.

We agree to represent the words one-third, one-fifth, and 1 1 1 one-twentieth by the symbols and we say that if

"
3' 5 20'

the Primary Unit be divided into three equal parts, will

represent one of these parts.

If we have to represent two of these subordinate units, we do so by the symbol; if three, by the symbol; if four, by

3

the symbol, and so on. And, generally, if the Primary Unit be divided into b equal parts, we represent a of those parts by the symbol.

a b

140. The symbol we call the Fraction Symbol, or more briefly a FRACTION. The number below the line is called the DENOMINATOR, because it denominates the number of equal parts into which the Primary Unit is divided. The number above the line is called the NUMERATOR, because it enumerates how many of these equal parts, or Subordinate Units, are taken.

141. The term number may be correctly applied to Fractions, since they are measured by units, but we must be careful to observe the following distinction:

An Integer or Whole Number is a multiple of the Primary
Unit.

A Fractional Number is a multiple of the Subordinate
Unit.

142. The Denominator of a fraction shews what multiple the Primary Unit is of the Subordinate Unit.

The Numerator of a fraction shews what multiple the Fraction is of the Subordinate Unit.

143. The Numerator and Denominator of a Fraction are called the Terms of the fraction.

144. Having thus explained the nature of Fractions, we next proceed to treat of the operations to which they are subjected in Algebra.

145. DEF. If the quantity x be divided into b equal parts, and a of those parts be taken, the result is said to be the fraction

α

of x.

If a be the unit, this is called the fraction b.

147. Next let us suppose that each of the b parts is subdivided into c equal parts: then the unit has been divided into bc equal parts, and

Let the unit be divided into b equal parts.

a

Then will represent a of these parts.

..(1).

Next let each of the b parts be subdivided into c equal parts.

Then the primary unit has been divided into bc equal parts,

ac

and will represent ac of these subdivisions.

bc

(2).

Now one of the parts in (1) is equal to c of the subdivisions

COR. We draw from this proof two inferences:

I. If the numerator and denominator of a fraction be multiplied by the same number, the value of the fraction is not altered.

II. If the numerator and denominator of a fraction be divided by the same number, the value of the fraction is not altered.

149. To make the important Theorem established in the preceding Article more clear, we shall give the following proof 4 16 that = by taking a straight line as the unit of length. 5 20

Let the line AC be divided into 5 equal parts.

Then, if B be the point of division nearest to C,

Next, let each of the parts be subdivided into 4 equal parts.

AC contains 20 of these subdivisions,

Then

150. From the Theorem established in Art. 148 we derive the following rule for reducing a Fraction to its lowest terms.

Find the highest common factor of the numerator and denominator and divide both by it. The resulting fraction will be one equivalent to the original fraction expressed in the simplest terms.