+ (68) a2x (a+bx2)*+ a2(a+bx2)* a(a+bæ2)* x When simple radicals, like ✔ or a3, are involved in a fraction, it may be reduced to its most simple terms, in the same manner as if all its terms had been rational. the common factor of the numerator and denominator, being Va- Vx. the common factor of the numerator and denominator, being x + xy+y. In a case like this, where a compound radical is involved, along with powers of the same letter which appears in that radical, the rule for detecting the common factors of the numerator and denominator, must obviously fail, inasmuch as we can no longer arrange the dividend and divisor according to the powers of that letter, upon which the application of the rule depends. In this instance, we know from other sources that +1 is a factor of the numerator and denominator, and that consequently the fraction is reducible to CHAP. VII. Numbers may be ex pressed in an algebraical form. ON THE THEORY OF DECIMAL FRACTIONS. 182. ANY number may be expressed in an algebraical form, by multiplying each digit into a power of 10 equal to the number of digits which succeed it, and connecting the results together with the sign + the number thus 31245 Decimals: expressed in this manner becomes 3 x 101 + 1 x 103 + 2 x 102 + 4 × 10 + 5, the truth of which conversion will be manifest, if we write the number in words at full length and keep in mind that 102, 103 and 10', are 100, 1000 and 10000 respectively. 183. If we divide this number by 103, the result ing. expressed algebraically, would be their mean 3 x 10+1+2 x 10-1+4 x 10-2 +5 x 10-5 the digits after the mark (.), which is called the decimal point, being divided by powers of 10, whose indices express their distance from the place of units. If we divide the same number by 105, the result expressed algebraically is where all the digits are decimals. If we had divided this number by 107 instead of 105, the algebraical result would have been where the first significant digit is in the third place from the place of units, 3 being the index of 10 in the denominator of the fraction 3 103 It follows from the notation just explained, that the digits of all numbers are to be multiplied or divided by powers of 10, whose indices express their distances to the left or to the right of the place of units. 3 Thus 300 means 3 x 10° and .03 means : 70000 7 10 means 7× 101 and .0007 means : 3250 means 3 x 105 104 184. Conversion of a decimal We are thus enabled to express any decimal into an number, by a series of equivalent fractions connected by fraction. equivalent X : the sign if we add those fractions together, reducing them to the lowest common denominator, which is a power of 10, we shall obtain a single fraction which is equivalent to the decimal. Inasmuch as the reduction of these partial fractions to the lowest common denominator which is a power of 10, leads to the multiplication of the several digits in their numerators, by powers of 10 equal to their respective distances from the last decimal place, it follows that the sum of these numerators (of the fractions with a common denominator) is equal to the integral number which arises from omitting the decimal point: we thus get the following rule for the conversion of a decimal into an equivalent single fraction. RULE. Omit the decimal point and divide the resulting integral number, by a power of 10, whose index is the number of decimal places. Thus, 754 = 754 |