1 3+1 2+1 1+1. Here must the division stop, the numerator of the last fraction being unity. If it be required to reduce a continued fraction to a simple one, we have only to apply the rule for the reduction of complex fractions (32); for, by beginning at the last, and reducing the mixed denominator to an improper fraction, we shall have a series of complex fractions, the denominators of which are fractional. In the example given above, take the last fraction, 1 13 -; by reducing the mixed denominator to an improper frac In this way, any continued fraction may be reduced to a simple fraction; but by inspecting the work, we can arrive at a more simple method. The denominator of the fraction = 1 11 is formed by multiplying the integral part of the de nominator by 3, the denominator of the fractional part, and adding the numerator, which is unity. Again the de 1 nominator of the fraction = is formed in like man 2+2 ner, by taking the product of the two denominators and add ing 3 to the result. So also the denominator of = 1 3+ is equal to 3 × 11+4. Hence it appears, that the denominator of the required simple fraction is found, by beginning at the last denominator of the continued fraction, and multiplying successively by all the denominators; adding unity to the first product, and to each of the others, always the last previous result. In the reductions, the same number becomes alternately the denominator of a complex fraction, and the numerator of the simple one; and in the last reduction we have 1 4 1 3+1 37 11 11 37 Hence it appears, that the required numerator is the number obtained previous to the last multiplication. For reducing a continued fraction to a simple one, we have, therefore, the following RULE. Beginning at the last denominator, multiply successively by all the denominators, adding unity to the first product, and to each of the others, always the last previous result. This will give the denominator of the required fraction, and the product immediately preceding it, the numerator. 2x2+1=5, 5×2+2=12, 12×2+5=29: 29 is the denominator, and 12, the result immediately preceding it, the numerator. 2. Reduce 1 1+1 1+1 2+1 1+1 1+1 7+to a simple fraction. 2×7+1=15, 15×1+2=17, 17 × 1 + 15 = 32, 32 × 2 Num. Denom. +17=81, 81x1+32=113, 113 x1+81= 194. therefore, the simple fraction. (43.) A decimal fraction is one which has for its denominator an unit, with as many cyphers annexed to it as the numerator has places of figures. It is customary to write the numerator only, since by the number of places it contains, the value of the denominator is known. Thus, is written,4,4,24 and,361. When the numerator does not contain as many figures as the denominator has cyphers, the deficiency is supplied by writing ciphers at the left hand; thus, written,053,032, and 10,0011. 10 is A mixed number, made up of a whole number and a decimal fraction, is written with a point called the decimal point, between the whole number and the fractional part; thus, 4,67 is the same as 4,67, or 467. 1007 Ciphers on the right hand of a decimal do not alter its value, for, and 400, are the same, since they can each be reduced to. But ciphers on the left hand do alter the value of the decimal; while,3 represents,,03 is but, and ,003 is but To.. From the method of notation adopted in decimal fractions, it is evident that decimals, like whole numbers, decrease from left to right in a tenfold proportion, each place, as we recede from the left, having but one tenth the value of that which precedes it. Thus, if we commence at the left in whole numbers, we find each place representing a unit of a lower order, till we arrive at the lowest, namely, units. Continuing the same law of decrease, the first figure to the right of units becomes 10th parts, the second, 100th parts, the third, 1000th parts, and so on. We have then only to apply the system of numeration for whole numbers, counting from the decimal point both ways; towards the left, units, tens, hundreds, &c.; and towards the right, tenth parts, hundredth parts, thousandth parts, &c., as is represented in the following table :— The expression in the table is read, 2 millions, 367 thousands, 431, and 347 thousands, 632 millionths. (44.) Decimal fractions result from the division of one number by another in the following manner :— Let it be required to divide 63 by 8. 8)63(7,875 70 64 60 56 40 40 I find 8 contained 7 times in 63, with a remainder of 7; I now conceive that each of the units in 7 is divided into 10 parts, |