ner, by taking the product of the two denominators and add ing 3 to the result. So also the denominator of = 1 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 Hence it appears, that the required have 1 1 = 7 = 37 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. 1. Reduce 1 2+1 EXAMPLES. 2+1 2+ to a simple fraction. 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 +17=81, 81×1+32=113, 113×1+81= 194. therefore, the simple fraction. 112 is, (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,4,3,24 and 361 ,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, is written,05,032, and 1,0011. 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 4,7, or 487. Ciphers on the right hand of a decimal do not alter its value, for, and 40%, 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 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, sands, 431, and 347 thousands, 632 millionths. 367 thou (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, which gives me 70 tenth parts of a unit for a new dividend; this is divisible by 8; and since this second dividend is tenth parts, the quotient will be tenth parts; and hence the 8 in the quotient is written with the decimal point before it. I now find a remainder of 6 tenth parts of a unit; each of these parts are also subdivided into ten parts, or, which is the same thing, this remainder is multiplied by 10, which gives 60 hundredth parts; the quotient of which, by the divisor, is 7 hundredths, which is put in the place of hundredths in the quotient. The next remainder is likewise multiplied by 10, which gives thousandth parts for a new dividend, and a figure in the place of thousandths, in the quotient. Hence it appears, that a division may be executed in decimals, when it cannot be in whole numbers. In the preceding example, the quotient of 63 by 8 is 7,875. Expressed by a whole number and vulgar fraction, it would have been 77. By inspecting the work, it will be seen, that the vulgar fraction, is reduced to a decimal by adding cyphers to the numerator (for this is multiplying by 10), and dividing by the denominator. The following rule will now require no expla nation. To reduce a vulgar fraction to a decimal. RULE. (45.) Annex ciphers to the numerator, and divide by the denominator, and point off as many places for decimals in the quotient, as there are ciphers annexed to the |