To reduce a decimal fraction to a whole number and more simple decimals, or to more simple decimals alone if it do not contain a whole number, mark off by a point as many figures from the numerator as there are ciphers in the denominator. If the numerator have not places enough for this, write as many ciphers before it as it wants places, and put the point before these ciphers. Then, if there be any figures before the point, they make the _whole number_ which the fraction contains. The first figure after the point with the denominator 10, the second with the denominator 100, and so on, are the _fractions_ of which the first fraction is composed. 135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. When there are more ciphers in the denominator than figures in the numerator, as many ciphers are placed before the numerator as will supply the deficiency, and the point is placed before the ciphers. Thus, ·7 will be used in future to denote ⁷/₁₀, ·07 for ⁷/₁₀₀, and so on. The following tables will give the whole of this notation at one view, and will shew its connexion with the decimal notation explained in the first section. You will observe that the numbers on the right of the units’ place stand for units _divided_ by 10, 100, 1000, &c. while those on the left are units _multiplied_ by 10, 100, 1000, &c. The student is recommended always to write the decimal point in a line with the top of the figures or in the middle, as is done here, and never at the bottom. The reason is, that it is usual in the higher branches of mathematics to use a point placed between two numbers or letters which are multiplied together; thus, 15. 16, _a_. _b_, (_a_ + _b_). (_c_ + _d_) stand for the products of those numbers or letters. 1234 4 4 I. 123·4 stands for ---- or 123--- or 123 + --- 10 10 10 1234 34 3 4 12·34 ---- or 12--- or 12 + --- + --- 100 100 10 100 1234 234 2 3 4 1·234 ---- or 1---- or 1 + --- + --- + ---- 1000 1000 10 100 1000 1234 1 2 3 4 ·1234 ----- or --- + --- + ---- + ----- 10000 10 100 1000 10000 1234 1 2 3 4 ·01234 ------ or --- + ---- + ----- + ------ 100000 100 1000 10000 100000