147326 147326 326 ------. By (115) ------- is 147----; 1000 1000 1000 and since 326 is made up of 300, and 20, and 6; by (112) ³²⁶/₁₀₀₀₀ = ³⁰⁰/₁₀₀₀ + ²⁰/₁₀₀₀ + ⁶/₁₀₀₀. But (108) ³⁰⁰/₁₀₀₀ is ³/₁₀, and ²⁰/₁₀₀₀ is ²/₁₀₀. Therefore, ¹¹⁴⁷³²6/₁₀₀₀ is made up of 147 + ³/₁₀ + ²/₁₀₀ + 6/₁₀₀₀. Now, take any number, for example, 147326, and form a number of fractions having for their numerators this number, and for their denominators 1, 10, 100, 1000, 10000, &c., and reduce these fractions into numbers and more simple decimals, in the foregoing manner, which will give the table below. DECOMPOSITION OF A DECIMAL FRACTION. 147326 ------ = 147326 1 147326 6 ------ = 14732 + --- 10 10 147326 2 6 ------ = 1473 + --- + --- 100 10 100 147326 3 2 6 ------ = 147 + --- + --- + ---- 1000 10 100 1000 147326 7 3 2 6 ------ = 14 + --- + --- + ---- + ----- 10000 10 100 1000 10000 147326 4 7 3 2 6 ------ = 1 + --- + --- + ---- + ----- + ------ 100000 10 100 1000 10000 100000 147326 1 4 7 3 2 6 ------- = --- + --- + ---- + ----- + ------ + ------- 1000000 10 100 1000 10000 100000 1000000 147326 1 4 7 3 2 6 -------- = --- + ---- + ----- + ------ + ------- + -------- 10000000 100 1000 10000 100000 1000000 10000000 N.B. The student should write this table himself, and then proceed to make similar tables from the following exercises. EXERCISES. Reduce the following fractions into a series of numbers and more simple fractions: 31415926 31415926 --------, --------, &c. 10 100 2700031 2700031 -------, --------, &c. 10 100 2073000 2073000 -------, --------, &c. 10 100 3331303 3331303 -------, -------, &c. 1000 10000 133. If, in this table, and others made in the same manner, you look at those fractions which contain a whole number, you will see that they may be made thus: Mark off, from the right hand of the numerator, as many _figures_ as there are _ciphers_ in the denominator by a point, or any other convenient mark. This will give 14732·6 when the fraction is 147326 ------ 10 1473·26 147326 ------ 100