21. Rule of precedence of the four simple rules in Arithmetical questions (with or without brackets). Multiplication and division precede addition and sub traction. If any of the quantities are in brackets, of course each bracket (being a single quantity with respect to the rest of the expression) must be simplified first, before the above rule applies to the whole quantity to be simplified. Ex. I. I +2 × 3+ 4 = 1 + 6 + 4 = II. By an 22. By means of the radix ten, we are able to express any integers from unity upwards without limit. extension of the method adopted in numeration and notation, we are able to get a series of "decimal fractions" from unity downwards without limit. The series of decimal fractions must have ten, or some power* of ten, for their denominators. The turning-points in this extension of the old system are as follows: Now in the old system for integers, in such a number as 1234, since that system commenced from a fixed point, unity, we know that the 4 of the above number represents units, the 3, tens, the 2, hundreds, and the 1, a thousand, but since the extension of the system for fractions does not begin at a fixed point, nor end at a fixed point, we must know the value of some particular digit, as unity, in order to know the real position and value of the rest. This is done by placing a point after the unit, commonly called a decimal point, as thus; 123 4567. Here then the 3 represents units, the 2, tens, the 1, a hundred; whereas, on the other hand, the 4 represents tenths, the 5, hundredths, the 6, thousandths, the 7, ten-thousandths (or tenths of thousandths). We may tabulate these results in the following manner. A figure in the first place before the decimal point represents units. A figure in the second place before the decimal point represents tens, (unity x 10). A figure in the third place before the decimal point represents hundreds, (ten x 10). A figure in the fourth place before the decimal point represents thousands, (hundred x 10). Again : A figure in the first place after the decimal point represents tenths, (unity ÷ 10). = A figure in the second place after the decimal point represents hundredths, (tenth ÷ 10). = A figure in the third place after the decimal point represents thousandths, (hundredth ÷ 10). A figure in the fourth place after the decimal point represents ten-thousandths, (thousandth÷ 10). Thus the number 1234567 would be read, one hundred and twenty three, decimal 4567; where "decimal 4567" would mean 4 tenths, 5 hundredths, 6 thousandths, and 7 ten-thousandths. Again, the number 007 would be read, decimal 007, and would mean 7 thousandths, i. e., 7 1000 23. From the above article, the rules for reducing a decimal to a fraction, and a fraction with some power of ten for its denominator to a decimal, may be at once deduced. Hence we are in a position to give the verbal results. To reduce a decimal to a fraction. RULE. Write down the figures after the decimal point, for the numerator, and I followed by as many cyphers as there are decimal places, for the denominator. |