tional part by a dot placed between them. Thus the line supposed above would be written as 547°369 times A B. The dot which distinguishes the integer from the fractional part is called the decimal point. It will be observed that the local value of any figure to the right of the decimal point, will be a fraction having I for its numerator, and its denominator 1 followed by a cypher for each place that the figure is removed from the point. Ex. (1) 2401609005 when expressed in words, would be,—Two thousand four hundred and one, six tenths, nine thousandths, and five millionths. Ex. (2) Seven, three hundredths, and four ten-thousandths, when expressed in figures, becomes 7*0304. (7) Two hundred and thirty-five, eight tenths, three hundredths and five thousandths. (8) Seventy-four, two tenths, three thousandths, and six millionths. (9) Twelve hundred, eight ten-thousandths, and three ten-millionths. (10) Five thousandths, six hundred-thousandths, and four millionths. (11) One million, and four tenths. (12) Six hundred-millionths. Cyphers occurring at the end of a decimal do not affec its value and may be neglected. For instance, 3'500 signifies three and five tenths, i. e. the same as 3'5. But cyphers occurring in any other part of a decimal affect its value by removing the figures which follow, farther from the point. For instance, 27 signifies two tenths and seven hundredths; but 207 signifies two tenths and seven thousandths; and 0027, two thousandths and seven ten thousandths. 35.-A change in position of the decimal point will affect the local value of each figure. (For example, if instead of 31297 we write 312'97, each figure clearly has its value altered.) If the point be moved towards the right, every figure will be increased tenfold for each place through which the point is moved. If towards the left, every figure will be decreased tenfold for each place. Move Hence, conversely, we shall have the following rules :RULES.—To multiply a decimal by a power of 10. the decimal point as many places to the right as there are cyphers in that power. To divide a decimal by a power of 10. Move the decimal point as many places to the left as there are cyphers in that power. Ex. (1) 379°268 multiplied by 100 becomes 37926*8. For it is plain that if every figure has its value increased 100-fold, the number itself will be so increased. Ex. (2) 12761°23 divided by 1000 becomes 12°76123. Ex. (3) 3'41 if multiplied by 100000 becomes 341000', and if divided by 10000, it becomes '000341. Exercise 33. Multiply and divide (1) 789 365 by 10, by 100, and by 100000. 004 by 100, by 10000, and by 1000. (2) (3) 436' (4) by 1000000, by 1000, and by 10. 'I by ten, and by ten millions. 36. The arrangement and method of working to be employed in the four simple rules will be precisely similar to that employed in integral numbers, the position of the decimal point being the only new consideration. Addition. RULE. Arrange the numbers one under another, so that the units, tens, &c., may severally fall in columns, and add. The same rule is to be adopted as regards the carrying on of numbers to the next column, as is used in integers, and for the same reason. For instance, in the columu of tenths, the sum is 14 tenths, i. e., I unit and 4 tenths, the i is therefore carried on the column of units. Subtraction. RULE. Arrange the numbers as in addition, the smaller one being placed below the other, and subtract. If the number of decimal places in the lower line exceed that in the upper, cyphers may be placed in the upper to correspond to them. Ex. (1) From 34°2869 take 8°4389. 34°2869 8°4389 25.8480 Ans. The same rule is to be adopted as regards the borrowing of numbers from the next column as is used in integers, and for the same reason. For instance, in the third place of decimals, we have 8 thousandths to be subtracted from 6 thousandths. Borrowing then I from the next column, i. e., I hundreth, or 10 thousandths, we subtract 8 from 16, leaving 8; and both this I and the 3 have to be then subtracted from the 8. Ex. (2) From 9°2 take '001685. 9'200000 *001685 9'198315 Ans. H Find the value of— Exercise 34. (1) 21°3706+15°243+1*8954+*026891 +5°328+29*74. (2) 189426+13·8465+671*89 + 5'68912 +20°756+19°52, 57+0057 +6°8+1200'+*847+159°2 +3. (4) 0012 +10+5°8281+5+39°43+*6827+I. (5) 23'9875-12°4764; 35°14732-27*62815. (6) 1021274-83°072; 39°801-17'9645. (7) 30-52817; 1'7-8469. (8) 1-54237; 100-'00176. (9) 24 271-3*6485+15°271—13°256—14°125. (10) 52+52-17'8946-30°254-5+21'12. 37.-Multiplication. RULE. Multiply as in integers; and mark off as many decimal places in the product as there are in the multiplier and multiplicand together. Ex. (1) Multiply 1394 by 2'71 I'394 2'71 1394 9758 2788 3'77774 Ans. This rule for the number of places to be marked off depends upon the principle, that if the multiplier or multiplicand be diminished by dividing by any factor, the product will be diminished in the same manner. In this example 1394×271=377774. But 1394=1394 divided by 1000 (35), and 2.71=271 divided by 100, therefore the product 377774 must be divided by 1000 and 100, i. e., 100000, and written 3.77774. Ex. (2) Multiply 00363 by '000052. *00363 '000052 726 1815 *00000018876 Ans. Exercise 35. Find the values of (1) 24°35×423; 71°651×3°37; 251 X.04. (5) (2°465+1°21) × (3°2—2'89). 38.-Division. RULE. Divide as in whole numbers: subtract the number of decimal places in the divisor from the number in the dividend, the remainder will be the number in the quotient But if the number of decimal places in the divisor exceed that in the dividend, annex to the quotient as many cyphers as make up the difference; the quotient being then an integer. Ex. (1) Divide 5°754 by 1°37. 137) 5754 (42 Ans. 5 48 274 274 Ex. (2) Divide 5'754 by 00137. *00137) 5'754 (4200. Ans. 5 48 274 This rule for the number of decimal places to be marked off in the quotient depends upon the principle, that if the dividend be diminished by dividing by any factor, the quotient will be diminished in the same manner; but if, on the other hand, the divisor be so diminished, the quotient will be increased. Now bearing in mind that the removal of the decimal |