four hundred and seventy-seven thousand, seven hundred and seventy.' 3. It is clear then that the value of any figure in a number depends on its distance from the right-hand figure, increasing tenfold for every move to the left, or decreasing tenfold for every move to the right. Suppose, now, that we cease to mark off our numbers by commas after every third figure, as above, and write them thus, 3472-569. The dot between the 2 and the 5 is employed to show that the figure 2 occupies the units' place. Consequently the 7 represents seven tens; the 4, four hundreds; the 3, three thousands. And proceeding from the dot to the right, the 5 represents ten times less than if it were in the units' place; the 6, a hundred times less; the 9, a thousand times less. In other words, the 5 represents five tenth-parts of a unit; the 6, six hundredth-parts of a unit; the 9, nine thousandth-parts of a unit. Also, since the tenth is ten times as much as the hundredth, and the hundredth is ten times as much as the thousandth, the three figures on the right of the dot represent 'five hundred and sixty-nine thousandths' of a unit. And similarly, if the dot be moved one place to the left, the value of every figure will be made ten times less; and if the dot be moved one place to the right, the value of every figure will be made ten times greater. 4. This is what is meant by the decimal scale of notation,' or the 'decimal system.' The figures on the left of the dot represent a whole number,' and the figures on the right of the dot represent a 'fraction' of one unit, or (when written in this form) a 'decimal fraction.' If no dot is present, the figures will represent a whole number, and the dot might be understood to be present on the right of the units' place. 5. EXAMPLES. NUMERATION AND NOTATION. I. Write out in words the following numbers: 17; 635; 8972; 13461; 333570; 1912568; 3015670; 2342601; 101010101; 325000010; 8570601101; 320560707; 32563010011672009. II. Write in words: 3.1; 12:31; 625-625; 56718.01; 312.5607; 1.0001; 523467·101; 10000 00001; 3; ·301. III. Make the number 325.607 ten times less; ten times greater; a hundred times less; a hundred times greater ;—writing out the results in words. IV. Multiply 567.25 by 100; and divide the same number by 10000;-writing out the results in words. V. Why are the final noughts in 6·100 unnecessary? VI. Write in figures the following numbers: seven hundred and seventeen; eleven hundred and ninety-one; three thousand and three; thirty-one thousand and thirty-one; one hundred and one thousand and one; one million one hundred thousand one hundred and one; one billion one million one thousand one hundred and one. VII. Write in figures: one and one tenth; fifteen and seven tenths; thirty-five and thirty-five hundredths; sixteen thousand and sixteen thousandths; one hundred and ten and fifty thousandths. The Fundamental Rules, Applied to Whole Numbers and Decimal Fractions. 6. RULE. To add together two or more numbers, set them down in a column, units under units, tens under tens, and so on toward the left; and if there be any decimal fractions, tenths under tenths, and so on to the right. Then, beginning at the right, add up the first column of figures. If the sum is less than 10, set the figure down underneath, and go on to the next column. If the sum is 10 or more, set down the right-hand figure, and carry the other to the next column. Now add up the second column, and proceed in like manner, up to the last figure-column on the left. 7. RULE. To subtract one number from another, set the least under the greatest-units under units, &c., as above. Subtract the right-hand figure of the bottom number from the figure above it, and set down the remainder underneath. If, however, the lower figure is greater than the one above it, increase the latter by 10; then subtract, set down the difference, and decrease the top figure of the next column by 1. Proceed as before to the last figure-column on the left. Thus 5681253 835185 4846068 1 1 11 3265.298 98.5493 3166-7487 111 11 Here in the first example, since we cannot take 5 from 3, we borrow ten from the fifty, and take 5 from 13: the remainder being 8. Then we have to take 8 (tens) from 4 (tens), instead of from 5 (tens), because we have previously transferred one ten to the units' column.-In the second example we begin by taking 3 from 10: for the fraction 298 means the same thing as 2980. 8. RULE. To multiply one number by another. If the multiplier be a single figure, multiply by it the units' figure of the number to be multiplied. If the product thus obtained be greater than 9, set down the units in the units' place, and carry the tens. Then multiply the tens' figure of the multiplicand by the multiplier, and add to the product the tens carried from the units' place; and so on. If there be a tens' figure in the multiplier, proceed with it in the same manner, but set down the first figure of the result in the tens' place. If there be a hundreds' figure in the multiplier, set down the first figure of the product in the hundreds' place. And so forth. Then add up the separate results. Thus (multiplicand) 345926 342.25 49 3 84 In the second example, since there are no hundredths nor tenths in the result, we dispense with the noughts; and as the multiplication of a fraction by a whole number has produced a whole number, we need not place the 'decimal point' after the units' figure 4. 9. RULE. To divide one number by another. Take as many figures from the dividend, beginning from the left, as are sufficient to make a number greater than the divisor; divide this number by the divisor, and set down the quotient as the first figure of the result. If this first division leaves any remainder, place the next figure of the dividend on the right hand of this remainder, and divide the number thus made up by the divisor, setting down the quotient on the right hand of the former quotient. And so forth. Thus-divisor dividend 11)6934237 divisor dividend quotient 11)6934237(630385 The first of these modes of arrangement is the shortest, and may be employed whenever the divisor is so small as to permit the successive acts of division and subtraction to be performed in the head. The student will do well to mark the identity of the process in the two cases. 10. When we break up the divisor into fractions, it often happens that several, or all of the successive steps give us a remainder; and the true remainder of the complete division must be found by combining the partial remainders in the manner indicated below: For the second dividend is a number of fours; the third, a number of twenties; the fourth, a number of one hundred and eighties. The accuracy of the result may be tested by working the example by long division, when the true remainder is the last result of the process. The accuracy of our work in any of the fundamental processes may be tested by employing the results in the reverse process; as in the following simple example. Division. 1260)69284325 (54987 Proved by 54987 quot. Multiplication. 1260 6300 |