EXERCISES IN THE SIMPLE RULES OF ARITHMETIC FOR MASTERS AND MATES OF HOME TRADE PASSENGER SHIPS. EXERCISES IN NUMERATION. Express in Figures :— I. Five hundred and ninety-eight. 2. One thousand, seven hundred and eighty-three. 3. Six thousand and eighty-six. 4. Eighty-nine thousand and sixty-three. 5. Six hundred and three thousand, two hundred and forty. 6. Twenty thousand, six hundred. 7. Ninety thousand and ninety-two. 8. Two hundred and four thousand, six hundred and forty-one. 9. Eight hundred thousand and eight hundred. 10. Three million, six thousand and four. 13. Ten million, ten thousand and ten. 14. Seventy million, seven hundred and four thousand, and thirty-two. 15. Forty-five million, three hundred and eighty-seven thousand, and twenty-five. 16. Three hundred and forty-nine million, four thousand and sixty-five. 17. One hundred million, ten thousand and one. 18. Eight hundred and forty-two million, two hundred and forty-eight thousand, four hundred and eighty-four. 19. Nine hundred and nine million, nine thousand and ninety-nine. 21. Three hundred and five million, forty thousand and eight. 22. Seven hundred million, seven hundred thousand and seven hundred. 23. Two hundred and two million, two hundred and two thousand, two hundred. 24. Nine hundred million, and nine hundred. D It is usual in the applications of Arithmetic to express the operation of ADDITION by signs invented for the purpose: thus, the sum of 4 and 5 is expressed in the form 4+5=9, wherein the sign + between 4 and 5 denotes the addition of the latter number to the former, and is read plus or more by; and the sign between 5 and 9 expresses the result of such addition to be 9, or the equality between the sum of the numbers 4 and 5, and the number 9; so that the arithmetical expression 4+ 5 = 9 is read 4 plus 5 equals 9. similar, 2 + 3 + 712, shows the sum of the three numbers 2, 3, 7, to be 12. 25. Add together three hundred and nine million, four hundred and seventeen thousand, and eighty-seven; six hundred and seventy-five thousand, and forty-nine; seven thousand and ninety-seven million, eight hundred and fourteen thousand, three hundred and five; seventy-nine million, five hundred and four thousand, and forty-nine; six thousand and seventy-eight million, four hundred and thirty-nine thousand, six hundred and forty-seven; seven thousand million, eight hundred and seventy-six thousand, four hundred and twenty-nine. EXERCISES IN SIMPLE SUBTRACTION. The operation of SUBTRACTION is indicated or expressed by the sign, which is read minus or less by, with the use of the sign; thus, the excess of 7 above 5 will be expressed in the form 7 -34, which is read 7 minus 3 equals 4; where the sign between 7 and 3 denotes the subtraction of the latter from the former, and the sign between 3 and 4 shows the equality of the excess to 4. EXERCISES IN SIMPLE MULTIPLICATION. The operation of MULTIPLICATION is expressed by the signs X, which is read into, or multiplied by; thus, 5 X 7 = 35 denotes the result of the multiplication of 5 by 7 to be 35; so again, 4 × 5 × 13 = 260 expresses the continued product of 4, 5, and 13. EXERCISES IN SIMPLE DIVISION. The operation of DIVISION is expressed by the sign÷, which is read by or divide by ; thus, 42 ÷ 7 = 6, implies that the result of the division of 42 by 7 is 6. 1. Express in figures, ten thousand and four. 2. 29483 +7648 + 32479 + 586 + 298364 + 98765 + 897 +789 + 5678 + 99. 3. From 6794006897 take 3985160534. 4. Multiply 94785830 by 78060. 5. Divide 5688208152 by 594. 6. Express in figures, one hundred million, one hundred thousand and one hundred. Add together 90473, 9456, 268, 59, 45694, 5437, 87668497, 2837, 9865, 3652, 999, 7. and 8888. 8. Find the difference between 100000000000 and 87649786. 9. Multiply 326904678 by 3060900. 10. Divide 236487698743 by 85409. 11. Express in figures, one hundred and three million, eighty thousand, two hun 12. 13. dred and seven. Add together 69074, 6745, 723, 29, 931648, 9005, 76245, 54267, 47096, and 7777. 14. Multiply 167409678 by 768900. 15. Divide 60000007006490088805 by 98706543. DECIMAL FRACTIONS. DECIMALS Occur so frequently in all computations relating to Nautical Astronomy, that it becomes absolutely necessary to have a knowledge of their application and their relation to Vulgar Fractions. In the Notation of Integers or common numbers, the actual value of each figure depends upon its position with respect to the place of units, its value in any one position being one-tenth of what it would be if it stood one place further to the left; thus the number 1111 denotes one thousand, one hundred, one ten, and one unit, or 1000 + 100 + 10 + 1, where the second unit beginning with the right hand one is ten times the first, the third is ten times the second, the fourth ten times the third, and so on; or beginning with the first on the left, the second is the tenth part of the first, the third the tenth part of the second, and so on, till we come down to the last unit, which is merely one; or in other words the figures decrease in a tenfold ratio from left to right. Now we may evidently extend this principle still further, and on the same plan may represent one-tenth of one, one-tenth of this, or onehundredth of one, one-thousandth of one, and so on, by simply putting some mark of separation between the integers and these fractions. The mark actually used is a dot or full stop and is called the decimal point, thus IIIIIIII. The unit (or 1) next the dot, on the left, is 1, the unit one place from this on the left is 10; the next is 100; the next 1000, and so on. In like manner, the unit next the decimal point, on the right, is %, the next too, the next roo, and so on. In other words, any figure one place to the right of the unit's place will be one tenth of what it would be if it were in the unit's place, and will thus really denote a decimal fraction; any figure two places to the right of the unit's place will be one-hundredth of what its value would be if it were in the unit's place, and so on for any number of figures. This being agreed upon, it follows that a decimal may either be considered as the sum of as many fractions as it contains digits, or as a single fraction; thus: Hence a decimal is always equivalent to the vulgar fraction whose numerator is the decimal considered as integral, that is, the number itself, when the decimal point is suppressed, and whose denominator is 1 followed by as many cyphers as there are decimal places in it. |