19. What is the denominator of .000309? What is the numerator, and how read? 20. What effect have the ciphers in .0083? 18. How is it found? 21. Write a rule for expressing a decimal by writing its denominator. 323. The relation of the orders of units in an integer and decimal will be seen from the following table: 1. The Unit is the standard in both cases. The integral orders are multiples of one unit, and the decimal orders are decimal fractions of one unit. 2. Figures that are equally distant from the units' place on the right or left, have corresponding names; thus, tenths correspond to tens, hundredths to hundreds, and so on. 3. In reading an integer and decimal together, "and" should not be used anywhere but between the integer and fraction. Thus, 9582.643 should be read, nine thousand five hundred eighty-two and six hundred forty-three thousandths. 4. Dimes, cents, and mills being respectively tenths, hundredths, and thousandths of a dollar, are written as a decimal. Thus, $.347 is 3 dimes, 4 cents, and 7 mills. In reading dimes, cents, and mills, the dimes are read as cents. Thus, $62.538 is read, 62 dollars, 53 cents, 8 mills. 28. Write with figures: Seventy-three thousandths; four hundred five millionths; eight ten-thousandths. 29. Three thousand nine hundred-millionths; ninety-one millionths; six hundred four thousand three billionths. 30. Eighty-four and seven ten-thousandths; nine thousand six and five hundred seven ten-millionths; six and three millionths. 31. Four thousand thirty-seven and nine hundred seven billionths; one million one and one thousand one ten-millionths. REDUCTION. PREPARATORY PROPOSITIONS. The following preparatory propositions should be very carefully studied. 325. PROP. I.—Annexing a cipher or multiplying a number by 10 introduces into the number the two prime factors 2 and 5. Thus, 10 being equal 2×5, 7x10 or 707 × (2 × 5). Hence a number must contain 2 and 5 as a factor at least as many times as there are ciphers annexed. 326. PROP. II.-A fraction in its lowest terms, whose denominator contains no other prime factors than 2 or 5, can be reduced to a simple decimal. Observe that every cipher annexed to the numerator and denominator makes each divisible once by 2 and 5 (325). Hence, if the denominator of the given fraction contains no other factors except 2 and 5, by annexing ciphers the numerator can be made divisible by the denominator, and the fraction reduced to a decimal. = Thus, 1888 (235-II). Dividing both terms of the fraction by 8 (235-III), we have 38885.875. Reduce to decimals and explain as above: 16. How many ciphers must be annexed to the numerator and denominator of to reduce it to a decimal ? 17. Reduce to a decimal, and explain why the decimal must contain three places. 18. If reduced to a decimal, how many decimal places will make? Will make? Will make, and why? 327. PROP. III.-A fraction in its lowest terms, whose denominator contains any other prime factors than 2 or 5 can be reduced only to a complex decimal. Observe that in this case annexing ciphers to the numerator and denominator, which (325) introduces only the factors 2 and 5, cannot make the numerator divisible by the given denominator, which contains other prime factors than 2 or 5. Hence, a fraction will remain in the numerator, after dividing the numerator and denominator by the denominator of the given fraction, however far the division may be carried. Thus, 1888 (235-II). Dividing both numerator = 1. How many tenths in ? In ? In ? In ? In ? 2. Reduce to hundredths &; †; ñ; †; &; H. 3. How many thousandths in ? In ? In ? In? 328. PROP. IV.—The same set of figures must recur indefinitely in the same order in a complex decimal which cannot be reduced to a simple decimal. Observe carefully the following: 1. In any division, the number of different remainders that can occur is 1 less than the number of units in the divisor. Thus, if 5 is the divisor, 4 must be the greatest remainder we can have, and 4, 3, 2, and 1 are the only possible different remainders; hence, if the division is continued, any one of these remainders may recur. 2. Since in dividing the numerator by the denominator of the given fraction, each partial dividend is formed by annexing a cipher to the remainder of the previous division, when a remainder recurs the partial dividend must again be the same as was used when this remainder occurred before; hence the same remainders and quotient figures must recur in the same order as at first. 3. If we stop the division at any point where the given numerator recurs as a remainder, we have the same fraction remaining in the numerator of the decimal as the fraction from which the decimal is derived. 329. PROP. V.-The value of a fraction which can only be reduced to a complex decimal is expressed, nearly, as a simple decimal, by rejecting the fraction from the numerator. 3 27 Thus, = (327). Rejecting the from the numer ator, we have, a simple fraction, which is only of smaller than the given fraction or Observe the following: 27. 100 1. By taking a sufficient number of places in the decimal, the true value of a complex decimal can be expressed so nearly that what is rejected is of no consequence. Thus, 3 27272727-1 11 ator, we have 100000000 1000000009 ; rejecting the from the numer 21212127, or .27272727, a simple decimal, which is only of 1 hundred-millionths smaller than the given fraction. 2. The approximate value of a complex decimal which is expressed by rejecting the given fraction from its numerator is called a Circulating Decimal, because the same figure or set of figures constantly recur. |
