41. Five hundred and twenty-nine thousand, three hundred and sixty-two, billionths. 42. Nine thousand, three hundred and sixty-two, millionths. 43. Nine billion, six hundred and one million, five hundred and twenty-nine thousand, three hundred and sixty-two, hundred-billionths. 44. Twenty-nine thousand, three hundred and sixty-two, millionths. 45. Sixty-one million, five hundred and twenty-nine thousand, three hundred and sixty-two, ten-billionths. 46. Decimal forty thousand and seventy-six millionths. 47. One million, five hundred and twenty-nine thousand, three hundred and sixty-two, hundred-millionths. 48. Two ten-millionths. 49. Three, and decimal four hundred and two, tenmillionths. 50. Sixty-two million, three hundred and twenty-eight thousand, five hundred and eleven, and decimal four thousand and six, ten-thousandths. 51. Sixty-two hundred-thousandths. 52. Decimal nineteen billion, six hundred and one million, five hundred and twenty-nine thousand, three hundred and sixty-two, trillionths. 53. Twenty-four ten-thousandths. 54. Three hundred and sixty-two, hundred-millionths. 55. One hundred and twenty-eight thousand, and decimal eight hundredths. 56. Twenty thousand and forty-eight, and decimal eight hundred and fifteen thousand, three hundred and sixty-four, hundred-millionths. SEE NOTE, PAGE 13. As 10 tenths equal 1, .1 is 10 tenths divided by 10, or 1 divided by 10, and may be expressed 1 ÷ 10, 1: 10, or .248 is 248 divided by 1000, and may be expressed 248÷1000, 248: 1000, or 248. is of the same general nature as a decimal fraction. But the number by which 7 is divided being neither 10 nor a number which is the product of two or more 10's multiplied together, is not a decimal fraction; being of the same general nature as a decimal fraction, it is a fraction. The number divided is called the numerator; the number by which it is divided is the denominator; the numerator and the denominator are terms of the fraction. From the very nature of the decimal scale, it follows that the decimal fraction has a numerical value. 10 x .33, 100 x .033, 1000 x .003 = 3. If any decimal be multiplied by its denominator, the product will be the numerator as a whole number. Hence, the decimal has a numerical value, for otherwise its product would have no value. But the decimal fraction is only a particular case of the fraction in general. Any decimal may be expressed in the common fractional form, and a fraction in the common form may be changed into a decimal. Thus, .75, for 3300 hundredths, and 300 hundredths ÷ 475 hundredths, or .75. Hence, any fraction has a numerical value. A fraction is a numerical quantity. The general name for numerical quantities is number. A fraction is a number. NOTE. It is only by a process of evolution that a fraction has become a number. Being an extension of primary ideas, underlying principles must be established and developed before this is possible. The idea of a fraction as a number having now been developed, any division may hereafter be made to come out even, and so multiplication and division may now be considered as the reverse of each other. The first three decimal places being tenths, hundredths, and thousandths, 3 is read three tenths; .03, three hundredths; .003, three thousandths. But .3, .03, and .003 are the same three fractions as 1, 180, and So, is read three tenths; 18, three hundredths; Too, three thousandths. Similarly, is read three fifths; 3, three fourths;, three thirds;, three halves. may also be read numerator 3, denominator 2; or, as already explained, 3 divided by 2. 21 is read two and one half divided by three and two thirds, 3 or numerator, two and one half; denominator, three and two thirds. of is not a fraction, as it has not a numerator and a denominator. NOTE. The fraction is of geometrical origin. The terms used make this evident. Fraction means a fragment or a piece. Following out this idea, the denominator denominates or names the pieces to indicate their size as compared with the whole; the numerator numerates or numbers the pieces. Whole means an entire thing, as distinguished from a fragment or piece. Applied to numbers, these terms have no significance. The only quantities that can be cut into pieces are geometrical quantities, or magnitudes, and the pieces are geometrical fractions. A numerical fraction is just as much an entire quantity as a whole number is. A fraction being a number divided by a number, the principles of dividend, divisor, and quotient, apply to numerator, denominator, and value of a fraction. 12 contains 3 four times; 24, or twice times, or twice as many times as 12 does. tains 3 one half as many times as 24 does. of the fraction, this becomes 12, contains 3 eight Conversely, 12 con- Multiplying the numerator of a fraction multiplies the value of the fraction; dividing the numerator, divides the value. 12 contains 3 four times; 12 contains 6 two times. Multiplying the denominator of a fraction divides the value of the fraction; dividing the denominator, multiplies the value. So, to multiply a fraction and a whole number, we multiply the numerator by the whole number, or divide the denominator by the whole number. To divide a fraction by a whole number, we divide the numerator by the whole number, or multiply the denominator by the whole number. From the same general principles, it follows that the value of the fraction (two halves) is twice as great as . = 1, and so is one half of 1. is one third of 1. , being twice as great as, is two thirds of 1. The name of a fraction indicates its value as compared with 1. 12 contains 3 four times; 24 contains 6 four times. Multiplying or dividing both numerator and denominator by the same number does not alter the value of the fraction. The least common multiple of 364 and 280 is 3640. A fraction whose numerator equals or exceeds the denominator is termed an improper fraction; one whose numerator is less than its denominator is a proper fraction. TO REDUCE AN IMPROPER FRACTION TO A WHOLE Perform the division; thus, 21. This is the ordinary case of division. |