NUMERATION OF DECIMAL FRACTIONS. By reversing the table for Notation of Decimal Fractions, (p. 17,) we obtain the SECOND NUMERATION TABLE: FOR DECIMAL FRACTIONS. To be read from decimal point downward. • DECIMAL POINT. Digits to the right of decimal point stand for decimals. Hundredths. Thousandths. Ten-thousandths. Hundred-thousandths. Digits in first place of decimals represent Tenths. &c. 66 Millionths. Ten-millionths. &c. tor may be named separately. 1. The value of each significant figure in the numera read in the name of the right hand one. 2. The whole of the digits in the numerator may be 1000, and 1000; or, 8432 may be read collectively, as Thus 8432 may be read either separately, as 10, 180, Every decimal fraction may be read in two ways. should always be named, whether written or not. In reading off decimal fractions their denominators ten-thousandths, 3. The latter method is the best, and therefore should be commonly employed, but the former is sometimes required.* EXERCISE XIX. Read, or write in words, first taking the digits separately and then collectively, each of the following decimal fractions: Write the above with their denominators restored. Read, or write in words, in the same manner, the following numbers, consisting of integers and decimals: DIVISION OR PARTITION OF AN INTEGER. The division of a unit is the cutting of it into equal parts, as explained on page 12. The division of an integer, or number of units, is the distribution of it into equal lots of units.† Halves of an Integer. Take two equal strips of paper, or other convenient units, and place them before you at a little distance from each other. The two units are now distributed into two equal portions. Each of these portions is one half of the two units, and consists of one unit. Take other two similar units. Put one of them with each of the former. Here are now two twos, or four, divided into two equal lots, each containing two units. .. One half of two twos = two units. Dispose similarly of a third two, and a fourth two, and * NOTES TO TEACHER. Decimal fractions should always be read with their denominators whether so written or not: for example, 2 should always be read "two tenths or twenty one-hundredths," and so on; and never (as is the practice with some arithmeticians) "decimal two." A still more objectionable habit is that of reading such an expression as 842, as "decimal eight hundred and forty-two." The true character of the decimal fraction is constantly kept before the pupil's mind by supplying its denominator in reading it. See subsequent remarks on the true idea of Division, note under elementary rules. The pupil must always read in words every arithmetical symbol he meets with. a fifth two, and so on until you have divided ten twos. And as you proceed, you will remark that of three twos of ten twos of five twos = 5. 6. = 10. of six twos So that the half of an integer is found by distributing it into two equal lots of units. And this is done by separating every two of its units into two portions of one unit each, and then collecting those units into two separate lots. PRINCIPLE III. There are as many units in the half of a number as there are twos in the whole of it. If, after distributing, as above directed, all the twos that can be got from a number, a single unit should remain, that unit must be divided into two halves, (page 12,) and one of these halves must be put with each of the two equal lots of units. Thus, to find the half of eleven, which consists of five twos and one unit: First. Distribute the five twos into two lots of 5 each. Second. Put with each of these lots one half of remaining unit. ..of 11. = = 5* This result quite agrees with Principle III. For 11 consists of five twos and one unit; that is, of five twos and the half of two. And the half of eleven consists of five units and the half of a unit. Therefore, there are as many units in the half of 11 as there are twos in the whole of it. Thirds of an Integer. The third of an integer is found by distributing all its threes into three equal lots of units, and putting with each of those lots of each unit remaining after all the threes are divided. Fourths, Fifths, Sixths, &c., of an Integer. The fourth part of an integer is found by distributing its units into four equal lots, of units and fourths. The fifth part of an integer is found by distributing its units into five equal lots, of units and fifths. The sixth part of an integer is found by distributing its units into six equal lots, of units and sixths. The seventh part of an integer is found by distributing its units into seven equal lots, of units and sevenths. And so on with other parts of an integer. The name of the part indicates how many of the units shall be distributed at once; *NOTE TO TEACHER. The decimal point is not required when the units are followed by a vulgar fraction. for, to obtain the third of an integer, we take its units three at a time; to obtain its fifth, we take its units five at a time; to obtain the tenth, we take the units ten at a time. The name of the parts into which the integer is to be divided determines also the fractions into which each of the remaining units is to be cut. EXTENSION OF PRINCIPLE III. There are as many units in the third of a number as there are threes in the whole of it; as many units in the fourth as fours in the whole; as many units in the tenth as tens in the whole; &c. &c. Concrete and Abstract Numbers. A number which expresses one particular kind of unit is a concrete or applicate number. A number which expresses no particular kind of unit is an abstract number. Thus, 24 is an abstract number, because no particular kind of unit is expressed by it; it represents 21 of any or every sort of unit. But 24 shillings, or 24 yards, or 24 pounds is an applicate or a concrete number, because the 24 is applied to the expression of 24 units of a stated kind; namely, shillings, or yards, or pounds. EXERCISE XX. QUESTIONS, AND EXAMPLES IN ABSTRACT NUMBERS. 1. What is meant by the division of a unit? 2. What does the division or partition of an integer mean? 3. How is the half of a unit found? 4. How is the half of an integer procured? 5. How many units in the half of an integer? 6. What is the half of nineteen twos? 7. What is the half of 6 of 9 of 7 of 14 of 15 of 20 of 23.? 8. How is the third of an integer found? 9. What is the number of units in the third of an integer ? 10. What is the third of nine threes? 11. How many units and thirds of a unit in one third of 7 of 8. of 10 of 12 of 14 of 16 of 20 of 22 of 25.? 12. What is the third of two units? 13. What is the third of 5 of 4 of 6 of 9 of 11 of 13 of 15.? 18. By Principle III. How many units are there in one tenth of a number? 19. If I distribute the units of a number into nine equal lots, what part of the integer is each of those lots? 20. What number of units will there be in each of those lots? 21. In dividing an integer, what is it decides how many equal lots the units shall be distributed into? 22. What else is decided by the same thing? 23. How many units are distributed at a time in finding the twelfth of an integer? 24. If I take the units of an integer nine at a time, and distribute each nine into units, what part of the integer shall I find? EXAMPLES IN CONCRETE OR APPLICATE NUMBERS. 25. What is one half of 10 pence, of 9 pence, of 5 pence, of 11. pence? 26. What is of 5 pounds of beef? What is of 7 yards of carpet? 27. If 14 pounds of sugar were divided into three equal parts, how much sugar would be in each lot? 28. Five sticks of equal length will reach, end to end, seven feet; how long is each stick? 29. Find of 8 ounces; of 9 gallons; of 10 minutes. 30 Find 1 of 14 pounds; of 11 shillings; r of 29 sheets of paper. 31 Share 5 apples of equal size equally among 3 boys. 32. Fourteen pounds of cherries are to be divided equally among 3 boys: what weight for each? 33. Nineteen hundred-weights of soap are packed in 7⚫ equal-sized boxes: how much in each box? THE PRINCIPLE OF DECIMAL NOTATION. Because, by Principle III., there are as many units in one tenth of a number as there are tens in the whole of it, then of 10,000. there are a thousand tens in 10,000. ... 1,000 is ...there are ten thousand tens in 100,000... 10,000 is of 100,000. And Table II. for comparison of decimal fractions, page 19, shews that •1 of 1. = 001 = -01 of 1-0001 of 01 of 001 = =of 001000001 Hence, in a line consisting of digit 1 repeated in adjoining places of integers and decimals, as 1111111 111111 each digit has one-tenth of the value of its left-hand neighbour, and tenfold the value of its right-hand neighbour. D |