CHAPTER VI.

DECIMALS.

72. Numbers which denote whole units are called Integral numbers; but it is often necessary to express parts of a unit.

If a unit is divided into two equal parts, each part is called one-half, and is expressed by . If a unit is divided into three equal parts, each part is called one-third, and is expressed by ; two of the parts are called two-thirds, and are expressed by . Again, if a unit is divided into four equal parts, each part is called one-fourth, and is expressed by; into five equal parts, each part is called one-fifth, and is expressed by ; into six equal parts, each part is called one-sixth, and is expressed by ; into seven equal parts, each part is called one-seventh, and is expressed by ; into eight equal parts, each part is called one-eighth, and is expressed by; into nine equal parts, each part is called one-ninth, and is expressed by ; into ten equal parts, each part is called one-tenth, and is expressed by

If AB (see page opposite) represent a unit of length, each division of the line next below AB represents onehalf of a unit; and each division of the second line below. AB represents one-third of a unit; and so on.

How many halves of a unit make a whole unit?

How many fourths make a half? how many make a whole unit?

How many sixths make a third? how many make a half? how many make a whole unit?

How many eighths make a half? a fourth? a whole unit? How many tenths make a fifth? a half? a whole unit?

73. When a unit is divided into ten equal parts, and we wish to express in figures one or more of these parts, we do not usually write them,, etc., but we write 1, 2, 3, etc., and separate the number which denotes parts of a unit from the number which denotes whole units by a decimal point. Thus, two units and three-tenths of a unit are

written, 2.3.

If each tenth of a unit is divided into ten equal parts, that is, the entire unit into a hundred equal parts, each part is called a hundredth of the unit; and if each hundredth is divided into ten equal parts, that is, the entire unit into a thousand equal parts, each part is called a thousandth of the unit; and so on.

These tenth-parts are called Decimal parts, from the Latin word decem, which means ten; and these parts are commonly called Decimal Fractions.

Let AB, for example, represent the unit of length by which a certain distance is to be measured. Suppose the given distance to contain AB 137 times, and a remainder LM to be left, which

is less than AB. Take AC, a tenth of AB, and suppose AC is contained in LM 4 times, with a remainder OM less than AC. Again, suppose AD, a tenth of AC (that is, a hundredth of AB), to be contained in OM 3 times, with a remainder less than AD. And again, suppose a tenth of AD (that is, a thousandth of AB), to be contained in this last remainder 9 times. Then the whole distance expressed in lengths of AB will be 137.439.

The series of figures 137.439 means 1 hundred 3 tens+7 units+4 tenths+3 hundredths 9 thousandths; as 1 hundred = 10 tens 100 units, and 3 tens = 30 units, the integral value is 137 units; so, 4 tenths=40 hundredths 400 thousandths, and 3 hundredths 30 thousandths; the decimal value therefore is 439 thousandths.

If the unit is the yard-stick, the whole is read "one hundred thirty-seven and four hundred thirty-nine thousandths yards"; if the unit is the meter-stick, the whole is read "137 and 439 thousandths meters."

NOTE. The pupil will get the clearest notions of decimals by taking a meter-stick (which is divided in tenths, hundredths, and thousandths) and measuring given lengths; such as, the length of the side of the room, of the platform, of the window-sill, etc., etc., and writing down the result in each case. Whenever the length measured is less than a meter, he should write down 0, and after it the decimal point, then the actual measure. Thus, if the length is found to be 8 tenths 2 hundredths and 7 thou

L

B

D

с

sandths, it is expressed by 0.827, and read "eight hundred twentyseven thousandths of a meter."

=

74. It will be seen that 1 tenth 10 hundredths, 1 hundredth 10 thousandths; and, conversely, 10 thousandths 1 hundredth, 10 hundredths 1 tenth, 10 tenths 1 unit; so that in decimal numbers, as in integral numbers, 10 in any place is equal to 1 in the next place to the left, and 1 in any place is equal to 10 in the next place to the right.

=

Hence figures in the first decimal place denote tenths, in the second place hundredths, in the third place thousandths, in the fourth place ten-thousandths, in the fifth place hundred-thousandths, in the sixth place millionths, and so on.

75. In reading decimals, read precisely as if the decimal were an integral number, and add the name of the lowest decimal place. It is best to pronounce the word "and" at the decimal point, and omit it in all other places. Thus, 100.023 is read one hundred and twenty-three thousandths. Ambiguity in reading, from having zeros at the end of a decimal, is avoided by a pause; thus, 0.300 is read three hundred . . . thousandths, while 0.00003 is read three... hundred-thousandths.

76. Read the following numbers:

0.3; 0.7; 0.65; 0.99; 37.5; 26.9; 425.312; 617.624; 94.57; 83.28; 0.9; 0.96; 57.09; 3.207; 2.03; 3.045; 40.7; 0.055; 0.074; 0.0215; 7.3945; 0.14875; 0.00005; 2.000375; 100.015625; 3.7525; 2.1136257.

77. Express in the decimal notation:

Seven tenths; nine tenths; eleven hundredths; eight hundredths; one hundred thirty-four thousandths; twenty-five thousandths; two hundred and thirty-four thousandths; nineteen and forty-one hundred-thousandths; twenty-five and sixteen ten-thousandths;

thirteen and two hundred one hundred-thousandths; six hundred fifty-eight thousand three hundred forty-two millionths; eighty-six and eight hundred three thousand three hundred four millionths; three and twenty-nine hundredths; fifteen and six hundred seventy-one thousandths; fifty-three ten-thousandths; twenty-two and sixty-seven hundredths; fourteen and two thousand three hundred fifty-one ten-thousandths; two and two hundred nineteen thousandths; three and one hundred fifty-seven thousandths.

78. Zeros occurring at the end of a decimal do not affect its value. Thus, 3.50700 means 3 units + 5 tenths +0 hundredths 7 thousandths +0 ten-thousandths + hundred-thousandths, and is, therefore, 3 and 507 thousandths, the same as 3.507.

79. The arrangement and method of working employed in decimals is precisely like that employed in integral numbers, the decimal point being the only new consideration.

ADDITION OF DECIMALS.

Add 17.5163, 236.3, 1.7162, 0.00132.

OPERATION.

17.5163

236.3

1.7162

0.00132

255.53382

Write the numbers in columns, units under units, tens under tens, tenths under tenths, and so on, so that the decimal points will fall in a vertical line, and add as in integral numbers.