DECIMALS.

224. A Decimal Fraction or a Decimal is a fraction having for its denominator ten or some power of ten; as 10, 100, 1000, 10000. It expresses one or more of the decimal divisions of a unit.

225. Decimals may be expressed in the same form as common fractions; that 18, with the denominator written. Practically, however, this is never done.

REMARK.-The two points of difference between common and decimal fractions are,

1. The denominator of a common fraction is always written, while that of a decimal is only indicated.

2. The denominator of a common fraction may be any number, while that of a decimal must be 10 or some power of 10.

226. The Decimal Point (.) is a period, and is used to limit the value of of a decimal expression, and to determine the denominator; in this latter relation it takes the place of the unit 1 of the denominator when fully written; as, in the decimal expression .3, read 3 tenths, the decimal point considered as 1 and placed before a cipher, represents the order of its units, and shows that the indicated denominator is 10.

REMARK-When the decimal point is used to separate the integral from the fractional part in mixed decimals, or dollars and cents in decimal currency, it is called a separatrix.

227. Decimals are either pure or mixed.

228. A Pure Decimal corresponds to a proper fraction, the value being less than the unit 1; as, .3, .17, .206, .5191.

A Mixed Decimal corresponds to an improper fraction, the value being greater than the unit 1; as, 17.4, 5.192, 32.3017.

229. The Value of a Decimal is computed from the decimal point, and the orders have the same scale as integers. A removal of the decimal point one place to the right, multiplies the expression by ten; removing it two places, by 100; three places, by 1000, and so on. A removal of the decimal point one place to the left, divides the expression by 10; two places, by 100; three places, by 1000, and so on.

230. From the above it will be observed that if a cipher be placed between the numerical expression of the decimal and the point, the expression being thereby removed one place further from the point, will be divided by 10. But as the value of the decimal expression is computed from the point to the right, it follows that one or more ciphers, placed after the decimal, will not alter its value. is expressed decimally .3; a cipher annexed to the decimal gives .30; two ciphers annexed gives .300 . By this it will be observed that the expressions, though unlike in form are of equal value. Each of the expressions .5, .50, .500, .5000, .50000, .500000, is equal to .

300 = 1000

231. Principles.-1. Decimals increase in value from right to left, and decrease from left to right, in a tenfold ratio.

2. A decimal should contain as many places as there would be ciphers in its denominator if written, the decimal point representing the unit 1 of such denominator.

3. The value of any decimal figure depends upon its place from the decimal point.

4. Prefixing a cipher to a decimal decreases its value the same as dividing it by ten.

5. Annexing one or more ciphers to a decimal does not alter its value.

232. For Notation and Numeration of Decimals we begin with the decimal point as a simple separatrix ; in the integral expression the first place to the left is units (corresponding to the decimal point in the decimal expression), the 2d tens, the 3d hundreds, etc., while from the separatrix to the right we have in order (the point standing for units), tenths, hundredths, thousandths, etc. 233. The Order of a Decimal may be found by numerating either from right to left or from left to right, only let it be remembered that the decimal point stands in the position of the unit 1 of the decimal denominator.

The order of a decimal may usually be determined by inspection, if the fact to be drawn from the following illustration be observed. If .35 be numerated from the right as in integers, the point is in hundreds place; hence, read 35 hundredths; in .1463 the point is in ten-thousands' place, read 1463 ten-thousandths; in .014065 the point is in millions place, and is read 14065 millionths 234. The value of a decimal may be determined by the same numeration as that employed in integers. The relation of orders in a mixed decimal is clearly shown by the following

The above number is read 111 million 111 thousand 111, and 11 million 111 thousand 111 hundred-millionths.

REMARK.-It is better, in reading mixed decimals, to connect the integral and fractional parts by and; as, 2.5 read 2 and 5 tenths; 17.016, 17 and 16 thousandths.

Rule.-I. Numerate from the decimal point, to determine the denom inator.

II. Read the decimal as a whole number, and give to it the denom ination of the right-hand figure.

237. The doubt which often arises in the mind of the pupil as to how a decimal should be written, may be entirely dispelled by keeping in mind the following facts:

1st. That they are fractions.

2nd. That both terms should be written or indicated.

3rd. That the denominator of any decimal (if written) would be 1, with as many ciphers to the right as the decimal contains places.

4th. When the numerator (or decimal) does not contain as many places as the denominator (if written) would contain ciphers, prefix ciphers to make the number of places equal.

EXAMPLE.-Write as a decimal three-tenths.

EXPLANATION.-Observe that in writing three-tenths as a common fraction, the mental operation is as follows: after writing 3, the numerator, you ask yourself 3 what? the answer is, 3 tenths; then the ten is written below as a denominator, thus obtaining. Now reason in the same way regarding the decimal, and after writing 3, the numerator, ask yourself 3 what? and answer, 3 tenths; and indicate it by placing before the three, a decimal point, which rep. resents the 1 of the decimal denominator; notice that the 3 occupies one place corresponding to the one cipher in the denominator.

Again, express decimally 416 thousandths.

EXPLANATION.-Write the 416 and ask what? answer, thousandths, which is determined by numerating from the right; units, tenths, hundredths, and (the point answering to the figure 1 of the denominator) thousandths; then place the point.

REMARK.-By extending and developing this method of writing decimals, the pupil can in a few minutes master the entire matter, so that he can write any decimal as readily and with as great certainty as if it were a whole number.

Rule.-I. Write the decimal the same as a whole number, prefixing ciphers when necessary, to give to each figure its true local value.

II. Place the decimal point before the left-hand figure of the decimal.

7. Eighty-three and five hundred four ten-thousandths.

8. Seven hundred ten and two hundred forty-three hundred thousandths.

10. Forty-five and forty-six thousandths.

11.

12.

One thousand one and one hundred ten-thousandths.

One thousand eight hundred ninety and ninety thousandths.

13. Eight hundred fifty and five hundredths.

8. Twenty-five thousand four hundred and eleven hundredths.

9. Twenty-one and fifteen thousand fifteen ten-millionths.

10. Eighteen thousand eighteen ten-billionths.

15. Fifty-four million, fifty-four thousand, fifty-four and fifty-four million fifty thousand fifty-four ten-billionths.

16.

One hundred three thousand five hundred eighty-seven thousandths. 17. Sixty-four thousand sixty-four hundredths.

18. Two million six hundred four thousand two hundred-thousandths.

19. Nine billion nineteen million twenty-nine thousand thirty-nine millionths.

20. Seventy-seven tenths.

21. Eighty-seven thousand one hundredths.

22. Four hundred seventy-nine million twenty seven thousand four and ninety-nine thousand four ten-billionths.

23. Seventy trillion and seven trillionths.

24. Eleven hundred and eleven ten-thousandths. Three thousand one billionths.

25.

29. Eighteen hundred ninety and eighteen hundred ninety hundred-billionths.

240. Write as decimals the following:

241. To Reduce Decimals to a Common Denominator.

EXAMPLE 1.-Reduce .021, .64, .03705, .5, .17272538, to equivalent decimals having the least common denominator.

OPERATION.

EXPLANATION.-Since the decimal having the greatest number of decimal .02100000 places is hundred-millionths its denominator is the least common denominator .64000000 of the given expressions; this highest decimal contains 8 places, and by adding 5 .03705000 places, or ciphers, to the first, 6 to the second, 3 to the third, and 7 to the fourth, .50000000 all are reduced to 8 places, or to hundred-millionths, which is the least common .17272538 denominator of the given expressions.