INTRODUCTION TO DECIMALS.

MENTAL EXERCISES.

1. If a unit is divided into 10 equal parts, what is one of these parts called?

2. If one-tenth is divided into 10 equal parts, what is one part called? 2 parts? 3 parts?

3. If is divided into 10 equal parts, what is one part called? ? parts? 4 parts?

4. What part of 4 is 4 tenths? of 5 is 5 tenths? of 5 tenths is 5 hundredths? of 6 tenths is 6 hundredths?

5. What is of 3 hundredths? What part of 3 hundredths is 3 thousandths? of 6 thousandths is 6 ten-thousandths?

6. In the number 4444, the 4 units is what part of the 4 tens? the 4 tens is what part of the 4 hundreds ?

7. A term in units place denotes what part of the value which it does in tens place?

8. A term in tens place denotes what part of the value which it does in hundreds place?

9. By the same law, a term to the right of units place would denote what part of its value in units place? Ans. One-tenth.

10. How may we indicate that a term is at the right of units place? Ans. By placing a dot (.) at the right of units place.

11. How will you express 2 in this manner? Ans. 2.5.

12. What does the dot between the 2 and the 5 denote? Ans. That 2 is in units place and 5 in tenths place.

13. In the expression 11.1, the 1 at the right of units denotes what part of a unit?

14. In the expression 11.11, the second term at the right of the period denotes what part of a tenth? what part of a unit?

15. How may we then write tenths, hundredths, etc., without a denominator? Ans. By writing them at the right of units.

16. What shall we call the first place at the right of units? ond place? Third? Fourth? Fifth?

Sec

17. Write without a denominator 2 tenths; 3 tenths; 4 tenths; € tenths; 8 tenths; 3 hundredths; 5 hundredths; 1 thousandth; 3 thou sandths; 6 thousandths.

18. Read the following expressions: 4.6; 25.45; 26.34; 18.05, 25.235; 36.205; 46.008.

19. These fractions arising from the successive division by 10 are called decimal fractions. The term decimal is derived from decem, meaning ten

SECTION V.

DECIMAL FRACTIONS.

195. A Decimal Fraction is a number of tenths, hun dredths, thousandths, etc.

196. A Decimal Fraction is usually expressed by plac ing a point before the numerator and omitting the denomina zor; thus .5 expresses; .05 expresses ; .005 10%, etc.

197. The Symbol of a decimal is the period, called the decimal point, or separatrix. It indicates the decimal, and separates decimals and integers.

198. The places at the right of the decimal point are called decimal places. The first place to the right of the decimal point is tenths, the second place is hundredths, etc. Thus, .245 expresses 2 tenths, 4 hundredths, and 5 thousandths.

199. The method of expressing decimal fractions arises from the method of notation for integers, and is a continua tion of it. This beautiful law, as applied to integers and fractions, is exhibited in the following

NOTATION AND NUMERATION TABLE.

200. A Decimal is a decimal fraction expressed by the method of decimal notation; as, .5, .25, etc.

201. A Pure Decimal is one which consists of decimal figures only; as, .345.

202. A Mixed Decimal is one which consists of an integer and a decimal; as 4.35.

203. A Complex Decimal is one which contains a common fraction at the right of the decimal; as, .45}.

NOTES.-1. The first treatise upon decimals was written by Stevinus, and published in 1585.

2. The decimal point, Dr. Peacock thinks, was introduced by Napier, the Inventor of logarithnis, in 1617, though De Morgan says that Richard Witt made as near an approach to it as Napier.

EXERCISES IN NUMERATION.

1. Read the decimal .45.

SOLUTION.-This expresses 4 tenths and 5 hundredths, or since 4 tenths equals 40 hundredths, and 40 hundredths plus 5 hundredths equal 45 hundredths, it may also be read 45 hundredths. Hence the following rules:

Rule I.-Begin at tenths, and read the terms in order towards the right, giving each term its proper denomination. Rule II.-Read the decimal as a whole number, and give it the denomination of the last term at the right.

NOTE.-In the second method we may determine the denominator by numerating from the decimal point, and the numerator by numerating towards the decimal point.

EXERCISES IN NOTATION.

1. Express 25 thousandths in the form of a decimal.

SOLUTION.-25 thousandths equal 20 thousandths plus 5 thousandths, or 2 hundredths and 5 thousandths; hence we write the 5 in the third or thousandths place, 2 in the second or hundredths place, and fill the va cant tenths place with a cipher, and we have .025. Hence the following rules:

Rule I.-Place the decimal point, and then write each term so that it may express its proper denomination, using ciphers when necessary.

Rule II.— Write the numerator, and then place the decimal point so that the right-hand term shall express the denom ination of the decimal, using ciphers when necessary.

NOTE.-To avoid ambiguity, where integers and decimals occur in the same written number, a comma should be placed between them; thus, three hundred and seven ten-thousandths should be written .0307, but three bundred, and seven ten-thousandths, 300.0007.

Express the following in decimal form:

2. Twenty-five hundredths. 3. 2 tenths and 8 hundredths. 4. 7 tenths and 9 hundredths. 5. Twenty-five thousandths. 6. 4 tenths and 7 thousandths. 7. Seven tenths and 8 thousandths.

8. Five hundred, and 25 thousandths.

9. Three tenths and 7 ten-thousandths.

10. Seven hundredths and 9 tenthousandths.

11. Three hundred, and 78 tenthousandths.

12. Five tenths, 6 hundredths, and 7 hundred thousandths.

13. Four hundredths, seven ten thousandths, and 6 hundred-thou sandths.

14. Nine hundred and sixtynine hundred-thousandths..

15. Two tenths and three millionths.

16. Five hundredths, six thousandths, and eight millionths.

17. Thirty-five thousand, and eight millionths.

18. Ninety-three hundred and seven ten-millionths.

19. Eighteen thousand and one hundred-millionths.

20. Two million, and 6 thou sand and 9 hundred-millionths.

1. Moving the decimal point one place to the right, multi plies the decimal by 10; two places, multiplies by 100; etc For, if the point be moved one place to the right, each figure will express ten times as much as before, hence the whole decimal will be ten times as great; etc.

2. Moving the decimal point one place to the left, divides the decimal by 10; two places, divides by 100; etc.

For, if the point be moved one place to the left, each figure will express 1 tenth of its previous value, hence the whole decimal will be only 1 tenth as great; etc.

3. Placing a cipher between the decimal point and the decimal, divides the decimal by 10.

For, this moves each figure one place to the right in the scale, in which case they express 1 tenth as much as before, and hence the decimal is only 1 tenth as great.

4. Annexing ciphers to the right of a decimal does not change its value.

For, each figure retains the same place as before, and hence expresses the same value as before, and consequently the value of the decimal is unchanged.

MENTAL EXERCISES.

1. How many tenths in 17 in ? in ? in ? 2. How many hundredths in ? in

3. How many hundredths in ? in

? in ? in? in ff?

? in? in? in ?

4. How many halves in .5? in .50? 5ths in .2? in .4?

5. How many 4ths in .25? in .75? eighths in .125? in .375? 6. Express as a common fraction and reduce to lowest terms, .5; .4; .8; 25; .50; .75; .80; .125; .625.

REDUCTION OF DECIMALS.

204. There are two cases of the reduction of decimals: 1st. To reduce decimals to common fractions.

2d. To reduce common fractions to decimals.

CASE I.

205. To reduce a decimal to a common fraction. 1. Reduce .45 to a common fraction SOLUTION.-.45 expressed in the form of a common fraction is, which reduced to its lowest terms equals. Hence the following

OPERATION.

.45%, Ans.

Rule. Write the denominator under the decimal, omit ting the decimal point, and reduce the common fraction to its lowest terms.