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28 | THe Marine REVIEW SCIENTIFIC LAKE NAVIGATION. By Clarence E. Long. Decimal Fractions. PART V. The decimal system is a system of reckoning by tens or tenths. Decimal arithmetic or decimal fractions' is one of the most important branches of mathematics used in connection with the practice of navigation; +and since calculating with decimals enters so largely in the prob- lems of navigation, the student should thoroughly master the subject. A decimal is founded on the number 10; proceeding by powers of 10, or one-tenth. A decimal fraction, is a fraction whose denominator is 10 or a power of. 10; it is usually written without the denominator; the number of ciphers in the denominator being indicated by the number of places occupied by the numerator preceded if necessary by ciphers, and placed after a point or a per.od. Thus, .5 = 5-10, read five-tenths; .05 = 5-100, read five-hundredths; .005 = 5-100, read five-thousandths, etc. Common fractions, as we have seen, are expressions for any assighable part of a unit; they are usually denoted by two numbers, placed the one above the other with a line between them; thus, 14 denotes the fraction one-fourth, or one part out of four of some whole quantity considered divisible into four equal parts. The lower number 4, is called the denominator of the fraction, showing into how many parts the whole is divided; and the upper number, 1, is called 'the numerator, and shows how many of those equal parts are contained in the fraction. And it is evident that if the numerator and denominator be varied in the same ratio, the value of the fraction will remain unaltered ; thus, if the numerator and denominator of-the fraction %, be multiplied by 2, 3; or 4, etc., the fractions arising will be 2-8, 3-12, 4-16, etc., which are evidently equal to \%. A decimal fraction is a fraction whose denominator is always a unit with some number of ciphers annexed, and the numerator any number whatever, as 3-10 (.3, read three-tenths); 7-100 (.07, read, seven-hundredths) ; 15-1000 (015, read fifteen-thousandths);-etc. And as the de- nominator of a decimal is always.one of the numbers 10, 100, 1,000, etc., the inconvenience of writing the denomin- ator may be avoided by placing a point between the whole and the fractional part of the number; thus 3-10 is written .3 and 14-100 is written .14; the mixed number 3 14-100, consisting of a whole number and a tractional one, is _ written 3.14. _ Decimal place, the place of a figure after the decimal (point, as first, second, third,: place, etc. Decimal point, a dot or period used to separate a de- cimal fraction from a whole number, or to indicate its fractional character when standing alone. In setting down a decimal fraction, the numerator must _consist of as many places as there are ciphers in the , denominator, and if it has not so many figures, the defect "must be supplied by placing ciphers before it; thus, 14-100 =.14; 14-1000--.014; 14-10000--.0014, etc. And as ciphers on the right hand side of whole numbers increase their value in a tenfold proportion, as.2, 20, 200, etc., so when set on the left hand of decimal fractions they decrease their value.in a tenfold proportion, as .2, .02, 002, etc., but ciphers set on the right hand of these fractions make no alteration in their value, neither of increase or de- » €rease;_thus, .2 is the same as .20 or .200. As we have seen in common fractions, the unit or whole number (represented by the denominator) may be divided into any conceivable number; but not so in decimal frac- tions--the denominator is based upon the power of 10, or the unit is divided into either 10's, 100's, 1000's, etc. The numerator may be the same in either case. This notation and numeration of the decimal denominator into 10, 100, 1000, etc., is what render decimals so serviceable, since the common arithmetical operations can be performed the same way as they are in whole numbers; regard' being had only to the particular notation to distinguish the whole from the fractional part of a sum. This is made possible on account of the denominator, which is omitted, but indicated by the proper placing of the decimal point. The decimal point with the decimal places following in- dicate 'what the denominator is--you read the denomin- ator but you do not write it. Thus, .7 is read seven- tenths; 7 is the numerator, 10 is the denominator; 10 isn't written but the period before the 7 indicates what the denominator is. It must be 10, since we know the. denominator consists of one more figure than the numer- ator. The denominator always consists of the figure 1 with a number of ciphers attached, the number of ciphers depending on the number of decimal places or figures in the numerator, or the number of figures to the right of the decimal point. In every case let the figure 1 of the denominator stand for the decimal point in the numerator; then for every decimal place to the right of the decimal point of the - numerator, place a cipher to the right of the figure 1 and 'the result is your denominator. Or, you read first the figure just as it stands; this is the numerator of your fraction, then you read your denominator, which depends for its amount on the number of decimal places to the right of the decimal point. The denominator will always be the figure I with a cipher for each decimal place in the numerator. The division of anything into tenths, hundredths, thousandths, etc., ate called decimal divisions. One or more of the decimal divisions of a unit are called a decimal fraction. Since tenths are equal to ten times as many hundredths, hundredths equal to ten times as many thousandths, etc., decimals have the same law of increase. and decrease as do whole numbers, and the denominator may be indi- cated by the position of the figures to the right of the decimal point, thus rendering it unnecessary to write the denominator. In the decimal system of notation, the representative value of a figure is decreased tenfold by each removal one place to the right; hence: The figure at the right of units expresses tenths. The figure at the right of tenths expresses hundredths. The figure at the right of hundredths expresses thou- sandths, : The figure at the right of thousandths expresses ten- thousandths, The figure at the right of ten-thousandths expresses hun- dred-thousandths, ete. The first place is tenths; 2 places hundredths; 3 places thousandths; 4 places ten-thousandths, etc., that is, the number of places or figures, to the right of the decimal point. Rule.--Read the decimal just as you would a whole number, and give it the denomination of the right-hand figure, that is, if the right-hand figure is only one place from the decimal point, the denominator is tenths; if two places from the decimal point the denominator is