The Decimal Point is a period (.), which must always be placed before or at the left hand of the decimal. is expressed .6 Thus, NOTE. The decimal point is also called the Separatrix. This is a correct name for it only when it stands between the integral and decimal parts of the same number. And universally, the value of a figure in any decimal place is the value of the same figure in the next left hand place. The relation of decimals and integers to each other is clearly shown by the following And any order of decimals by one figure less than the corre sponding order of integers. 145. Since the denominator of tenths is 10, of hun What is the decimal point? What is it sometimes called? What is the value of a figure in any decimal place? dredths 100, of thousands 1000, and so on, a decimal may be expressed by writing the numerator only; but in this case the numerator or decimal must always contain as many decimal places as are equal to the number of ciphers in the denominator; and the denominator of a decimal will always be the unit, 1, with as many ciphers annexed as are equal to the number of figures in the decimal or numerator. The decimal point must never be omitted. EXAMPLES FOR PRACTICE. 1. Express in figures thirty-eight hundredths. 2. Write seven tenths. 3. Write three hundred twenty-five thousandths. 4. Write four hundredths. 5. Write sixteen thousandths. Ans. .04. 6. Write seventy-four hundred-thousandths. Ans. .00074. 7. Write seven hundred forty-five millionths. 8. Write four thousand two hundred thirty-two ten-thousandths. 9. Write five hundred thousand millionths. 10. Read the following decimals: NOTE. To read a decimal, we first numerate from left to right, and the name of the right hand figure is the name of the denominator. We then numerate from right to left, as in whole numbers, to read the numerator. 146. A mixed number is a number consisting of integers and decimals; thus, 71.406 consists of the integral part, 71, and the decimal part, .406; it is read the same as 71, 71 and 406 thousandths. EXAMPLES FOR PRACTICE. 1. Write eighteen, and twenty-seven thousandths. How many decimal places must there be to express any decimal? 3. Write fifty-four, and fifty-four millionths. 147. From the foregoing explanations and illustrations we derive the following important PRINCIPLES OF DECIMAL NOTATION AND NUMERATION. 1. The value of any decimal figure depends upon its place from the decimal point: thus .3 is ten times .03. 2. Prefixing a cipher to a decimal decreases its value the same as dividing it by ten; thus, .03 is the value of .3. 3. Annexing a cipher to a decimal does not alter its value, since it does not change the place of the significant figures of the decimal; thus,, or .6, is the same as fo, or .60. 4. Decimals increase from right to left, and decrease from left to right, in a tenfold ratio; and therefore they may be added, subtracted, multiplied, and divided the same as whole numbers. 5. The denominator of a decimal, though never expressed, is always the unit, 1, with as many ciphers annexed as there are figures in the decimal. 6. To read decimals requires two numerations; first, from units, to find the name of the denominator, and second, towards units, to find the value of the numerator. 148. Having analyzed all the principles upon which the writing and reading of decimals depend, we will now present these principles in the form of rules. RULE FOR DECIMAL NOTATION. I. Write the decimal the same as a whole number, placing What is the first principle of decimal notation? Second? Third? Fourth Fifth? Sixth Rule for notation, first step? ciphers where necessary to give each significant figure its true local value. II. Place the decimal point before the first figure. RULE FOR DECIMAL NUMERATION. I. Numerate from the decimal point, to determine the de nominator. II. Numerate towards the decimal point, to determine the numerator. · III. Read the decimal as a whole number, giving it the name or denomination of the right hand figure. EXAMPLES FOR PRACTICE. 1. Write 425 millionths. 2. Write six thousand ten-thousandths. 3. Write one thousand eight hundred fifty-nine hundredthousandths. 4. Write 260 thousand 8 billionths. 5. Read the following decimals: 6. Write five hundred two, and one thousand six millionths. 7. Write thirty-one, and two ten-millionths. 8. Write eleven thousand, and eleven hundred-thousandths. 9. Write nine million, and nine billionths. 10. Write one hundred two tenths. Ans. 10.2. 11. Write one hundred twenty-four thousand three hundred fifteen thousandths. 12. Write seven hundred thousandths. 13. Write seven hundred-thousandths. Second? Rule for numeration, first step? Second? Third ? REDUCTION. CASE I. 149. To reduce decimals to a common denominator. 1. Reduce .5, .375, 3.25401, and 46.13 to their least common decimal denominator. OPERATION. .50000 .37500 3.25401 ANALYSIS. A common denominator must contain as many decimal places as is equal to the greatest number of decimal figures in any of the given decimals. We find that the third number contains five decimal places, and hence 100000 must be a common denominator. As annexing ciphers to decimals does not alter their value,(144.,3)we give to each number five decimal places by annexing ciphers, and thus reduce the given decimals to a common denominator. Hence, 46.13000 RULE. Give to each number the same number of decimal places, by annexing ciphers. NOTES. 1. If the numbers be reduced to the denominator of that one of the given numbers having the greatest number of decimal places, they will have their least common decimal denominator. 2. A whole number may readily be reduced to decimals by placing the decimal point after units, and annexing ciphers; one cipher reducing it to tenths, two ciphers to hundredths, three ciphers to thousandths, and so on. EXAMPLES FOR PRACTICE. 2. Reduce .17, 24.6, .0003, 84, and 721.8000271 to their least common denominator. 3. Reduce 7 tenths, 24 thousandths, 187 millionths, 5 hundred millionths, and 10845 hundredths to their least common denominator. 4. Reduce to their least common denominator the following decimals: 1000.001, 841.78, 2.6004, 90.000009, and 6000. What is meant by the reduction of decimals? Case I is what? Give explanation. Rule. |