common fractions. To read 0.305, we read "three hundred five thousandths." The denominator is always 1 with as many zeros as decimal places. Thus the denominator for 0.14 is 1 with two zeros, or 100. For 0.003 it is 1,000; for 0.101 it is 1,000; and for 0.3 it is 10. The denominator may also be determined by counting off place values of the decimal. For 0.13 we may think "tenths, hundredths" and the fraction is in hundredths.">
To read a decimal fraction in full, we readboth its numerator and denominator, as in reading common fractions. To read 0.305, we read "three hundred five thousandths." The denominator is always 1 with as many zeros as decimal places. Thus the denominator for 0.14 is 1 with two zeros, or 100. For 0.003 it is 1,000; for 0.101 it is 1,000; and for 0.3 it is 10. The denominator may also be determined by counting off place values of the decimal. For 0.13 we may think "tenths, hundredths" and the fraction is in hundredths. In the example 0.1276 we may think "tenths, hundredths, thousandths, ten-thousandths." We see that the denominator is 10,000 and we read the fraction "one thou- sand two hundred seventy-six ten-thousandths." A whole number with a fraction in the form of a decimal is called a MIXED DECIMAL. Mixed decimals are read in the same manner as mixed numbers. We read the whole number in the usual way followed by the word "and" and then read the decimal. Thus, 160.32 is read "one hundred sixty and thirty-two hundredths." The word "and" in this case, as with mixed numbers, mean6 plus. The number 3.2 means three plus two tenths.
It is also possible to have a complex decimal. A COMPLEX DECIMAL contains a common fraction. The number 0.3 1/3 is a complexdecimal and is read "three and one-third tenths." The number 0.87 1/2 means 87 1/2 hundredths. The common fraction in each case forms a part of the last or right-hand place. In actual practice when numbers are called out for recording, the above procedure is not used. Instead, the digits are merely called out in order with the proper placing of the decimal point. For example, the number 216.003 is read, "two one six point zero zero three." The number 0.05 is read, "zero point zero five."
Decimal fractions may be changed to equivalent fractions of higher or lower terms; as isthe case with common fractions. H each decimal fraction is rewritten in its common fraction form, changing to higher terms is accomplished by multiplying both numerator and denominator by 10, or 100, or some higher power of 10. For example, if we desire to change s to hundredths, we may do so by multiplying both numerator and denominator by 10. Thus,
In the decimal form, the same thing may be accomplished by simply annexing a zero. Thus,
0.5 = 0.50
Annexing a 0 on a decimal has the same effect as multiplying the common fraction formof the decimal by 10/10. This is an application of the fundamental rule of fractions. Annexing two 0’s has the same effect as multiplying the common fraction form of the decimal by 100/100; annexing three O’s has the same effect as multiplying by 1000/1000; etc.
REDUCTION TO LOWER TERMS
Reducing to lower terms is known as ROUND-OFF, or simply ROUNDING, when dealing with decimal fractions. If it is desired to reduce 6.3000 to lower terms, we may simply drop as many end zeros a6 necessary since this is equivalent to dividing both terms of the fraction by some power of ten. Thus, we see that 6.3000 is the same as 6.300, 6.30, or 6.3.
It is frequently necessary to reduce a number such as 6.427 to some lesser degree of precision. For example, suppose that 6.427 is to be rounded to the nearest hundredth. The question to be decided is whether 6.427 is closer to 6.42 or 6.43. The best way to decide this question is to compare the fractions 420/1000, 427/1000, and 430/1000. It is obvious that 427/1000 is closer to 430/1000, and 430/1000 is equivalent to 43/100; therefore we say that 6.427, correct to the nearest hundredth, is 6.43. A mechanical rule for rounding off can be developed from the foregoing analysis. Since the digit in the tenths place is not affected when we round 6,427 to hundredths, we may limit our attention to the digits in the hundredths and thousandths places. Thus the decision reduces to the question whether 27 is closer to 20 or 30. Noting that 25 is halfway between 20 and 30, it is clear that anything greater than 25 is closer to 30 than it is to 20.
In any number between 20 and 30, if the digit in the thousandths place is greater than 5, then the number formed by the hundredth6 and thousandths digits is greater than 25. Thus we would round the 27 in our original problem to 30, as far as the hundredths and thousandths digit6 are concerned. This result could be summarized as follows: When rounding to hundredths, if the digit in the thousandths place is greater than 5, increase the digit in the hundredths place by 1 and drop the digit in the thousandths place.
The digit in the thousandths place may be any one of the ten digits, 0 through 9. If these ten digits are split into two groups, one composed of the five smaller digits (0 through 4) and the other composed of the five larger digits, then 5 is counted as one of the larger digits. Therefore, the general rule for rounding off is Stated as follows: If the digit in the decimal place to be eliminated is 5 or greater, increase the digit in the next decimal place to the left by 1. If the digit to be eliminated is less than 5, leave the retained digits unchanged.
The following examples illustrate the rule for rounding off:
1. 0.1414 rounded to thousandths is 0.141.
Observe carefully that the answer to example 2 is not 3.2. Some trainees make the error of treating the rounding process as a kind of chain reaction, in which one first rounds 3.147 to 3.15 and then rounds 3.15 to 3.2. The error of this method is apparent when we note that 147/1000 is closer to 100/1000 than it is to 200/1000.
Problems of the following type are sometimes confusing: Reduce 2.998 to the nearest hundredth. To drop the end figure we must increase the next figure by 1. The final result is 3.00. We retain the zeros to show that the answer is carried to the nearest hundredth.
Practice problems. Round off as indicated:
1. 0.5862 to hundredths