Lesson 24 Section 2 ## FRACTIONS INTO DECIMALS## Exact versus inexact decimalsIn the previous section we saw the most frequent |
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"Let 4 fall into the house" 11 does not go into 4. Write 0 in the quotient, place a decimal point, and add a 0 onto the dividend. (Lesson 12) "11 goes into 40 three (3) times (33) with 7 left over. "11 goes into 70 six (6) times (66) with 4 left over." Since we are dividing 11 into 40 again, we see that this division will never be exact. We will have 36 repeated as a pattern:
will ever be complete or exact. However we can approximate it with as many decimal digits as we please according to the indicated pattern; and
That explanation illustrates the view that, in the mathematics of computation and measurement, what What is more, if the decimal really did not end, it would not be a number. Why not? Because a decimal, to be useful, has a name. It is not that we will never finish naming an infinite sequence of digits. We cannot even begin.
want to use that number as a decimal, we must approximate it. Let us approximate it with three decimal digits (Lesson 12):
Exact versus inexact decimals
which is
Fractions, then, when expressed as decimals, will be either exact or inexact. Inexact decimals nevertheless exhibit a pattern of digits. The
Which fractions -- in lowest terms -- will have exact decimals? Only those whose denominators could be multiplied to become a power of 10. For they are the denominators that a decimal fraction is understood to have. They are the numbers whose only factors are 2's and/or 5's; which are the only factors of the powers of 10. Here are a few of the numbers that are composed only of 2's or 5's: A fraction in lowest terms with denominator 6 will not have an exact decimal, because 6 = 2 × 3. It is not possible to multiply 2 × 3 so that it becomes a power of 10. Example 3.
9 goes into 1 9 goes into 10 Again, 9 goes into 10 And so on. This division will never be exact -- we will keep getting 1's in the quotient.
See Problem 15 at the end the Lesson.
Displayed is
Therefore, to three decimal digits,
Example 5. In a class of 52 students, 29 were women. a) What fraction were women?
b) Use a calculator to express that fraction as a decimal.
See
This is approximately c) What percent were women?
(Lesson 4.) In summary, look at what we have done:
"Out of," with a calculator, always signifies division: Division of a smaller number by a larger. We will return to this in Lesson 30.
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