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 is an example of mathematical positivism. It asserts that in the mathematics of computation and measurement, what exists is what we actually observe or name, now. That 0.363636 never ends is a doctrine that need not concern us because it serves no useful purpose. Such actual infinities have no practical effect on calculations in arithmetic or calculus. 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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