Lesson 24 Section 2
FRACTIONS INTO DECIMALS
Exact versus inexact decimals
In the previous section we saw the most frequent
"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 viewpoint that, in the mathematics of computation and measurement, we may say that something exists when we can actually experience it; when we can observe it or name it. Actual infinities -- ".363636 goes on forever" -- we cannot experience. Actual infinities are not required to solve any problem in arithmetic or calculus; they serve no useful purpose, and therefore are not necessary.
What is more, if the decimal really did not end, it would not be a number. Why not? Because, like any number, a decimal has a name. It is not that we will never finish naming an infinite sequence of digits. We cannot even begin.
Answer. According to what we just saw:
Exact versus inexact decimals
which is .25, is exact.
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.
9 goes into 1 zero (0) times.
9 goes into 10 one (1) time with 1 left over.
Again, 9 goes into 10 one (1) time with 1 left over.
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.
Answer. Divide 73 by 96. Press
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.
This is approximately .558.
c) What percent were women?
Answer. To change a number to a percent, multiply it by 100.
.558 = 55.8%
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.
Please "turn" the page and do some Problems.
Continue on to the next Lesson.
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