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 is a fact. It is possisble to witness that decimal approximation and produce it. We will now explain why we do not say that .363636 goes on forever.
To begin with, representing a number as a decimal began as an idea. "Let's represent a fraction as a decimal." And there is a method. It does not take long to realize that will never be exact, and that .363636 is an approximation. That is in contrast to what is only an idea, namely that .363636 goes on forever. Obviously, we cannot witness that nor can we produce an infinite sequence of digits.
We are taught, of course, that .363636 goes on forever, and so we think that's mathematics -- that's the way things are. We do not realize that it is a human invention. It is an idea. This writer asserts that an idea that we can actually bring into this world -- ".363636"-- has more being for mathematics than what is only an idea.
What is more, infinite decimals are not required to solve any problem in arithmetic or calculus; they have no consequences and therefore they are not even necessary.
Even if we imagine that the decimal did go on forever, then 1) it would never be complete and would never equal ; and 2) 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:
Solution. We are to divide 309 by 16.
Exact versus inexact decimals
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 6. 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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