Transcript Decimals - California State University, Fullerton

Decimals
To Do: A restaurant offers a 10% discount. Do you
prefer getting the discount before or after the 8%
sales tax is added on?
First the discount, then the tax:
P  0.9  1.08
First the tax, then the discount:
P  1.08  0.9
Same
Think about how you might respond to a
student who asks you the following question?
Adding and
Why, when we add decimals, we have to
line up the decimal points, while, when we
multiply decimals, we don’t have to line up
the decimal points?
Multiplying Decimals
To Do: Add 0.3 + 0.4 by first
expressing each as a fraction, then
adding the fractions, and finally
convert back to a decimal.
How does this compare with the
way decimal addition usually is
done?
To Do: Add 0.6 + 0.9 by first expressing
each as a fraction.
How does this compare with the
way decimal addition usually is
done?
To Do: Add 0.3 + 0.57 by first expressing
each as a fraction.
How does this compare with the
way decimal addition usually is
done?
To Do: Multiply 0.7×0.43 by first expressing
each as a fraction.
How does this compare with the
way decimal multiplication usually
is done?
m
is in lowest terms. Under
n
m
what circumstances does
have a terminating
n
decimal representation?
To Do: Suppose
Terminating and
Repeating Decimals
Answer: If the only prime divisors
of n are 2 and/or 5.
m
To Do: Suppose
is in lowest terms. Under
n
m
what circumstances does
have a repeating
n
decimal representation?
Answer: If there is a prime divisor
of n other than 2 and 5.
1
To Do: How do we know that
has a
7
repeating decimal representation rather than a
non-repeating decimal representation?
When 10, 20, 30, 40, 50 or 60 is divided
by 7, the remainder can only be one of
the numbers 1, 2, 3, 4, 5 or 6, which
means we’ll again be dividing into either
10, 20, 30, 40, 50 or 60.
m
is a proper fraction for which n is not divisible by 2 or
n
5, then the period of its decimal expansion is how many 9’s
are in the smallest number of the form 9999 for which n
is a divisor.
If
To Do: What is the smallest number of the form 9999 for
which 7 is a divisor?
7 is not a divisor of 9 or 99 or 999 or 9999 or 99999.
However, 7 is a divisor of 999999 since
1
999999 = 7142857. So, has period 6. Indeed,
7
1
= 0.142857142857142857
7
To Do: What is the smallest number of the form
9999 for which 13 is a divisor?
Indeed,
5
=0.384615 (period =6).
13
To Do: What is the smallest number of the form
9999 for which 37 is a divisor?
12
=0.324
Indeed,
37
15
= 0.8823529411764705
17
m
If
is a proper fraction for which n is not divisible by 2 or
n
5, then the period of its decimal expansion is how many 9’s
are in the smallest number of the form 9999 for which n
is a divisor.
1
has period 5 for what n?
n
99999 = 32  41  271
1
 0.02439
41
1
 0.00369
271
Convert 0.29 to a fraction
Let x = 0.29
Then 100x = 29.29
So 99x = 29
29
x
99
To Do: Convert 0.756 to a fraction
Let x = 0.756
Then 1000x = 756.756
So 999x = 756
756 28
x

999 37
m
If
is a proper fraction for which n is not divisible by 2 or
n
5, then the period of its decimal expansion is how many 9’s
are in the smallest number of the form 9999 for which n
is a divisor.
Why does this rule work?
1
Suppose we didn’t know the period of .
7
1
Write = 0.abcdefghijkl
7
1000000
= abcdef.ghijkl
Then,
7
1000000 1 999999
But,
- =
= a whole number
7
7
7
1
= 0.abcdefghijkl
7
1000000
= abcdef.ghijkl
7
1000000 1 999999
- =
= a whole number
7
7
7
Thus, their decimal parts must be equal.
That is, g = a, h = b, i = c, j = d, k = e, and l = f.
Therefore, the decimal repeats every 6 decimal
places.