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INTRODUCTION TO DECIMALS
In this section, we will explore relationships between
fractions and decimals and see how decimals
are an extension of the baseten number system . The word decimal comes
from the Latin decem,
meaning “ten”.
RELATING TO MONEY
One of everyone’s first encounters with a decimal is in the context of money. A
child sees the
price of an IPOD Shuffle is $79.84. Let’s look at different ways to express this
money amount.
$79.84 = “______________________________________________________” (words)
$79.84 = ______________________ Because $0.84 is __________________of a dollar.
$79.84 = __________ dollars + __________ dimes + __________ pennies
__________ + _______ . ____________ + _____________ . _______________
Because a dime is _____________ of a dollar and a penny is _________________ of
a dollar.
EXPANDED NOTATION
$79.84 = __________ + __________ + ________ . _________ + ____________ .
________________
= _____________ + _____________ + _____________ + _____________
READ THE FOLLOWING NUMBERS. THEN WRITE IN EXPANDED NOTATION USING
POWERS OF 10.
12.659 = _____________________________________________________________
409.0783 = _____________________________________________________________
0.54321 = _____________________________________________________________
2,084.08 = _____________________________________________________________
CONCRETE REPRESENTATIONS
We can use Base10 blocks to help model decimals. However, we might need to
change what each
block represents from problem to problem. The unit block should always represent
the “smallest
place” in the number. Draw the Base 10 block representation of each.
CONVERTING FRACTIONS TO DECIMALS
1. DENOMINATOR IS A POWER OF 10
Read the fraction and write as a decimal.
2. DENOMINATOR IS NOT A POWER OF 10
If the denominator is not a power of 10, as in 3/5, we can convert it so it
does.
which is ________ (as a decimal)
Before we take this further, let’s look at the powers of 10 and see what they
have in
common . Write the prime factorization of each of the following :
10
100
1,000
10,000
What do we see in common?
How is the exponent (s) related to the number of zeros in the original number?
Using what we just concluded above, we can easily change the following fractions
to decimals.
Notice that in all the above examples we were able to
change the denominator to some power of
10. The original denominators were all combinations of 2’s and/or 5’s and we
simply incorporated
in through multiplication enough 2’s or 5’s to make a power of 10. They are all
examples of
terminating decimals – decimals that can be written with a finite number of
places to the
right of the decimal point.
But what about:
What can we multiply 3 by to get a power of 10?
(This is an example of a decimal that does not terminate.)
The leads us to the following theorem:
THEOREM A rational number in simplest form
can be written as a terminating decimal if, and only if, the prime factorization of the denominator contains no primes other than 2 or 5. 
NOTE: To determine whether a rational number can be
represented as a terminating
decimal, we only need to consider the prime factorization of the denominator,
BUT
The fraction must be in simplest ( reduced ) form first.
Which of the following can be written as a terminating decimal? Explain why or
why not.
ORDERING TERMINATING DECIMALS
1. Line up the numbers by place value
2. Start at the left and find where the digits differ
3. Compare these digits. The number with the larger digit is greater
4. NEGATIVES ARE IN REVERSE
0.543 _______ 0.563 2.765 _______ 3.765 0.0087 _______ 0.087
2.34 _______ 2.35 43.566 _______ 34.566
Order from greatest to least :
3.45
3.35
3.42
3.451
3.358
3.405
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