SOLUTION: write the decimal expansions for these rational number. tell whether the expansions are terminating or repeating. Find the rational number equivalents for these decimal expansio

You didn't give any problems,
so I'll make some up. I hope they are
like the ones you were asked to solve.

All rational numbers expressed
as decimals will either terminate
or repeat a block of digits
forever.  That's because in long
division all the remainders will
always be less than the divisor,
so sooner or later the remainder
will either be 0 or will be the
repeat of a remainder obtained
earlier.

write the decimal expansions for these 
rational numbers. tell whether the 
expansions are terminating or repeating. 

#1.   9/32

Use long division:

    .28125 
32)9.00000
   6 4
   2 60
   2 56
      40
      32
       80
       64
       160
       160
         0

Terminating because the division 
eventually has 0 remainder

#2. 15/22

Again use long division:

     .6818 
22)15.0000 
   13 2
    1 80
    1 76
       40
       22
       18 

This is a repeating decimal because the same 
non-zero remainder 18 occurred twice in the 
long division. These two 18's are indicated in 
red above.


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Find the rational number equivalents for these 
decimal expansions. tell whether they are 
rational or irrational

#3.   .185185185185···

Let N = .185185185185···
There are 3 digits in the repeating block
"185", so multiply by 103 or
1000

1000N = 185.185185185···

Now place the first equation underneath
and subtract the two equations

1000N = 185.185185185···
    N =    .185185185185···
 999N = 185.000000000
 999N = 185
    N = 185/999
that reduces to
    N = 5/27

This is RATIOnal because it is the
RATIO of two integers

#2.  2.9373737837···

Let N = 2.937373737··· 

There are 2 digits in the repeating block
"37", so multiply by 102 or
100

100N = 293.737373737···

Now place the first equation underneath
and subtract the two equations

100N = 293.737373737···
   N =   2.93737373737···
 99N = 290.800000000···
 99N = 290.8
Clear of decimals by multiplying thru
by 10

990N = 2908
   N = 2908/990
that reduces to
    N = 1454/495

This is RATIOnal because it is the
RATIO of two integers
  
#3.  .72772277722277772222···

This is IRRATIONAL because  
there is no block of repeating
digits.

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simplify the following 
radicals. 

#4.  

#5. 

Edwin