Exploring Decimal Fractions

2225 days ago by kcrisman

Decimal Fractions - Our First Exploration

Now we will dispense with preliminaries and start our explorations.   Hooray!

What we will be exploring is a very simple topic - decimal fractions.  Like this.

1/7 
       

These are fractions in base 10.  Just regular old fractions.  Good ol' fractions.

We want to know about their decimal representations.  There is really only one tool you will need to explore this.

n(1/7) 
       
0.142857142857143
0.142857142857143

The "n" function numerically approximates.  If you want more digits, just ask for them:

n( 1/7 , digits=50 ) 
       
0.14285714285714285714285714285714285714285714285714
0.14285714285714285714285714285714285714285714285714

You might want to be able to do a few of them at a time.  I don't care how you do this.  

  • You could do them one at a time, just below each other in different cells.  Cut-and-paste makes this not too tedious.
n(1/7,digits=50) 
       
0.14285714285714285714285714285714285714285714285714
0.14285714285714285714285714285714285714285714285714
n(2/7,digits=50) 
       
0.28571428571428571428571428571428571428571428571429
0.28571428571428571428571428571428571428571428571429
n(3/7,digits=50) 
       
0.42857142857142857142857142857142857142857142857143
0.42857142857142857142857142857142857142857142857143
n(4/7,digits=50) 
       
0.57142857142857142857142857142857142857142857142857
0.57142857142857142857142857142857142857142857142857
  • You could use a "loop" syntax, if it looks familiar.  "[1..6]" just means the integers 1 to 6.
for i in [1..6]: n(i/7,digits=50) 
       
0.14285714285714285714285714285714285714285714285714
0.28571428571428571428571428571428571428571428571429
0.42857142857142857142857142857142857142857142857143
0.57142857142857142857142857142857142857142857142857
0.71428571428571428571428571428571428571428571428571
0.85714285714285714285714285714285714285714285714286
0.14285714285714285714285714285714285714285714285714
0.28571428571428571428571428571428571428571428571429
0.42857142857142857142857142857142857142857142857143
0.57142857142857142857142857142857142857142857142857
0.71428571428571428571428571428571428571428571428571
0.85714285714285714285714285714285714285714285714286
  • Or you could use a "list comprehension", if you feel comfortable with that.
[ n(i/7,digits=50) for i in [1..6] ] 
       
[0.14285714285714285714285714285714285714285714285714,
0.28571428571428571428571428571428571428571428571429,
0.42857142857142857142857142857142857142857142857143,
0.57142857142857142857142857142857142857142857142857,
0.71428571428571428571428571428571428571428571428571,
0.85714285714285714285714285714285714285714285714286]
[0.14285714285714285714285714285714285714285714285714, 0.28571428571428571428571428571428571428571428571429, 0.42857142857142857142857142857142857142857142857143, 0.57142857142857142857142857142857142857142857142857, 0.71428571428571428571428571428571428571428571428571, 0.85714285714285714285714285714285714285714285714286]

I don't care which one you do.  Eventually you will get curious and ask how to use programming to help make you more efficient!  But for now, just do what feels comfortable.

 
       

Your goal:

  • To find out absolutely everything you can about patterns in the decimal expansions of fractions.
    • For instance, it's pretty clear that all of the fractions above repeat every six numbers.
    • You might notice another pattern about the digits which actually occur. 

As you come up with ideas, you can let me know and I'll put investigation of that here.  

  • But I want the patterns and questions YOU find, not ones I guide you into.
  • The only other thing I'll say is that you should look for patterns that are general.  
    • Don't just say something about $\frac{i}{7}\ldots$
    • Say something about numbers with an odd denominator, or prime denominator, or about all numbers with the same denominator.  
    • That's the sort of thing that's worth proving.

Good luck!  It should be a lot of fun.