Division

Date: 11/25/97 at 22:54:55
From: Cyndi Gustafson
Subject: Division

How do you divide 1,000,000,000,000 by 999,999,999,999,999,999,999?

Date: 12/01/97 at 12:47:52
From: Doctor Mark
Subject: Re: Division

Hi Cyndi,

Well, you divide these two numbers the same way you divide any two 
numbers! Of course, this one will be pretty complicated because the 
divisor (the one with the 9's) is so much bigger than the dividend 
(the one with the 0's). Here's how it would go:

                             .000 000 001
           _________________________________________
999...999 | 1,000,000,000,000.000 000 000 000 000 000 000 000
              999,999,999,999,999,999,999
            ______________________________
                                        1

Now we would have to put down a bunch of 0's to the right of that 1 on 
the bottom, and keep going with the division. Unfortunately, my 
computer screen (and yours) won't be able to fit all those zeros 
without really strange things happening, though you could do it on a 
pretty big piece of paper. Let's see if there is another way of 
figuring out what happens.  

I should warn you right away that since your question was so good, the 
answer will take a while to give; you might want to try reading little 
pieces of it at a time. If you are just interested in what the number 
you asked for looked like, it is at the end of the message.  

First off, I don't want to keep writing that big number with all the 
9's, so let's give it a name: I'll call it Cyndi's number.

Do you know that  a fraction is just a division?  If you want to find 
the decimal equivalent of 3/5, for example, you divide 3 by 5 to get
0.600000... (the three dots at the end (...) just mean "goes on like 
this forever"). So 3/5 is just the same thing as 3 divided by 5.

Now suppose we wanted to find what 300/5 was. We could do that on a
calculator, or by hand, to find the answer (60), or we could say 
that since 3/5 is just 0.6000..., and 300 is 100 times bigger than 3, 
300/5 should just be 100 times bigger than 0.6000....  But it's easy 
to multiply 0.6 by 100: just move the decimal point two places to the 
right, one place for every 0 in the 100 we are multiplying by:

   300/5 = 100 x (0.6000...) = 60.0000... = 60.

What does this have to do with your problem?  Lots!  Suppose that 
instead of dividing 1,000,000,000,000 by Cyndi's number, we divided 1 
by Cyndi's number. That looks even more complicated than before, but 
it turns out to make things easier, not harder.

Since 1,000,000,000,000 is 1,000,000,000,000 times bigger than 1, if 
we knew what 1 divided by Cyndi's number was, all we would have to do 
is multiply that answer by 1,000,000,000,000 to get the answer to the 
problem we really want (your original question).

But it's easy to multiply by any power of 10: just move the decimal 
point to the right by a number of places equal to the number of 0's 
in the power of 10. (As in the example of finding 300/5 from the 
answer to 3/5; since 300 is 100 times bigger than 3, and 100 has *two* 
zeros, find the answer to 0.6000..., then move the decimal point *two* 
places to the right to get 60.000 = 60).

So in this case, since 1,000,000,000,000 has 12 zeros, we divide 1 by
Cyndi's number, then move the decimal point 12 places to the right, 
and we will get the answer.

So if we could find out what

          1
  -------------------
    Cyndi's number

was, we would just move the decimal point 12 places to the right, and 
we would find the answer to your question. But the new problem looks 
almost as complicated as the original problem, so what should we do?

One thing we could do is try some smaller numbers made up all of 9's, 
and see if we can find a pattern. Let's try it. If you use your 
calculator, you will find that

   1
   -  = .111111111... (goes on forever, with the 1 repeating)
   9

   1
  --  = .0101010101... (goes on forever, with the 01 repeating)
  99

   1
 ---  = .001001001001001001... (goes on forever with 001 repeating)
 999

If your calculator has enough digits in its display, you would also 
find that

    1
 ---- = .00010001000100010001... (goes on forever with 0001 repeating)
 9999

    1
----- = .00001000010000100001...(goes on forever with 00001 repeating)
99999

Do you see the pattern?

If the divisor has one 9, then the repeating pattern is just 1.
If the divisor has two 9's, then the repeating pattern is just 01.
If the divisor has three 9's, then the repeating pattern is just 001.
If the divisor has four 9's, then the repeating pattern is just 0001.
If the divisor has five 9's, then the repeating pattern is just 00001.

So we see that if the divisor has a certain number of 9's, the 
repeating pattern is just (that number minus one) zero's, ending 
with a 1.

[Do you see that the pattern for 1/9 fits this?  There is one 9 in the
divisor, so we would expect there to be 1 - 1 = 0 "zeros" in the 
repeating pattern: that is, *no* zeros, then a 1: that's just the 
number 1!]

So we expect that if we were to find 1/999999 (the divisor has six 
9's), we would get a repeating pattern with *five* zeros (since 
6 - 1 = 5), then a 1, and that's what happens. You might not be able 
to see that on your calculator, but if you do it out by hand on a 
piece of paper, it's pretty easy to convince yourself this is true.  
You might also try it on a computer at home or at school.

So now back to Cyndi's number, 999,999,999,999,999,999,999. That 
number has twenty-one 9's, and so if we do the division, we would 
expect to find a repeating pattern of 20 zeros, then a 1:

000 000 000 000 000 000 001,

so we would expect to get

          1
------------------- =
   Cyndi's number

   = .000 000 000 000 000 000 001 000 000 000 000 000 000 001...

where the pattern "(20 zeros)1" repeats forever.

Do you remember that we really wanted to find 1,000,000,000,000/
Cyndi's number? That means we want to take the answer we just found, 
and move the decimal point 12 spaces to the right, since there are 
12 zeros in 1,000,000,000,000. If we do that, we find the answer to 
your question:

1,000,000,000,000/999,999,999,999,999,999,999 =

   = .000 000 001 000 000 000 000 000 000 001 000 000 000 000 000 000 
001...

or, written in another way:

= .(8 zeros)1(20 zeros)1(20 zeros)1... (repeats forever with the 
pattern "(20 zeros)1" ).

Whew!  A really good question, Cyndi, and I'm sorry that it took so 
long to explain, but sometimes really simple questions have answers 
that are more complicated than one might expect.

Later on, when you get to algebra, you will see a simpler way to do 
all of this, and will find out that any fraction can be represented by 
a decimal that has a repeating pattern. For instance, the fraction 1/7 
has a repeating part that is 142857:

  1
----- = .142857142857142857142857142857142857...
  7

There's a lot of interesting math here, much of it discovered by a 
German mathematician named Gauss, who got interested in these problems 
when he was about 6 years older than you, so you've already got a head 
start on him!

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/