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Converting from Base 6 to 10 and Back


Date: 09/26/2001 at 17:33:10
From: leah 
Subject: Converting base 6 to 10 and the reverse

I need to know how to convert 2411 in base 6 to base 10.


Date: 09/27/2001 at 12:18:56
From: Doctor Greenie
Subject: Re: Converting base 6 to 10 and the reverse

Hello, Leah -

There are two basic methods for converting a number from base 6 (or 
any other base) to base 10; and there are two basic methods for 
converting a number the other way - from base 10 to another base.  
Converting in either direction, one of the two methods is rather slow 
and inefficient, but it is very good for helping to understand the 
process, because it helps teach place values in different number 
bases; in both cases the second method is much faster and more 
efficient.

I will work an example in both directions using both methods. It is 
probably best to use the slower method when you are first learning how 
to convert numbers from one base to another; then, when you have done 
enough problems that you understand the process, you can use the 
faster method to save time. (I work with computers, and I frequently 
have to convert numbers from base 10 to base 16 or the other way 
around; since this is something I do often, I use the faster method to 
save time. But when I was first learning how to convert numbers from 
one base to another, I used the slower method.)

Here is the problem I am going to work: Convert 1524 (base 6) to base 
10; and then convert my result in base 10 back to base 6. I will first 
work the problem using the two slow methods; then I will work the same 
problem using the more efficient methods.

SLOW METHODS

In base 10, the place values are

     1
     10
     10^2 = 10x10 = 100
     10^3 = 10x10x10 = 1000
     etc.

In base 6, the place values are

     1
     6
     6^2 = 6x6 = 36
     6^3 = 6x6x6 = 216
     etc.

*** Base 6 to Base 10, Slow Method
----------------------------------

Let's start by considering the base 10 number 1524; let's analyze it 
using the place values for base 10:

    1524 (base 10) =

    1000 (base 10) = 1x1000 = 1000 (base 10)
   + 500 (base 10) = 5x 100 =  500 (base 10)
   +  20 (base 10) = 2x  10 =   20 (base 10)
   +   4 (base 10) = 4x   1 =    4 (base 10)
   ---------------------------------------
                            = 1524 (base 10)

To convert the number 1524 (base 6) to base 10, we need to use the 
place values for base 6:

    1524 (base 6) =

    1000 (base 6) = 1x216 = 216 (base 10)
   + 500 (base 6) = 5x 36 = 180 (base 10)
   +  20 (base 6) = 2x  6 =  12 (base 10)
   +   4 (base 6) = 4x  1 =   4 (base 10)
   ---------------------------------------
                          = 412 (base 10)

So the base 6 number 1524 is equivalent to the base 10 number 412.

This completes the process of converting a number from base 10 to 
base 6 using the slow place-value method. Now let's convert the base 
10 number we have obtained, 412, back to base 6, using the slow 
place-value method. We should, of course, come up with our original 
base number 1524 (base 6) as our result.

***Base 10 to Base 6, Slow Method
---------------------------------

To convert 412 (base 10) to base 6, we again need to use the base-6 
place values:  1, 6, 36, 216, ....  The next place value in base 6 
(after 216) is 6 times 216, which is bigger than the number we are 
trying to convert (412), so we don't need to use larger base-6 place 
values.

In this slow place-value method of converting from base 10 to base 6, 
we use the largest base-6 place values first. So our first step is to 
find out how many 216s there are in the number we are converting, 412:

    412/256 = 1 remainder ???

This tells us that the digit in the 216s place of our base-6 number is 
1.  In base 10, the value of 1 in the 216s place is 1x216 = 216. So 
our base 6 number is 1XXX, and with the 1 in the 216s place, the 
base-10 number we have left is

    412
   -216
   ----
    196

The next base-6 place value is 36; we next need to find out (in base 
10) how many 36s there are in 196:

   196/36 = 5 remainder ???

So the base-6 digit in the 36s place is 5; 5 in the 36s place has the 
base-10 value 5x36 = 180. Our base 6 number is now 15XX; with these 
two base-6 digits in place, the base-10 number we have left is

    196
   -180
   ----
     16

The next base-6 place value is 6; the base-10 number we have left, 16, 
is equal to two 6's with 4 left over, so the last two digits of our 
base-6 number are 24 and our final base-6 number is 1524 - which is 
(as it must be) the base-6 number we started with.

This slow process for converting from base 10 to base 6 is made even 
slower by all the explanation that goes with it. Here is the process 
again, without all the words:

    412/216 = 1 remainder 196
    196/36  = 5 remainder 16
     16/6   = 2 remainder 4
      4/1   = 4 remainder 0


FAST METHODS

These methods are faster because we don't need to deal with the place 
values for base 6; the only number we need to use is the base, 6.

*** Base 6 to Base 10, Fast Method
----------------------------------

To convert 1524 (base 6) to base 10 with the fast method, we start 
with the leftmost digit and move right; at each step we multiply by 6 
and add the next base-6 digit, until we have used all the digits. For 
the conversion of 1524 (base 6) to base 10, the process is as follows:

                                                  base-6 equivalent
       action                   base-10 value   of this base-10 value
   -------------------------------------------------------------------
   get first base-6 digit (1)         1                1
   multiply by 6                      6                10
   add next digit (5)                11                15
   multiply by 6                     66                150
   add next digit (2)                68                152
   multiply by 6                    408                1520
   add last digit (4)               412                1524

In this process, we find the base-10 equivalent of our base-6 number 
by "building" the base-6 number from left to right. We start with the 
leftmost base-6 digit, 1; when we multiply by 6, we get the base-10 
equivalent of the base-6 number 10, and when we then add the next 
base-6 digit (5), we get the base-10 equivalent of the base-6 number 
15; when we again multiply by 6 and add the next base-6 digit (2), we 
get the base-10 equivalents of the base-6 numbers 150 and then 152; 
and, finally, when we multiply again by 6 and then add the last base-6 
digit (4), we get the base-10 equivalents of the base-6 numbers 1520 
and finally 1524.

*** Base 10 to Base 6, Fast Method
----------------------------------

Finally, here is the conversion of 412 (base 10) back to base 6, using 
the fast method:

    412/6 = 68 remainder 4...  so the base 6 number is XXX4
    68/6  = 11 remainder 2...  so the base 6 number is XX24
    11/6  =  1 remainder 5...  so the base 6 number is X524
    1/6   =  0 remainder 1...  so the base 6 number is 1524

When we divide the base-10 number 412 by 6 to get '68 remainder 4', we 
know that all but 4 of the 412 can be grouped into groups of 6; this 
tells us the the right-most digit of the base-6 equivalent of 412 
(base 10) is 4. We now know that our base-10 number 412 is 68 groups 
of 6 with 4 left over. Next we divide the 68 by 6 to find that those 
groups can themselves be grouped into 11 groups of 6 with 2 left over; 
this tells us that the next digit to the left in our base-6 number is 
2. And finally the 11 groups we now have can be grouped into 1 group 
of 6 with 5 left over, so the two leftmost digits of our base-6 number 
are 15.


I hope you can use the above examples to figure out how to do your 
problem.

Write back if you have any further questions on this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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