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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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