Counting: Base 6, 12, 16

Date: 5/31/96 at 14:19:30
From: Scott Asplund
Subject: Base Six Number System

1) In a base six system how do you count to 25? 
2) Do you ever use the numeral 7? 
3) In a duodecimal system (base twelve) why are letters sometimes   
   used instead of numbers? 

Date: 5/31/96 at 15:36:40
From: Doctor Darrin
Subject: Re: Base Six Number System

In the base six number system, you would never use the numeral 7.  You 
would only use numerals from 0 to 5. The idea of base six is just like 
the normal base ten system, except that instead of using the digits 0 
to 9, we use digits 0 to 5, and instead of having a ones digit, a tens 
digit, a hundreds digit and so on, we use a ones digit, a sixes digit, 
a thirty-sixes digit, and so on. So in base 6, the number 321 means 1 
one plus 2 sixes plus 3 thirty-sixes, or 121. So to count to 25, we 
would need:

Base 6                 Base 10

1                          1    
2                          2
3                          3
4                          4
5                          5
10  (1 six and 0 ones)     6
11  (1 six and 1 one)      7
12  (1 six and 2 ones)     8
13                         9
14                        10 
15                        11
20  (2 sixes and 0 ones)  12 
21  (2 sixes and 1 one)   13
22                        14
23                        15
24                        16
25                        17
30                        18
31                        19 
32                        20
33                        21
34                        22
35                        23
40                        24
41  (4 sixes and 1 one)   25


For base twelve, we would want to use digits from 0 to eleven, but 
there are no digits which mean ten or eleven, so we generally use A 
and B for these digits. Also the place values would be ones, twelves, 
one-hundred-forty-fours, etc. So for instance, A2 in base twelve would
mean ten twelves plus two ones, or one-hundred-twenty-two. In fact, in 
the hexadecimal system (base 16), which is used a lot in computer 
science, the letters A through F are used for digits (since we need 
digits for each number from 0 to 15).

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

Date: 5/31/96 at 15:29:17
From: Doctor Byron
Subject: Re: Base Six Number System

Hi Scott,

To count in base six, you generally use the first 6 digits starting 
with 0:

0 1 2 3 4 5

Notice that because zero is included, six in not used.

Before I talk about counting in base six, let's think about how you 
count in base 10 (the normal decimal system we see everyday).
 0  1  2  3  4  5  6  7  8  9
10 11 12 13 14 15 16 17 18 19
20 ...
.
.
.
90 91 92 93 94 95 96 97 98 99
100 ...

You start by counting up to the highest digit available (9 in this 
case). To advance another number, you add one to the next column, and 
reset the column to zero.  So in base 6, you would count like this:
 0  1  2  3  4  5
10 11 12 13 14 15
20 21 22 23 24 25
30 ...
.
.
.
50 51 52 53 54 55
100 101 102 103 104 105
110 ...

In a system with more than ten numerals, such as the duodecimal 
system, we need to add some other characters in order to count. For 
example, the hexidecimal system (base 16), which is often used in 
computer programming. It uses 10 digits plus 6 letters.
 0  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
20 21 22 23 ...
.
.
.
90 91 92 93 94 95 96 97 98 99 9A 9B 9C 9D 9E 9F
A0 A1 A2 ...
.
.
.
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 FA FB FC FD FE FF
100...

Now, at this point you may be wondering how to convert numbers from 
these other bases back to good old familiar base 10. To do that, let's 
again take a look at the meaning of numbers in base 10:

2453 = 3 x 10^0  +  5 x 10^1  +  4 x 10^2  +  2 x 10^3

(Where 10^2 means 10 to the second power.)

Now in some other base, you break the number up in the same way, but
replace the 10s you had before with whatever base you are using.  For 
example, let's see what 243 (base 6) is in base 10:

243 (b6) = 3 x 6^0  +  4 x 6^1  +  2 x 6^2 = 3 + 4 x 6 + 2 x 36 = 99

I hope this will help you explore bases a little more easily. Good 
luck!

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

Date: 5/31/96 at 18:4:27
From: Scott Asplund
Subject: Re: Base Six Number System

Thanks for your quick reply!! One more question about your answer... 
how do you count to 25 in a base 16 using letters?

Date: 6/3/96 at 9:54:4
From: Doctor Darrin
Subject: Re: Base Six Number System


In base 16, the letters A through F would stand for the numbers 10 
through 15 (A=ten, B=11, ... , F=15). So you would count as follows:

Base sixteen                          Base 10
1                                     1
2                                     2
3                                     3
...                                   ...
9                                     9
A                                     10  
B                                     11 
...                                   ...
E                                     14     
F                                     15
10 (One sixteen and 0 ones)           16
11 (One sixteen and one one)          17
...                                   ...
18 (One sixteen and 8 ones)           24
19 (One sixteen and 9 ones)           25

If you wanted to continue, then:

1A  (One sixteen and 10 ones)         26  (remember A stands for 10)
1B  (One sixteen and 11 ones)         27  (B stands for 11)
1C
1D
1E
1F  (One sixteen and 15 ones)         31
20  (two sixteens and 0 ones)         32   

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