Base Number Equivalence Tables

Date: 10/30/2001 at 13:51:39
From: Laurianne Brown
Subject: Base Numbers Tables

My son needs to show a table for base 3 and base 7. Help, please.
I have looked at your examples on converting bases but I am still not 
clear.

Date: 10/31/2001 at 11:30:09
From: Doctor Greenie
Subject: Re: Base Numbers Tables

Helo, Laurianne -

I don't know whether you are looking for a table of addition or 
multiplication facts for base 3 and base 7, or just a table of the 
base-3 and base-7 equivalents of the first several counting numbers.  
I will guess that for a 12-year-old you are only looking for a table 
of the equivalents of our familiar base-10 numbers.

One way to get a hands-on approach to understanding different number 
bases is to consider counting money. For the purposes of this 
exercise, for counting in base b, we will have bills in the 
denominations of 1 dollar, b dollars, b^2 (b times b) dollars, 
b^3 (b times b times b) dollars, and so on; and we will have only 
(b-1) bills of each denomination to work with.

So in the familiar base 10, we have 9 1-dollar bills, 9 10-dollar 
bills (10 times $1), 9 100-dollar bills (10 times $10), and so on.  
To represent the counting numbers from 1 to 20 using these bills, 
we have

   number  10's / 1's    total
  ---------------------------------------
      1        0 1     0(10) + 1(1) =  1
      2        0 2     0(10) + 2(1) =  2
      3        0 3     0(10) + 3(1) =  3
    ...
      8        0 8     0(10) + 8(1) =  8
      9        0 9     0(10) + 9(1) =  9
     10        1 0     1(10) + 0(1) = 10
     11        1 1     1(10) + 1(1) = 11
     12        1 2     1(10) + 2(1) = 12
    ...
     18        1 8     1(10) + 8(1) = 18
     19        1 9     1(10) + 9(1) = 19
     20        2 0     2(10) + 0(1) = 20

When we reached the number 9, we had used all 9 of the 1-dollar bills 
we have; so when we went to the next counting number, we "traded in" 
the 1-dollar bills for one bill of the next denomination. Similarly, 
when we reached 19 (represented with 1 10-dollar bill and 9 1-dollar 
bills), we had again used all 9 of the 1-dollar bills we have, so when 
we went to the next counting number we again traded in the 1-dollar 
bills for a second bill of the next larger denomination.

Now let's count from one to twenty in base 7. For base 7, we have 6 
1-dollar bills, 6 7-dollar bills (7 times 1 dollar), 6 49-dollar bills 
(7 times 7 dollars), and so on. To make the numbers one to twenty 
using these bills, we have

     number                number
   (base 10)  7's / 1's   (base 7)     total (base 10)
  ---------------------------------------------------------
      1          0 1          1     0(7) + 1(1) =  0+1 =  1
      2          0 2          2     0(7) + 2(1) =  0+2 =  2
      3          0 3          3     0(7) + 3(1) =  0+3 =  3
      4          0 4          4     0(7) + 4(1) =  0+4 =  4
      5          0 5          5     0(7) + 5(1) =  0+5 =  5
      6          0 6          6     0(7) + 6(1) =  0+6 =  6
      7          0 0         10     1(7) + 0(1) =  7+0 =  7
      8          1 1         11     1(7) + 1(1) =  7+1 =  8
      9          1 2         12     1(7) + 2(1) =  7+2 =  9
     10          1 3         13     1(7) + 3(1) =  7+3 = 10
     11          1 4         14     1(7) + 4(1) =  7+4 = 11
     12          1 5         15     1(7) + 5(1) =  7+5 = 12
     13          1 6         16     1(7) + 6(1) =  7+6 = 13
     14          2 0         20     2(7) + 0(1) = 14+0 = 14
     15          2 1         21     2(7) + 1(1) = 14+1 = 15
     16          2 2         22     2(7) + 2(1) = 14+2 = 16
     17          2 3         23     2(7) + 3(1) = 14+3 = 17
     18          2 4         24     2(7) + 4(1) = 14+4 = 18
     19          2 5         25     2(7) + 5(1) = 14+5 = 19
     20          2 6         26     2(7) + 6(1) = 14+6 = 20

In base 7, we used up all our 1-dollar bills when we reached the 
number 6; when we went to the next counting number, we traded in the 
1-dollar bills for a single bill of the next higher denomination ($7).  
And when we reached the counting number 13 (base 10), represented in 
base 7 by 1 7-dollar bill and 6 1-dollar bills, we had again used all 
the 1-dollar bills, so when we went to the next counting number we 
again traded in the 1-dollar bills for a second bill of the next 
higher denomination.

And now let's count from one to twenty in base 3. For base 3, we have 
2 1-dollar bills, 2 3-dollar bills (3 times 1 dollar), 2 9-dollar 
bills (3 times 3 dollars), 2 27-dollar bills (3 times 9 dollars) and 
so on. To make the numbers one to twenty using these bills, we have

     number       9's /     number
   (base 10)   3's / 1's   (base 3)     total (base 10)
  --------------------------------------------------------------------
       1         0 0 1         1     0(9) + 0(3) + 1(1) =   0+0+1 =  1
       2         0 0 2         2     0(9) + 0(3) + 2(1) =   0+0+2 =  2
       3         0 1 0        10     0(9) + 1(3) + 0(1) =   0+3+0 =  3
       4         0 1 1        11     0(9) + 1(3) + 1(1) =   0+3+1 =  4
       5         0 1 2        12     0(9) + 1(3) + 2(1) =   0+3+2 =  5
       6         0 2 0        20     0(9) + 2(3) + 0(1) =   0+6+0 =  6
       7         0 2 1        21     0(9) + 2(3) + 1(1) =   0+6+1 =  7
       8         0 2 2        22     0(9) + 2(3) + 2(1) =   0+6+2 =  8
       9         1 0 0       100     1(9) + 0(3) + 0(1) =   9+0+0 =  9
      10         1 0 1       101     1(9) + 0(3) + 1(1) =   9+0+1 = 10
      11         1 0 2       102     1(9) + 0(3) + 2(1) =   9+0+2 = 11
      12         1 1 0       110     1(9) + 1(3) + 0(1) =   9+3+0 = 12
      13         1 1 1       111     1(9) + 1(3) + 1(1) =   9+3+1 = 13
      14         1 1 2       112     1(9) + 1(3) + 2(1) =   9+3+2 = 14
      15         1 2 0       120     1(9) + 2(3) + 0(1) =   9+6+0 = 15
      16         1 2 1       121     1(9) + 2(3) + 1(1) =   9+6+1 = 16
      17         1 2 2       122     1(9) + 2(3) + 2(1) =   9+6+2 = 17
      18         2 0 0       200     2(9) + 0(3) + 0(1) =  18+0+0 = 18
      19         2 0 1       201     2(9) + 0(3) + 1(1) =  18+0+1 = 19
      20         2 0 2       202     2(9) + 0(3) + 2(1) =  18+0+2 = 20

When we count in base 3, we only have 2 bills of each denomination; we 
use up both of the 1- and 3-dollar bills to make the counting number 
8, so when we go to the next counting number, we need to trade in all 
the smaller bills for a single bill of the third denomination, $9.

I hope this helps.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/