Exploring Number Bases

     As pointed out in the discussion Why Study Math various schemes were used to represent numbers throughout the ages. All of these systems did not support fundamental operations such as addition and subtraction. It was not until the idea of a base to define a number system was developed that number representation and computation became straight forward.

     Probably because we have 10 fingers, the base ten system naturally came about. Using the digits 0,1,2,3,4,5,6,7,8, and 9 and noting what position the digit occupies relative to some point we are able to represent an unlimited number of numbers. This reference point for base 10 is called the decimal point.

     Before we investigate other number systems, let's study the one we are most familiar with, the base 10 system. Let's count starting from 1.

				decimal point
			1	●
			2	●
			3	●
			4	●
			5	●
			6	●
			7	●
			8	●
			9	●
				●

   We've used all our numbers, 1,2 ,3 ..., and 9. The next number will be ten. If we place a 1 in the next column, that is the second position away from the decimal, we can mean that we have ten objects, like so:

		1		●

  The empty space between the 1 and the decimal point needs a place holder, and it is customary to use the symbol '0'. The meaning are “no” ones. So we have:

		1	0	●

   Which we recognize as the familiar '10' for the number 10. What we've done here is count up to 9 then carry a 1 into the next column and write a zero in the first column. From this point forward we count again as usual with the 1 in the second column riding along to remind us that we already have 10 from which we are counting forward. Of course we'll count until we reach 9 more than 10, that is 19, and one more requires a carry of 1 into the next column. We already have a 1 there so we add this one to the previous 1 and get 2, place a zero in the first column and we have twenty '20'. Like so:

				decimal point
			1	●
			2	●
			3	●
			4	●
			5	●
			6	●
			7	●
			8	●
			9	●
		1	0	●

		1	1	●
		1	2	●
		1	3	●
		1	4	●
		1	5	●
		1	6	●
		1	7	●
		1	8	●
		1	9	●
		2	0	●

   In the first column we count by singles, that is ones. Every ten numbers we add a single to the second column. So this column represents the number of tens. Carrying this further, once we arrive to 99 then one more will require a carry from the first column over to the second column, but that is a 9, so we carry into the third column and leave 0's behind in the first and second column. We recognize that number to be 100.

Like so:

		9	9	●
	1	0	0	●

   Moving from left from the decimal point the columns represent: ones, tens, hundreds, thousands, ten thousands, etc.

ten thousands	thousands	hundreds	tens	ones	●

    or number of:

    10,000's    1000's    100's    10's   1's

So, the number 354 represents 3 100's, 5 10's and 4 1's.

The number 21,081 represents 2 10,00's, 1 1000, no 100's, 8 10's and one 1.

   Each column represents groups of its previous column. For example, the number 21 represents 2 groups of 10 ones plus 1. The number 816 represents 8 groups of 10 tens of 10 plus 1 group of 10 ones plus 6. A great way of demonstrating these groups is by counting money in the denominations: penny, dime, and dollar. Group the pennies by tens, the dimes by tens, etc.

Other Bases

   In the previous section there are a few noteworthy items. First of all, as we counted 1, 2, 3, ..., 9 we found that the next number started a 1 in the next column and we placed a 0 in the first column as a placeholder. So, we used 10 symbols altogether, 0,1,2,3,4,5,6,7,8,9, and we say this is a base 10 numbering system.

   Secondly, as we moved from column to column we first made use of all ten of these symbols and each column represents 10 of the previous column, that is the first column counted 1's, the second counted 10's, the third column counted 100's, the fourth column counted 1000's etc.

So, what we have is one of the symbols 0,1,2,3,4,5,6,7,8,9 multiplied by the value of the column, that is 1, 10, 100, 1000, etc., determined by the column that symbol occupies. Example, the number 19204 is:

          4 * column 1 == 4 * 1 +

          0 * column 2 == 0 * 10 = 0 +

          2 * column 3 == 2 * 100 = 200 +

          9 * column 4 == 9 * 1000 = 9000 +

          1 * column 5 == 1 * 10000 = 10000

   Now, there is nothing special about a base 10 numbering system except that it's the system with which we are most comfortable. Again, we have 10 fingers and 10 toes, so base 10 is natural for us. Suppose we had 6 fingers on each hand. Then I suppose it would be natural to use a base 6 numbering system, Let's investigate that system now.

   Recalling from our base 10 discussion, there we needed 10 symbols, likewise here we'll need 6 symbols. These symbols can be anything whatsoever, but since we're counting let's use regular numbers we had in the base 10 system.

Ok, let's begin,

     1

     2

     3

     4

     5      Now, here we need to move to the next column like before

  1 0

     Wait a minute! Already we have ten? No, we wouldn't call this 'ten'. It looks like ten, but it is not ten. If we were only counting objects, we'd have 6 objects. So, maybe we should call this 'six'. This seems odd, but is is correct. We could call it 10 but we'd have to qualify it so there is no confusion and say 10 base 6 and write it this way: 10₆.

So, let's continue counting,

      1

      2  

      3

      4

      5

   1 0

   1 1

   1 2

   1 3

   1 4

   1 5           We need to move to the next column now

   2 0

  and we have 20 base 6, written 20₆.

   Well, we are not use to this number system so 20 base 6 makes no intuitive sense to us, however 12 objects is more clear (12 base 10).

   Furthermore, the normal operations, addition, subtraction, multiplication and division require the use of special tables just for this base, base 6. As an exercise we will develop these tables for base 6. But before we move on, let's discuss converting back and forth between other bases and base 10. Let's discuss these conversions now.

   Suppose we have the base 6 number 2051. We can directly convert this number to base 10, but we need to recall the value of each column. Now column 1 always has the value 1. For base 6, column 2 to has the value 6, column 3 has the value 36 (=6*6), and column 4 has the value 216 (=6*6*6).

     So 2051₆ = 2*216 + 0*36 + 5*6 + 1*1 =  463 base 10.

This procedure works for all base conversions to base 10.

   Now, suppose we have the base 10 number 83. The task is to determine the base 6 representation. Basically, we need to know how many times 6 divides the base ten number for each column.

 Like this:

      83 / 6 = 13 remainder 5

      13 / 6 = 2 remainder 1

       2 / 6 = 0 remainder 2

   Once we reach 0 we are done. What we do now is use the remainders reading backwards getting 215₆ .

(The reason we take the remainders backwards is, each time we divide we repeatedly divide the original number by 6, then 36, then 216, but these are groups of 1, 6, and 36 respectively. So, 1, 6, and 36 match 5, 1, and 2 respectively).

Addition Table base 6

Using this table,

for example, in base 6, 4 + 4 = 12 (which you can verify in base 10, 6 + 2 = 8)

another example in base 6,   4 + 5 + 2

add like base 10, carrying 1 over as needed

working from the bottom up:

         2  <--  add carry in next position   (step 5 )

              4      = 2 carry 1   (put the 2 below the line and carry the 2 (step 4)

           + 5         4 carry 1   4 + 4 =  1 2  ( 2  + carry 1)   (step 3)

           + 3         5 no carry  5 + 5 = 1 4   (4 + carry 1)    (step 2)  

              2          2 + 3 =5         (step 1)

     ---------

         2 2 base 6 (we had 2 carries )

Multiplication Table base 6

Example: 3 * 4 = 20

Example:    13 * 45 =

                1 3

                4 5

             ---------

              1 1 3      (5*3 = 23 --> write the 3, carry the 2, then  5*1 + 2 = 11)

           1 0 0         (4*3 = 20 --> write the 0 carry the 2, then 4*1 + 2   = 10)

           -----------

           1 1 1 3             (= 9 * 29 = 261 base 10)

Right of the decimal Point

   All of the numbers we've discussed so far are whole numbers in that they contain no fractional part. We use the right hand side of the decimal to represent fractional parts of a number.   To handle these numbers we need to understand what these digits right of the decimal represent.   This will be covered at a later date.