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Other Number Bases

Our number system is so cool because it's based on the number 10.   However, there are other bases, most notably base 2 (binary) and base 16 (hexadecimal).  Both base 2 and base 16 are used in computers and computer science!

Base 10 (decimal)

First of all, we have to review base 10 before we go any further.

    The first place is the one's place (right side of the number).  The next place (to the left) is the 10's place because each place is ten times as big (10's, 100's, 1000's, etc.).  You could look at it as 10 to the zero power (1), 10 to the first power (10), 10 to the second power (100), etc.  It's called base 10 because 10 is the base and then each place is a higher (or lower) power of 10.

            1,000,000    100,000   10,000  1,000    100   10    1

                   10⁶           10⁵          10⁴      10³       10²  10¹ 10⁰
 

Base 2 (binary)

   Well, in base two, things work pretty much the same.   The first place (on the right) is the ones place (two to the zero power).  The second place (as you go left), is the two's place (twice as big as one).  The third place is the four's place (two to the second power).  So the

                    64     32   16     8     4      2      1

Instead of having a 10's place, we have a 2's place (kind of weird, huh?)   

Now we have to worry about digits.  In base ten, there are 10 individual digits we may use in one place (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).  If we want to go over 9, then we have to use TWO digits (10).

In base 2, there are only two digits we may use (0, 1).  If we want to represent 2, we have to go over to the next place.

        10 (base 2) =  2     because the 1 is in the 2's place (2) and the 0 is in the 1's place (0).

    1010 (base 2) = 10   because the first 1 (on the left) is in the 8's place, the other 1 is
                                    in the 2's place, thus  8 + 2 is 10.  Don't think of 1010 as one
                                    thousand ten.

Base 16 (hexadecimal)

     The first place is still the one's place, and each place is sixteen times as big as the preceding place.

        ...         4,096         256         16            1

     There is also another problem.  In base 16, we need sixteen different symbols to go into each place.  We already have 10 easy ones (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)  but we need six more.   We can't use 10 because that's a combination of TWO digits and we would be over into another place.  Therefore (whoever thougth this up),  used letters to represent the two digits we need.  A = 10,  B = 11, C = 12,  D = 13,  E = 14,  F = 15.

        Thus   AA (base 16) =    10 * 16  +  10 * 1,  or 170.

Base prime

     One of the more unusual ideas along these lines is base prime.  As expected, each place is a prime number.  Please remember that 1 is neither prime nor composite!

        ...    17         13        11         7        5         3        2

Unlike all of these other bases, this system is based on the prime factorization of the number.

For instance,  120  is   2^3  *  3 * 5, and would be   113  _(prime)  because you need three 2's, one 3, and one 5 , all multiplied together (prime factorization).

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(last updated October 2, 2007)