Base-ten, Octal, and Hexadecimal Numbers

Base-ten numbers are the numbers we're used to seeing in our everyday life. Base-ten math seems natural to us because we've been doing using base-ten for our entire lives. Humans use base-ten because we have 10 fingers and it is easy for us to count to ten. If humans had 8 fingers, we would probably use base-8 (aka octal) numbers.

The following lines show the digits and numbering of the base-2 number:

digits:    1 0 1
numbering: 2 1 0

Every digit must be multipled by 2 raised to the numbering. All of these products are summed to compute the result.

1*(2**2) + 0*(2**1) + 1*(2**0) = 5

This link has an awesome description on how to convert number bases.

We will again map out the digits and numbering of the digits to aid with the conversion.

digits:    1 1 1 0
numbering: 3 2 1 0

1*(2**3) + 1*(2**2) + 1*(2**1) + 0*(2**0) = 14

The divide by two algorithm can be used to convert a base-ten number into a base-2 number. The algorithm starts by dividing the base-ten number by 2. The remainder is the last digit of the binary number and the remainder is divided by 2 again. The algorithm keeps repeating until the result of the division is 0 or 1 - this final result is prepended to the binary number.

1                     => 1
2 / 2 => remainder 0  => 0
4 / 2 => remainder 0  => 0
8 / 2 => remainder 0  => 0

8 base-ten is equivalent to 1000 base-2.

This page has a great description of the divide by two algorithm. This video describes the algorithm as well.

We will use the divide by two algorithm again to solve this problem. We are using integer division when running the algorithm. Integer division is when the fractional part of the division is discarded, so 3 / 2 is 1 with integer division (not 1.5).

 1                     => 1
 3 / 2 => remainder 1  => 1
 6 / 2 => remainder 0  => 0
12 / 2 => remainder 0  => 0

The result is 1100.

Base-16 numbers use the following digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A is 10, B is 11, C is 12, and so forth. D base-16 is equivalent to 13 base-ten.

We will use a generalized version of the divide by two algorithm again to solve this problem (we want to convert to a hexadecimal number, so we'll divide by 16). We start by dividing 357 by 16 to get 22 with a reminder of 5. The remainder is prepended to the result and then 22 is divided by 16 to get 1 with a remainder of 6. The divisor (1) is now less than 16, so it is prepended to the result to get 165.

 1                       => 1
 22 / 16 => remainder 6  => 6
357 / 16 => remainder 5  => 5

The result is 165.

15                          => F
  249 / 16 => remainder  9  => 9
 3995 / 16 => remainder 11  => B
63933 / 16 => remainder 13  => D

The result is F9BD.

A two digit base-16 number can represent 256 different numbers 16*16. A two digit base-ten number can only represent 100 different numbers 10*10.