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The Decimal, Binary, and Hexadecimal Numbering Systems

Basics of Numbering Systems

There are many different numbering systems, each one based on a number of symbols. In assembly language programming, we are interested in more the one numbering system. We normally work with decimal. The decimal system has 10 symbols. The computer, on the other hand, normally works with the binary system, which only has two symbols. The computer is real good with binary, but humans are not, so we use a "shorthand" of the hexadecimal numbering system for human representation of binary values. Hexadecimal number system has 16 symbols. To make sure that there is no confusion about which base we are using, we will assume that in the absence of other indications, it is decimal. We use a subscript for the base for all other systems: 10, 102, 1016 See how it could be confusing without a system of notation!

Each symbol represents a unique value. You are familiar with the decimal system, using the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The decimal system is based on the powers of 10.When we want to represent a value greater than 9, we use a compound system, where the right-most symbol is one power of ten lower that its neighbor on the left.

Decimal System

Thus, we have the number 397 is really: 

3 times 100 102      300
9 times  10 101       90
7 times   1 100        7
                     ---
                     397
These are mathematical concepts we use daily, even if we do not think about them when we use those concepts.

Binary System

The computer uses a different numbering system. One that is based on two symbols (which also is based on two states, on and off, just like the light switch on the wall.) The two symbols are 0 and 1. In this system, everything is based on a power of two.

The most common powers of two and values (which are the ones you are expected to know) are:

20 1
21 2
22 4
23 8
24 16
25 32
26 64
27 128
28 256
29 512
210 1024
211 2048
212 4096
213 8192
214 16384
215 32768
216 65536

To convert a binary number to decimal, simply add up the decimal equivalents of the positions in the value that are non-zero This leads to 10112 being converted as: 

1 times  8 23         8 (decimal)
0 times  4 22         0 (decimal)
1 times  2 21         2 (decimal)
1 times  1 20         1 (decimal)
                    ---
                     11 (decimal)

Examination of this example should reveal that we used the same concepts, but substituted two for ten as the base.

To convert a decimal number to binary, we can use subtraction to reverse the process. (Note: There are other methods, but this only involves adding and subtractings.)

First, find the largest power of two that does not exceed the value to be converted. If we wish to convert the number 24010 to binary, we can see from the chart the we are looking for 12810 or 27. That means that there are eight positions in the binary equivalent (have to account for 20) and the left-most is set to 1. 

 1 _ _ _ _ _ _ _2
Now subtract 128 from 240. 
 240
-128
----
 112

And Continuing

Now the next power of 2 is check to see if it can be subtracted without the answer being a negative number. If it is possible, we put a 1 in the binary result, otherwise we put a 0. This continues until the last position is filled.

Well, 64 can be subtracted, so: 

 112        11 _ _ _ _ _ _2
- 64
----
  48
32 can be subtracted, so: 
  48        111 _ _ _ _ _2
- 32
----
  16
16 can be subtracted, so: 
  16        1111 _ _ _ _2
- 16
----
   0
8 can not be subtracted, so: 
   0        11110 _ _ _2
4 can not be subtracted, so: 
   0        111100 _ _2
2 can not be subtracted, so: 
   0        1111000 _2
1 can not be subtracted, so: 
   0        111100002

Hexadecimal System

When working with large binary values, humans have a tendency to make mistakes, so we use a different numbering system which is more convenient to use, the hexadecimal numbering system. Hexadecimal means sixteen. So, what sixteen symbols are used for this? Simple, use the ten from the decimal system and add the first six letters of the alphabet: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

This leads to FE08: 

F times  4096 163 61440 (decimal)
E times   256 162  3584 (decimal)
0 times    16 161     0 (decimal)
8 times     1 160     8 (decimal)
                  -----
                  65032 (decimal)
We can compare the first sixteen binary, decimal, and hexadecimal values in the following table:

BinaryDecimalHex
000000
000111
001022
001133
010044
010155
011066
011177
100088
100199
101010A
101111B
110012C
110113D
111014E
111115F

One of the things to note, the value 0000 is exactly the same value as 0. Leading zeroes do not alter a value. When working with binary and hex values, it is common to write them as 8-, 16-, or 32-bit values by adding the necessary leading zeroes.

Hexadecimal to Binary

Hex to binary and binary to hex are the simplest, since you only have to worry about sixteen possibilities! To convert FE08 to binary, look up each hex digit and put in the four bits that are the same value: 
F    E    0    8
1111 1110 0000 1000

Binary to Hexadecimal

This is the reverse of Hex to Binary. To convert 1111111000001000, first divide the number into groups of four bits, starting from the left side:

1111 1110 0000 1000

Then convert each four bit group to hex: 

1111 1110 0000 1000
F    E    0    8

Hexadecimal to Decimal

Forget it! Convert hex to binary and then binary to decimal. Trust me, it is easier! (Calculators are handy for this also!)

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©2004, Gary L. Burt