We humans use a number system that is called "base ten". That number system uses a set of symbols to represent a group of things, and in many countries around the world, the Arabic number symbols are what is used. These symbols are 0 1 2 3 4 5 6 7 8 9.

We humans use a ten number system that starts with zero and works it's way to nine. When we reach nine we start a new column to the left marking down a one, and then put a zero to the right of it (09 + 01 = 10).

Computers think in binary, but display their binary data for humans in various formats such as octal or hexadecimal. The octal system is also called 'base eight' and hexadecimal is called 'base sixteen'.

To make any discussion of number systems meaningful, you have to understand that all number systems are composed of symbols. We humans in most English speaking nations use the symbols called 'zero' through 'nine' (0 - 9). These symbols are arranged in columns. Every time you move one column to the left, it's worth ten times what it would be worth one column to the right. (100 is ten times greater than 10). We humans have gotten so used to using the symbols that we don't even think about them as symbols anymore. We see a "4" and think: FOUR: * * * *

Here are four sections on the various number systems. I promise I will get them all online, at some point...


This is the number system I'll spend most of my time on. It even has it's own tutorial. Binary data is represented using the symbols "0" and "1" (zeroes and ones). This is directly stored on computers using miniature 'on/off' switches called 'transistors'. A switch that is turned "on" represents a "1", and a switch that is turned off represents a "0". To make things easier on the humans, computers are programmed to display things in octal, decimal, or hexadecimal. However, understanding binary is at the heart of understanding computers themselves. Computer logic (Boolean Logic) is in binary and uses AND and OR. You will have a great deal of trouble understanding IP addressing, subnetting and supernetting if you don't understand binary.


Not a great many things are represented in octal any more. System core dumps on some computers used to be in octal, but memory dumps have advanced over the years. Most systems use hexadecimal or binary today. The octal numbering system counts from zero to seven (0 - 7), and starts over by placing a one followed by a zero (10). Therefore symbol pattern one-zero (10) in octal is actually what we humans call eight. Each octal digit represents three binary bits (octal 7 = binary 111).


OK, this is base ten, or the good old counting system you learned in school. You already know everything you need to know about it, right? Well, maybe not. Base ten doesn't really convert well to binary and back again, as ten doesn't fit nicely into binary's way of thinking. Since we humans use base ten, and computers use binary (base two) you have to become quite skilled at converting between the two. A wanabe brags about understanding binary. A true guru-hacker can think and convert between them in his sleep. This is because they learned to do this while exhaused from becoming phase-shifted during 72-hour hacking sessions.

HEX (hexadecimal)

Hexadecimal counts from zero to fifteen (0 - F), and then rolls over to a one followed by a zero (16 = 10). The difficulty most folks have with hexadecimal is that the number system counts all the way up to the last symbol we humans use (9), but continues on afterwards using letters instead of numbers. This is simply because we humans only have ten fingers, and we don't have number symbols for anything past 9. We humans count from nine to ten by moving to a second column, drawing a one, and putting a zero behind it.

Network Interface devices have a MAC address that is in hexadecimal. Colors in HTML are done in hexadecimal. Hex-dumps are made of computer memory when a computer crashes. If you are reading something, and come across a set of characters preceded by '0x', you're probably dealing with something in hexadecimal.

Here is chart showing a count through the first set of symbols in each numbering system.

 Decimal   Binary   Octal    Hex  
00 0 0000 00 00
01 0 0001 01 01
02 0 0010 02 02
03 0 0011 03 03
04 0 0100 04 04
05 0 0101 05 05
06 0 0110 06 06
07 0 0111 07 07
08 0 1000 10 08
09 0 1001 11 09
10 0 1010 12 0A
11 0 1011 13 0B
12 0 1100 14 0C
13 0 1101 15 0D
14 0 1110 16 0E
15 0 1111 17 0F
16 1 0000 20 10


Basic Concepts | Number Systems | Hexadecimal | Decimal | Octal | Binary


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