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The '''binary system''' is a way of representing numbers in base 2, i.e. using only the digits 0 and 1. The term 'Binary' means ''composed of two parts'' and comes from the Latin, originally meaning "two by two".  Binary, or base 2, is one of many possible [[Number Systems]].

A number written in the system can be denoted by following it with a subscript 2, i.e. <sub>2</sub>. Each digit represents the number of a power of 2 in the complete number, similarly to in the [[decimal system]], where each digit represents the number of a power of 10. The power is defined by the number of digits in the number from right to left through the digit, minus 1, e.g. 100<sub>2</sub>, where the digit 1 is the third digit from the right, and thus represents 2<sup>2</sup>, or 4. While it is generally impractical for [[human]] use, it is the mainstay of modern [[computing]].  A binary system is also used in [[electronics]], which commonly uses '''0''' to mean "no voltage is present" '''1''' to mean "a voltage is present".  Binary notation is used in circumstances in which a thing is in one of two possible conditions and no other condition is possible; the switch is on or the switch is off, the page has data on it or the page has no data.

==Operations==

===Successor Function===

To increment a binary number, follow this rule:

#Current digit is the end digit
#Change the current digit
# If current digit = 1
##Then:
###Shift current digit to away from the end digit
###Go to step 2
##Else:
###You're done.

===Addition===

Binary addition is fairly simple, making for efficient use in computers. To add one bit (digit) binary numbers, use the following table:
{| class="wikitable"
|-
! X
! Y
! X + Y
! Carry
|-
| 0
| 0
| 0
| 0
|-
| 1
| 0
| 1
| 0
|-
| 0
| 1
| 1
| 0
|-
| 1
| 1
| 0
| 1
|-
|}

to add a multiple digit number, add the bits (digits) individually using the table, and add the carry if necessary. Consider the following example:

   000000110-  Previous Addition's carry
   1000110110  X
   +
   0110001010  Y
-----------------
   1110111000

===Subtraction===
Subtraction in binary can be carried out similar to decimal notation, however, there is a more efficient way, which is usually used in computers. Negative numbers a written in "Two's complement notation". In this notation, You "flip" the digits in the binary number and add one. For example,

<blockquote>
-10011 = 01100 + 1 = 01101
</blockquote>

To do subtraction, you add a number to its complement and ignore the final carry.

<blockquote>
5 - 2 = 101 - 010 = 101 + (101 + 1) = 101 + 110 = [1]011 = 011 = 3
</blockquote>

Note that this method requires the amount of digits to be fixed.

===Multiplication===
Multiplication in binary is far more simple than multiplication in decimal. To multiply two binary numbers X * Y, use the following algorithm:
#set a to 0
#start at the rightmost digit of the X
#for the nth digit (from the right, starting with 0)
##If the digit is zero, continue
##If the digit is one,
###Append n zeros to the end of Y and add this to a
#a is now equal to X * Y

For Example:

     1010 X *
     1101 Y
 ----------------
     '''0000'''
    '''1101'''0
   '''0000'''00
  '''1101'''000
 ---------------
 10000010

=First Numbers=
The first 16 binary digits:
{|style="text-align:center"
!Decimal!!Colspan=2 |Binary
|-
|width=50px|0
|width=50px align="right"|0
|width=10px|  
|-
|1
|align="right"|1
|-
|2
|align="right"|10
|-
|3
|align="right"|11
|-
|4
|align="right"|100
|-
|5
|align="right"|101
|-
|6
|align="right"|110
|-
|7
|align="right"|111
|-
|8
|align="right"|1000
|-
|9
|align="right"|1001
|-
|10
|align="right"|1010
|-
|11
|align="right"|1011
|-
|12
|align="right"|1100
|-
|13
|align="right"|1101
|-
|14
|align="right"|1110
|-
|15
|align="right"|1111
|-
|16
|align="right"|10000
|}

== Boolean operations ==
Because each bit can be considered a true/false value, [[Boolean algebra]] operations are easily done with binary values.  For this reason, they are often referred to as bit-wise operations.  The operations are NOT, AND, OR, NAND, XOR, SHL (shift left), and SHR (shift right) - and variations such as ROL (rotate left) and ROR (rotate right).

==See also==
* [[Binary Code]]
* [[Binary Function]]
* [[Binary Compound]]

== External links ==
*[http://forums.cisco.com/CertCom/game/binary_game_page.htm?site=celc A binary numbering game]
*[http://woodgears.ca/marbleadd/index.html= How to make binary numbers]

{{nobots}}
[[Category:Mathematics]]