# Binary system

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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. 2. 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. 1002, where the digit 1 is the third digit from the right, and thus represents 22, 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:

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

Binary addition is fairly simple, making for efficient use in computers. To add one bit (digit) binary numbers, use the following table:

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,

-10011 = 01100 + 1 = 01101

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

5 - 2 = 101 - 010 = 101 + (101 + 1) = 101 + 110 = 011 = 011 = 3

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:

1. set a to 0
2. start at the rightmost digit of the X
3. for the nth digit (from the right, starting with 0)
1. If the digit is zero, continue
2. If the digit is one,
1. Append n zeros to the end of Y and add this to a
4. a is now equal to X * Y

For Example:

    1010 X *
1101 Y
----------------
0000
11010
000000
1101000
---------------
10000010


# First Numbers

The first 16 binary digits:

Decimal Binary
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
16 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).