General

I'd like to write about bits, and how I understand them. We want from computer something, and write a program for it, in languages like Python, C, PHP or another, which is understandable for humans. But a computer doesn't understand such written code, and first, compilator translates this code into binary data. Binary data is a data, formatted specially for a computer. E.g. we want to do a simple operation like addition, 5+3. Or print something in the console. Or run the game, no matter, in every case compiler prepares binary data. To be true, compilator prepares byte-code, but by meaning it is a binary data. Binary data, in simple words, is a consequence of 2 values - 0 and 1. 1 byte consisting of 8 bits. As I remember from school, 1 byte has 8 bits, because such a number of bits is needed to store 1 symbol. 1 char is 1 byte. But it is not true, using python language I can show that 1 letter takes much more bytes:

import sys
>>> sys.getsizeof('a')
50

So, 1 char in python takes 50 bytes, or 50x8=400 bits. Interesting note, historically in different platforms there was bytes with 6,7,32,36 bits.

But my article will be about bits as mathematical numbers.

Arithmetic

As we know, any number can be represented by base 10 (decimal), 2 (binary), 16 (hexadecimal), and any other. 1 bit can accept only two values - 0 or 1. Number, of course, can consist of any number of bits. Addition, Subtraction, Multiplication, and Division can be done on bin numbers like on regular decimal numbers. In python bin numbers starts with prefix 0b.

>>> 0b0
0
>>> 0b1
1
>>> 0b10
2
>>> 0b11
3
>>> 0b11010
26
>>> 0b1+0b10
3
>>> bin(3)
'0b11'
>>> type(bin(3))
<type 'str'>
>>> int('11', base=2)
3
>>> int('0b11', base=2)
3
# we can prepend zeros on binary, it won't change it's value:
>>> int('01') ==  int('001') ==  int('000001') == 1
True 
1

In addition to ariphmetic operations, binary numbers also support boolean operations

Boolean operations on bits

With some restrictions, boolean operations on bits came from Bool algebra. You may consider 1 as True, and 0 as False. Operations are:

• binary AND
• binary OR
• excluding binary OR (XOR)
• binary NOT

See tables below to understand rules of this operations.

AND

Operator for OR for binary operations in python - &. If one of bits is 0, then resulted bit will be 0.

In practice this operation can be used for quick determination wheather number odd or not, by x & 1 (if result is 0, then number is even, otherwise number is odd)

>>> bin(0b1100 & 0b0101)
'0b100'
>>> bin(4)
'0b100'
>>> int(bin(4), base=2) & 1
0  # 4 is odd
>>> int(bin(5), base=2) & 1
1 # 5 is even

OR

Operator for OR for binary operations in python - |. In simple words, this operation returns 1 (True) when at least one of bits is 1 (True).

>>> bin(0b1100 | 0b0101)
'0b1101'

Excluding OR (XOR)

Operator for XOR for binary operations in python - ^. Similar to OR, but when two bits are 1, resulted bit will be 0.

>>> bin(0b1100 ^ 0b0101)
'0b1001'

NOT

Operator for binary NOT in python - ~. Reverse bits: 0 becomes 1, 1 becomes 0.

>>> bin(~0b01)
'-0b10'
>>> -0b10
-2

Need to say, that when we use bitwise NOT on a number, it changes its sign to the opposite.

Shifts

There's also shift operations: In these operations, the digits are moved, or shifted, to the left or right by N indexes.

Left shift

In this case, to the right end zeros added:

>>> bin(0b10110100 << 1)
'0b101101000'
>>> bin(0b10110100 << 2)
'0b1011010000'
>>> bin(0b10110100 << 3)
'0b10110100000'

Right shift

Bits shift to the right, from right side certain number of bits being dropped:

>>> bin(0b10110100 >> 1)
'0b1011010'
>>> bin(0b10110100 >> 2)
'0b101101'
>>> bin(0b10110100 >> 3)
'0b10110'

Where to use boolean operations on bits?

Mainly in programming bitwise operations are not commonly used. 1. Bitwise operations are used in cryptography. 2. If we have an IP-address, e.g. 192.168.8.2 and we know mask of a subnet, e.g. 255.255.248.0, we may determine bounds of IP addresses range for this subnet, and its address. To do this we must apply the mask to this IP address using logical AND:

>>> bin(192)
'0b11000000'
>>> bin(255)
'0b11111111'
>>> 192 & 255
192
>>> 168 & 255
168
>>> 11 & 248
8
>>> 2 & 0
0
>>>

and as a result, we have 192.168.8.0, which is a subnet address.

1. Left and right shifts may be used to quick multiplication or division to 2^N. In this case speed will be faster, compared to using *, /, ** operators.

>>> 0b1
1
>>> 0b10
2
>>> 0b100
4
>>> 0b1000
8

>>> 50 >> 1
25
>>> 25 >> 1
12

Of course, there's another use cases of bitwise operations.