Binary, also known as the base-2 number system, uses only two symbols: 0 and 1. It is the fundamental number system used in digital electronics and computer systems because it directly corresponds to the binary logic used in electronic circuits (on/off, true/false).
Binary Digits:

0: Represents the value zero.
1: Represents the value one.

Binary to Decimal Conversion:

To convert a binary number to decimal, multiply each binary digit by 2 raised to the power of its position (starting from 0) from right to left.

Example:

Decimal to Binary Conversion:

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainders. The remainders, read in reverse order, form the binary number.

Example:
Convert 11 to binary:

11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1

Reading the remainders from bottom to top, we get:
11=1011 (binary)

Binary to Hexadecimal Conversion:

To convert binary to hexadecimal, group the binary digits into sets of four (starting from the right). Then, convert each group to its hexadecimal equivalent.

Example:
Binary 10111011:

Group into sets of four: 1011 1011
Convert each group to hex: B B

So, 10111011 in binary is BB in hexadecimal.
Hexadecimal to Binary Conversion:

To convert hexadecimal to binary, convert each hex digit to its four-bit binary equivalent.

Example:
Hexadecimal BB:

Convert each digit to binary: B (1011), B (1011)
Combine the groups: 10111011

So, BB in hexadecimal is 10111011 in binary.
Binary to Octal Conversion:

To convert binary to octal, group the binary digits into sets of three (starting from the right). Then, convert each group to its octal equivalent.

Example:
Binary 10111011:

Group into sets of three: 101 110 11 (pad with leading zeros if necessary: 010 111 011)
Convert each group to octal: 2 7 3

So, 10111011 in binary is 273 in octal.
Octal to Binary Conversion:

To convert octal to binary, convert each octal digit to its three-bit binary equivalent.

Example:
Octal 273:

Convert each digit to binary: 2 (010), 7 (111), 3 (011)
Combine the groups: 010111011

So, 273 in octal is 10111011 in binary.
Summary Table:
Decimal Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
Uses of Binary:

Digital Electronics: Binary is used in all digital circuits and logic gates.
Computer Systems: Binary underlies all data representation, processing, and storage in computers.
Networking: IP addresses and subnet masks are often expressed in binary.

Understanding binary is essential for working with computers and digital systems, as it forms the basis of all modern computing technologies.

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