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Number Systems Guide
Binary System (Base-2)
The binary number system uses only two digits: 0 and 1. Each digit represents a power of 2.
Example: Binary 1011 = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11 in decimal
Hexadecimal System (Base-16)
The hexadecimal system uses 16 symbols: 0-9 and A-F (where A=10, B=11, ..., F=15).
Example: Hex 2A3 = (2 × 16²) + (10 × 16¹) + (3 × 16⁰) = 512 + 160 + 3 = 675 in decimal
Conversion Techniques
Decimal to Binary
Divide the decimal number by 2 repeatedly and note the remainders from bottom to top.
Example: 22 ÷ 2 = 11 remainder 0
11 ÷ 2 = 5 remainder 1
5 ÷ 2 = 2 remainder 1
2 ÷ 2 = 1 remainder 0
1 ÷ 2 = 0 remainder 1
Reading bottom to top: 10110
Binary to Decimal
Multiply each digit by its corresponding power of 2 and sum the results.
Example: 10110 = (1 × 2⁴) + (0 × 2³) + (1 × 2²) + (1 × 2¹) + (0 × 2⁰) = 16 + 0 + 4 + 2 + 0 = 22
Hexadecimal to Binary
Convert each hexadecimal digit to its 4-bit binary equivalent:
0 = 0000, 1 = 0001, 2 = 0010, ..., A = 1010, ..., F = 1111
Example: Hex 2A3 = 0010 1010 0011 = 001010100011 binary
Binary to Hexadecimal
Group the binary digits into sets of 4 (starting from the right) and convert each group to its hexadecimal equivalent.
Example: Binary 001010100011 = 0010 1010 0011 = 2A3 hex
Binary Addition
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry the 1)
Binary Subtraction
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1 from the next column)