Decimal ↔ Binary Converter
The binary system (base 2) is the foundation of all computers. Using only 0 and 1, all numbers can be represented. Our Binary Calculator converts decimal to binary and vice versa – perfect for computer science studies, school, and programming.
Place Values (from right):
2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32...
Decimal → Binary Conversion:
42 = 32 + 8 + 2 = 2⁵ + 2³ + 2¹
→ 101010
Binary → Decimal Conversion:
101010 = 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1
= 32 + 8 + 2 = 42
Bit Capacities:
• 4 Bit (Nibble): 0-15 (0000-1111)
• 8 Bit (Byte): 0-255 (00000000-11111111)
• 16 Bit: 0-65535
• 32 Bit: 0-4,294,967,295
Common Values:
• 128 = 10000000
• 255 = 11111111
• 256 = 100000000
• 1024 = 10000000000
Electronic circuits have two states: power on (1) or off (0). By combining these two states, all information can be represented. More states would be technically unreliable.
Bit = smallest unit (0 or 1). Byte = 8 bits. Kilobyte = 1024 bytes (not 1000!). Megabyte = 1024 KB. Gigabyte = 1024 MB. The 1024 comes from 2¹⁰.
Like decimal, but carry at 2: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (carry). Example: 1011 + 0111 = 10010 (11 + 7 = 18).
Computers usually use "two's complement". The highest bit indicates the sign. For 8 bits: 01111111 = +127, 10000000 = -128. This allows normal addition to work.
Pro Tip: Memorize powers of two: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024. This lets you quickly convert between binary and decimal in your head!
Computers use the binary system (Base 2) because transistors only have two states: On (1) and Off (0).