About the Binary Calculator

The binary number system (base-2) is the foundation of all modern computing. Every piece of data in your computer—photos, videos, documents, and programs—is ultimately stored as sequences of 0s and 1s. Our binary calculator helps you perform arithmetic operations on binary numbers and convert between binary, decimal, hexadecimal, and octal systems.

What you can do:

Binary arithmetic: Addition, subtraction, multiplication, and division
Base conversions: Binary ↔ Decimal ↔ Hexadecimal ↔ Octal
Bitwise operations: AND, OR, XOR, NOT, shift left, shift right
Two's complement: Work with signed (negative) binary numbers

Understanding Binary Numbers

Binary uses only two digits: 0 and 1. Each position represents a power of 2:

Position (from right)	Value	Power of 2
0	1	2⁰
1	2	2¹
2	4	2²
3	8	2³
4	16	2⁴
5	32	2⁵
6	64	2⁶
7	128	2⁷

Example: Binary 1011 = 1×8 + 0×4 + 1×2 + 1×1 = 11 in decimal

Binary Addition

Binary addition follows these rules:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (write 0, carry 1)
1 + 1 + 1 = 11 (write 1, carry 1)

Example: 1011 + 1101

  ¹ ¹   (carries)
  1011  (11 in decimal)
+ 1101  (13 in decimal)
------
 11000  (24 in decimal)

Binary Subtraction

0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (with borrow from next column)

Example: 1101 - 1011

  1101  (13 in decimal)
- 1011  (11 in decimal)
------
  0010  (2 in decimal)

Binary Multiplication

Binary multiplication is similar to decimal but simpler since you only multiply by 0 or 1:

Example: 101 × 11

    101  (5 in decimal)
  ×  11  (3 in decimal)
  -----
    101  (101 × 1)
   101   (101 × 1, shifted left)
  -----
  1111  (15 in decimal)

Converting Binary to Decimal

Multiply each bit by its corresponding power of 2 and sum:

Example: Convert 11010 to decimal

1 × 16 = 16
1 × 8 = 8
0 × 4 = 0
1 × 2 = 2
0 × 1 = 0
Total: 16 + 8 + 0 + 2 + 0 = 26

Converting Decimal to Binary

Divide by 2 repeatedly and record the remainders:

Example: Convert 45 to binary

45 ÷ 2 = 22 remainder 1
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

Read remainders bottom to top: 101101

Binary to Hexadecimal Conversion

Group binary digits into sets of 4 (from right) and convert:

Example: Convert 10110110 to hex

1011 = B
0110 = 6
Result: B6

Real-World Binary Applications

Application	How Binary Is Used
IP addresses	IPv4 addresses are 32-bit binary numbers (e.g., 192.168.1.1 = 11000000.10101000.00000001.00000001)
File sizes	1 KB = 1024 bytes (2¹⁰), 1 MB = 1,048,576 bytes (2²⁰)
Color codes	RGB values are 8-bit binary (0-255 per channel)
Permissions	Unix file permissions use 3-bit binary (rwx = 111 = 7)
Subnet masks	Network masks like /24 = 11111111.11111111.11111111.00000000
ASCII	Characters encoded as 7 or 8-bit binary

Frequently Asked Questions

Why do computers use binary?

Electronic circuits have two states: on (1) and off (0). Binary maps perfectly to these physical states, making it reliable and efficient.

What's the largest number an 8-bit binary can represent?

11111111 = 255 (unsigned). With two's complement (signed), the range is -128 to +127.

What is a bit, nibble, and byte?

Bit: Single binary digit (0 or 1)
Nibble: 4 bits (half a byte)
Byte: 8 bits

How do I represent negative numbers in binary?

Computers use two's complement: invert all bits and add 1. For example, -5 in 8-bit: 00000101 → 11111010 + 1 = 11111011

What is bitwise AND/OR/XOR?

AND: Output 1 only if both inputs are 1
OR: Output 1 if either input is 1
XOR: Output 1 if inputs are different

Related Calculators

Hex Calculator - Hexadecimal arithmetic and conversions
Decimal to Binary Calculator - Focused decimal-to-binary converter
Scientific Calculator - Advanced mathematical operations
ASCII Converter - Text to binary/ASCII conversions
Data Storage Converter - Convert between bits, bytes, KB, MB, GB, TB

Binary Calculator — Free Base-2 Math & Bitwise Tool

About this calculator