Binary Calculator: Convert & Calculate Binary Numbers
A binary calculator performs arithmetic operations (addition, subtraction, multiplication, division) and conversions on binary numbers—the base-2 numeral system using only digits 0 and 1—enabling computation with the fundamental number system of digital computers, programming, and digital electronics. This essential tool converts binary to decimal (base-10), hexadecimal (base-16), and octal (base-8) formats, converts decimal numbers to binary representation, performs bitwise operations (AND, OR, XOR, NOT), calculates two's complement for negative numbers, and provides comprehensive binary arithmetic for computer science students, programmers, digital engineers, mathematics learners, and anyone working with binary data, computer architecture, digital logic circuits, or low-level programming requiring precise binary calculations and conversions.
🔢 Binary Calculator & Converter
Perform binary operations and conversions
Binary Arithmetic Operations
Enter binary numbers (0s and 1s only)
Binary Number Converter
Convert binary to decimal, hexadecimal, and octal
Bitwise Operations
Perform bitwise AND, OR, XOR, NOT operations
Binary to Decimal Converter
Convert binary numbers to decimal (base-10)
Decimal to Binary Converter
Convert decimal numbers to binary (base-2)
Understanding Binary Numbers
Binary is a base-2 numeral system that uses only two digits: 0 and 1. Each digit in a binary number represents a power of 2, starting from the rightmost digit (2⁰ = 1). Binary is the foundation of all modern computing because digital circuits can easily represent two states: on (1) or off (0).
Binary Number System
Binary Place Values:
\[ \text{Binary: } 1011_2 = (1 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) \]
\[ = 8 + 0 + 2 + 1 = 11_{10} \]
Each position represents a power of 2:
...2⁴ 2³ 2² 2¹ 2⁰ = ...16 8 4 2 1
Binary Arithmetic Operations
Binary Addition
Binary Addition 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)
Binary Subtraction
Binary Subtraction Rules:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1 from next position)
Conversion Examples
Example 1: Binary to Decimal
Problem: Convert binary 1101 to decimal
Method: Multiply each digit by its place value
Step 1: Identify place values (right to left)
Position 0: 1 × 2⁰ = 1 × 1 = 1
Position 1: 0 × 2¹ = 0 × 2 = 0
Position 2: 1 × 2² = 1 × 4 = 4
Position 3: 1 × 2³ = 1 × 8 = 8
Step 2: Add all values
8 + 4 + 0 + 1 = 13
Answer: 1101₂ = 13₁₀
Example 2: Decimal to Binary
Problem: Convert decimal 25 to binary
Method: Divide by 2 repeatedly, track remainders
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Read remainders bottom to top: 11001
Answer: 25₁₀ = 11001₂
Verification: 16 + 8 + 0 + 0 + 1 = 25 ✓
Example 3: Binary Addition
Problem: Add 1011 + 0110
Solution:
1011
+ 0110
------
10001
Step-by-step:
Rightmost: 1 + 0 = 1
Next: 1 + 1 = 10 (write 0, carry 1)
Next: 0 + 1 + 1(carry) = 10 (write 0, carry 1)
Leftmost: 1 + 0 + 1(carry) = 10 (write 10)
Answer: 10001 (decimal 17)
Binary Conversion Table
| Decimal | Binary | Hexadecimal | Octal |
|---|---|---|---|
| 0 | 0000 | 0 | 0 |
| 1 | 0001 | 1 | 1 |
| 2 | 0010 | 2 | 2 |
| 3 | 0011 | 3 | 3 |
| 4 | 0100 | 4 | 4 |
| 5 | 0101 | 5 | 5 |
| 6 | 0110 | 6 | 6 |
| 7 | 0111 | 7 | 7 |
| 8 | 1000 | 8 | 10 |
| 9 | 1001 | 9 | 11 |
| 10 | 1010 | A | 12 |
| 15 | 1111 | F | 17 |
| 16 | 10000 | 10 | 20 |
Bitwise Operations
AND Operation
Bitwise AND (&):
Returns 1 only if both bits are 1
1 AND 1 = 1
1 AND 0 = 0
0 AND 1 = 0
0 AND 0 = 0
Example: 1100 AND 1010 = 1000
OR Operation
Bitwise OR (|):
Returns 1 if either bit is 1
1 OR 1 = 1
1 OR 0 = 1
0 OR 1 = 1
0 OR 0 = 0
Example: 1100 OR 1010 = 1110
XOR Operation
Bitwise XOR (^):
Returns 1 if bits are different
1 XOR 1 = 0
1 XOR 0 = 1
0 XOR 1 = 1
0 XOR 0 = 0
Example: 1100 XOR 1010 = 0110
Powers of 2 Reference
| Power | Binary | Decimal Value | Common Use |
|---|---|---|---|
| 2⁰ | 1 | 1 | Bit 0 |
| 2¹ | 10 | 2 | Bit 1 |
| 2² | 100 | 4 | Bit 2 |
| 2³ | 1000 | 8 | Byte (nibble) |
| 2⁴ | 10000 | 16 | Half-byte |
| 2⁸ | 100000000 | 256 | 1 Byte |
| 2¹⁰ | 10000000000 | 1,024 | 1 KB |
| 2¹⁶ | - | 65,536 | 16-bit value |
Real-World Applications
Computer Science
- Digital logic: Binary represents circuit states (on/off)
- Data storage: Files stored as binary data
- Memory addressing: RAM locations in binary
- Network protocols: IP addresses, subnet masks
- Programming: Bitwise operations, flags, masks
Digital Electronics
- Logic gates: AND, OR, NOT, XOR gates
- Circuit design: Digital circuit analysis
- Microcontrollers: Pin configuration, registers
- Signal processing: Digital signal representation
- Hardware programming: FPGA, embedded systems
Common Mistakes to Avoid
⚠️ Binary Calculation Errors
- Forgetting carries: In addition, carry 1 when sum exceeds 1
- Wrong place values: Always start from rightmost digit (2⁰)
- Invalid digits: Binary uses only 0 and 1, no 2-9
- Borrowing errors: In subtraction, borrow from next position
- Leading zeros: Don't confuse 0101 with 101 (same value)
- Bit order confusion: Most significant bit (MSB) is leftmost
- Signed vs unsigned: Negative numbers need two's complement
Tips for Binary Calculations
Quick Conversion Techniques:
- Memorize powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256
- Work right to left: Start with least significant bit
- Use grouping: Group by 4 bits for hex conversion
- Check reasonableness: Binary 1111 = 15 (maximum 4-bit)
- Practice patterns: Recognize common binary patterns
- Use calculators: Verify complex calculations
Frequently Asked Questions
What is binary and why is it important?
Binary is base-2 number system using only 0 and 1. Important because digital computers use binary—transistors are either on (1) or off (0). All data (numbers, text, images, programs) ultimately stored and processed as binary. Foundation of digital electronics, computer architecture, programming. Understanding binary essential for computer science, software development, hardware engineering. Enables efficient data representation, logical operations, memory addressing, and communication in digital systems.
How do you convert binary to decimal?
Multiply each binary digit by its place value (power of 2), then sum. Example: 1011₂. From right: (1×2⁰)+(1×2¹)+(0×2²)+(1×2³) = 1+2+0+8 = 11₁₀. Place values: rightmost is 2⁰=1, next is 2¹=2, then 2²=4, 2³=8, etc. Easier: know powers of 2 (1,2,4,8,16...), add positions with 1s. Practice makes conversion quick and intuitive.
How do you add binary numbers?
Like decimal addition but simpler rules. 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0, carry 1). Example: 1011+0110. Right column: 1+0=1. Next: 1+1=10 (write 0, carry 1). Next: 0+1+carry=10 (write 0, carry 1). Left: 1+0+carry=10. Result: 10001. Always start rightmost column, work left, track carries like decimal addition.
What is the difference between binary and hexadecimal?
Binary is base-2 (digits 0-1), hexadecimal is base-16 (digits 0-9, A-F). Hex is shorthand for binary—each hex digit represents 4 binary bits. Example: binary 11111111 = hex FF = decimal 255. Programmers use hex because it's compact: 8-bit byte written as 2 hex digits vs 8 binary digits. Converting: group binary by 4 bits, convert each group to hex. Both represent same data, hex just more human-readable.
What are bitwise operations used for?
Bitwise operations (AND, OR, XOR, NOT) manipulate individual bits. Uses: setting flags (turn specific bits on/off), masking (extract bit ranges), permissions (file access rights), graphics (color manipulation), encryption, data compression, network protocols. Fast and efficient for low-level programming. Example: AND with mask 00001111 extracts lower 4 bits. XOR useful for toggling bits, simple encryption. Essential for systems programming, embedded systems, game development.
How many bits are in a byte?
8 bits = 1 byte. Byte is fundamental storage unit. Can represent 2⁸=256 different values (0-255 unsigned, -128 to 127 signed). 4 bits = nibble (half byte). Common sizes: 16-bit (2 bytes), 32-bit (4 bytes), 64-bit (8 bytes). Kilobyte (KB) = 1,024 bytes (2¹⁰), not 1,000. Megabyte (MB) = 1,024 KB. Understanding bit/byte relationships crucial for programming, memory management, data representation.
Key Takeaways
Binary is the foundation of all digital computing, representing data using only 0s and 1s. Mastering binary arithmetic, conversions, and bitwise operations is essential for computer science, programming, digital electronics, and understanding how computers work at the fundamental level.
Essential principles to remember:
- Binary uses base-2: only digits 0 and 1
- Each position represents a power of 2
- Binary addition: 1+1=10 (carry 1)
- Convert binary→decimal: sum of (digit × 2^position)
- Convert decimal→binary: divide by 2, track remainders
- Bitwise operations manipulate individual bits
- 8 bits = 1 byte = 256 possible values
- Hexadecimal is compact notation (4 bits per digit)
- Leading zeros don't change value (0101 = 101)
- Practice with small numbers first, build confidence
Getting Started: Use the interactive calculator at the top of this page to perform binary operations, convert between number systems, and learn through practice. Choose your operation type (arithmetic, conversion, bitwise), enter your values, and receive instant results with detailed explanations showing step-by-step how binary calculations work.
