Binary Calculator: Add, Subtract, Multiply & Divide Binary Numbers
Free binary calculator for addition, subtraction, multiplication, and division. Supports arbitrary-precision binary arithmetic with real-time decimal and hexadecimal conversions. Perfect for computer science education.
Binary Addition: Step-by-Step Guide
Understanding Binary Addition with Carry Logic
- Step 1: Rightmost BitsAdd the rightmost bits. If the sum is 0 or 1, write it down. If the sum is 2 (1+1), write 0 and carry 1 to the next position.
- Step 2: Next PositionAdd the next bits plus any carry from the previous position. Again, if the sum is 2 or more, write the remainder and carry 1.
- Step 3: Continue LeftwardRepeat this process for each bit position, propagating carries as needed. The final carry, if any, becomes the leftmost bit of the result.
- Example: 1011 + 1101Rightmost: 1+1=10 (write 0, carry 1). Next: 1+0+1=10 (write 0, carry 1). Next: 0+1+1=10 (write 0, carry 1). Leftmost: 1+1+1=11 (write 1, carry 1). Result: 11000 (binary) = 24 (decimal).
Binary addition follows the same principles as decimal addition, but uses only two digits (0 and 1). The key concept is the carry mechanism that propagates from right to left.
Why Binary Addition Matters
- Processor Design:Modern processors use carry-lookahead adders and other optimizations to speed up binary addition, which is performed billions of times per second.
- Computer Science Education:Learning binary addition is essential for understanding how computers work at the lowest level, from simple calculators to supercomputers.
- Error Detection:Understanding binary arithmetic helps in error detection and correction algorithms used in data transmission and storage.
Binary addition is the fundamental operation that all computer arithmetic is built upon. Understanding how carries propagate helps you understand processor design and computer architecture.
Binary Subtraction Using 2's Complement
How 2's Complement Subtraction Works
- Method:To subtract B from A: Find the 2's complement of B, then add it to A. The result is A - B.
- 2's Complement Steps:1) Flip all bits (1's complement), 2) Add 1 to the result. For example, 2's complement of 1101: Flip to 0010, add 1 to get 0011.
- Why It Works:In n-bit arithmetic, the 2's complement represents the negative value. Adding it is equivalent to subtraction, and overflow is handled automatically.
- Example: 1011 - 11012's complement of 1101 is 0011. Add: 1011 + 0011 = 1110. In decimal: 11 - 13 = -2, which matches the result when interpreted as signed binary.
Computers perform subtraction by adding the 2's complement of the subtrahend. This eliminates the need for separate subtraction circuitry.
Binary Multiplication and Division
Binary Multiplication Process
- Process:For each bit in the multiplier, if it's 1, add a shifted copy of the multiplicand. Shift left for each position.
- Example: 1011 ร 11011011 ร 1 = 1011 (no shift), 1011 ร 0 = 0000 (shift 1), 1011 ร 1 = 101100 (shift 2), 1011 ร 1 = 1011000 (shift 3). Sum: 10001111 = 143 in decimal.
- Efficiency:Processors optimize binary multiplication using hardware multipliers, but the fundamental logic remains the same.
Binary multiplication uses the same long multiplication method as decimal, but simpler since you only multiply by 0 or 1.
Binary Division Process
- Process:Repeatedly subtract the divisor from the dividend, building the quotient bit by bit. Similar to long division in decimal.
- Example: 1101 รท 1011101 (13) รท 101 (5) = 10 (2) remainder 11 (3). The calculator shows the quotient in binary, decimal, and hexadecimal.
- Use Cases:Understanding binary division helps in processor design, algorithm analysis, and computer science education.
Binary division uses repeated subtraction or shift-and-subtract algorithms, similar to long division in decimal.
Number System Conversions
Binary to Decimal Conversion
- Method:Write powers of 2 from right to left (2โฐ, 2ยน, 2ยฒ, ...), multiply each bit by its power, then sum. Example: 1011 = (1ร2ยณ) + (0ร2ยฒ) + (1ร2ยน) + (1ร2โฐ) = 8 + 0 + 2 + 1 = 11.
- Shortcut:Start from the right, double and add: 1 โ 1, double+1=3, double+0=6, double+1=11.
Convert binary to decimal by summing powers of 2 for each 1 bit.
Binary to Hexadecimal Conversion
- Method:Group binary into 4-bit chunks from right, convert each to hex digit (0-9, A-F). Example: 1011 1101 = BD (hex).
- Why Use Hex:More compact representation for debugging, memory addresses, and bit manipulation. Hex is especially common in programming and computer science.
Hexadecimal groups 4 bits, making conversions straightforward and compact.
FAQ
? How do I add binary numbers?
Binary addition works like decimal addition, but you only use 0 and 1. Add each bit from right to left, carrying over when the sum is 2 or more. For example, to add 1011 and 1101: Start from the right: 1+1=10 (write 0, carry 1), 1+0+1=10 (write 0, carry 1), 0+1+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1). The result is 11000. Our calculator performs this automatically and shows the result in binary, decimal, and hexadecimal formats.
? How do I subtract binary numbers?
Binary subtraction can be done using the same borrowing method as decimal subtraction, or using 2's complement addition (which is how computers perform subtraction). Our calculator uses 2's complement: to subtract B from A, it finds the 2's complement of B and adds it to A. For example, 1011 - 1101 becomes 1011 + (2's complement of 1101). The result is displayed in binary, decimal, and hexadecimal for easy verification.
? How do I multiply binary numbers?
Binary multiplication uses the same long multiplication method as decimal, but simpler since you only multiply by 0 or 1. For each bit in the multiplier, if it's 1, add a shifted copy of the multiplicand. Shift left for each position. For example, 1011 ร 1101 = 1011 + 00000 + 101100 + 1011000 = 10001111. Our calculator handles this automatically and supports numbers of any length.
? How do I divide binary numbers?
Binary division uses repeated subtraction or shift-and-subtract algorithms, similar to long division in decimal. The calculator repeatedly subtracts the divisor from the dividend, building the quotient bit by bit. For example, 1101 รท 101 = 10 remainder 11. The calculator shows the quotient in binary, decimal, and hexadecimal formats.
? How do I convert binary to decimal?
To convert binary to decimal, multiply each bit by its corresponding power of 2 (starting from 2^0 on the right) and sum the results. For example, 1011 = (1ร2ยณ) + (0ร2ยฒ) + (1ร2ยน) + (1ร2โฐ) = 8 + 0 + 2 + 1 = 11. Our calculator shows decimal, hexadecimal, and binary conversions in real-time as you type.
? How do I convert binary to hexadecimal?
To convert binary to hexadecimal, group the binary digits into sets of 4 from right to left, then convert each group to its hexadecimal equivalent (0-9, A-F). For example, 1011 1101 = BD in hexadecimal. If the leftmost group has fewer than 4 bits, pad with zeros. Our calculator automatically displays the hexadecimal equivalent of all inputs and results.
? What happens if I enter invalid characters?
The calculator only accepts 0 and 1. If you type any other character, it will be automatically filtered out and an error message will appear briefly to remind you that only binary digits are allowed. This ensures your calculations are always valid.
? Can I work with binary numbers of any length?
Yes! Unlike calculators that limit you to 8, 16, 32, or 64 bits, our binary calculator supports arbitrary-precision arithmetic. You can add, subtract, multiply, or divide binary numbers of any length, making it perfect for computer science education and large binary calculations.