Binary Calculator
Convert binary to decimal and do base-2 addition, subtraction, multiplication, and division with step-by-step results.
Calculation Examples
How to Use the Binary Calculator
The binary calculator converts numbers and does full arithmetic in the base-2 number system. To convert, type a binary number using only 0s and 1s and the tool returns its decimal value.
For arithmetic, enter two binary values and choose addition, subtraction, multiplication, or division. The calculator applies the carry and borrow rules and shows a positional breakdown, so you can see where each bit comes from. For example, 1010 + 1101 = 10111, which is 10 + 13 = 23 in decimal.
How Binary Calculations Work
Binary is a base-2 positional system where each digit represents a power of 2. The binary number \(101_2\) equals \((1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 4 + 0 + 1 = 5_{10}\). In general, a binary number with digits \(d_i\) has the value \(\sum d_i \times 2^i\).
In addition, \(1 + 1 = 10_2\) (write 0, carry 1). Subtraction borrows in units of \(2^n\) instead of 10. Processors store signed integers in two's complement: to negate a number, invert every bit and add 1. In 8 bits, 5 is 00000101 and -5 is 11111011. An n-bit two's complement integer ranges from \(-2^{n-1}\) to \(2^{n-1} - 1\), so 8 bits cover -128 to 127, while unsigned 8-bit values cover 0 to 255. Because subtraction becomes addition, the same adder circuit handles both. IEEE 754 is a different standard: it defines floating-point formats, not integers.
Useful Tips 💡
- Pad shorter binary numbers with leading zeros so both inputs have equal length before you add or subtract by hand.
- Check that your input contains only 0s and 1s. Spaces and prefixes such as 0b can cause an error.
- Verify a result by converting it back: 10111 equals 16 + 4 + 2 + 1, which is 23.
📋Steps to Calculate
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Select the operation: convert, add, subtract, multiply, or divide.
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Enter your binary numbers using only 0s and 1s.
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Click Calculate to view the result with a step-by-step positional breakdown.
Mistakes to Avoid ⚠️
- Dropping leading zeros when comparing values. 00101 and 101 are both 5, but alignment matters in operations.
- Applying decimal carrying to binary addition. In binary, 1 + 1 = 10, not 2.
- Assuming every number fits in 8 bits. The unsigned maximum is 255, and 256 needs 9 bits (100000000). Adding 1 to 255 in 8 bits wraps to 0.
- Confusing signed and unsigned values. The same 8 bits 11111011 are 251 unsigned but -5 in two's complement.
Practical Applications📊
Verify bitwise operations and register values during low-level programming and hardware debugging.
Check hand calculations in digital electronics, logic gate design, and computer architecture coursework.
Work out subnet masks. A /26 mask is 255.255.255.192, which is 11000000 in the last octet, giving 64 addresses and 62 usable hosts per IPv4 subnet.
Size data fields. n bits hold values up to 2 to the power of n minus 1, so 8 bits hold 255 and 16 bits hold 65,535.