Start With the Question the Bits Must Answer
A binary expression can describe pure integer mathematics or behavior inside a fixed register. Those are related but different questions. Exact 11111111 + 1 equals 100000000. In an unsigned 8-bit word, the same operation stores 00000000 and overflows because the ninth bit does not fit.
Write down the operation, whether the operands are exact or fixed width, the number of bit positions, and whether a fixed pattern is unsigned or signed. This short specification prevents most binary-calculator disagreements before any arithmetic begins.
- Operation and operand bit strings
- Exact integer or fixed-width word
- Word size when fixed
- Unsigned or signed two's-complement interpretation
- Required output bases and overflow evidence
Convert Binary Place Values Exactly
Binary position zero represents 2^0, the next position represents 2^1, and each move left doubles the place value. The bit string 10110 therefore equals 1 x 16 + 0 x 8 + 1 x 4 + 1 x 2 + 0 x 1 = 22. Conversion is an integer sum, so no decimal floating-point approximation is required.
Hexadecimal is a compact companion to binary because one hex digit represents exactly four bits. Group 10110110 as 1011 0110 to obtain B6. Octal groups three bits at a time. Group separators improve scanning but never alter the represented value.
Work Through Binary Arithmetic and Remainders
Addition follows the same place-value idea as decimal arithmetic, but every column has base two. One plus one writes 0 and carries 1; one plus one plus a carry writes 1 and carries 1. Subtraction borrows one binary place, which contributes two to the current position. Multiplication forms shifted partial products for multiplier bits equal to 1.
Integer division finds a whole quotient and remainder. The calculator truncates the quotient toward zero, then verifies remainder = dividend - quotient x divisor. Division by zero has no defined integer result and is rejected rather than replaced with an infinity value.
- 1010 + 0011 = 1101
- 1010 - 0011 = 0111
- 1010 x 0011 = 11110
- 10101 / 00100 = 00101 remainder 00001
Apply AND, OR, XOR, NOT, and Shifts by Position
Bitwise logic aligns operands at their rightmost positions. AND selects shared 1 bits, OR combines 1 bits, and XOR selects positions where the operands differ. For 1100 and 1010, AND gives 1000, OR gives 1110, and XOR gives 0110. These operations are useful for masks, flags, permissions, color channels, and low-level fields.
NOT must know the word width because it inverts every included position. In an 8-bit word, NOT 00001111 is 11110000. Left shift moves bits toward higher places; exact left shift by k multiplies by 2^k. Right shift removes low positions, while signed fixed-width right shift extends the sign bit.
Read Unsigned and Signed Two's-Complement Words
An unsigned w-bit word covers 0 through 2^w - 1. A signed two's-complement word uses the highest bit as part of the encoding and covers -2^(w - 1) through 2^(w - 1) - 1. The pattern 11111101 is unsigned 253 or signed -3 at eight bits; neither reading is meaningful until the width and interpretation are stated.
To read a signed pattern whose highest bit is 1, subtract 2^w from its unsigned value. To encode a negative value, add 2^w and write the resulting low w bits. This representation gives one zero pattern and makes low-bit addition consistent across positive and negative fixed-width values.
Separate Overflow From the Stored Word
A processor-sized word can contain a valid pattern even when the exact mathematical result does not fit. Unsigned overflow occurs below zero or above 2^w - 1. Signed overflow occurs outside -2^(w - 1) through 2^(w - 1) - 1. Retaining the low w bits is wrapping; it is not evidence that the exact result changed.
Audit both rows. Signed 8-bit 01111111 + 00000001 has exact result 128, outside the signed range. The stored pattern is 10000000, interpreted as -128. A trustworthy report labels overflow and preserves 128 instead of showing only the wrapped negative value.
Audit the Calculator Result
First reproduce small inputs by place value or column arithmetic. Next check that the selected representation matches the problem statement. Compare binary and hexadecimal in four-bit groups, verify the decimal interpretation, and inspect the exact-result row before the stored-word row. For division, reconstruct dividend = quotient x divisor + remainder.
For bitwise operations, compare several positions directly. For shifts, verify the direction and count. When overflow is Yes, confirm that the stored word equals the low w exact-result bits and that the exact value is genuinely outside the displayed range.
- Check every input contains only 0 and 1 after separators are removed.
- Verify word size and signedness before interpreting a leading 1.
- Reconstruct the arithmetic with decimal or hexadecimal as an independent view.
- Treat grouped spaces as formatting only.
- Use the target language or hardware specification for implementation-specific promotion and shift rules.
Know the Boundary Between the Model and an Implementation
This calculator provides exact integer operations and explicit fixed-width clamping. Real languages and processors can add rules for operand promotion, shift-count masking, unsigned right shift, overflow traps, wrapping modes, and division signs. A result that is correct for this stated model may need adaptation before it reproduces a particular compiler or instruction.
Use the calculator for learning, inspection, and scenario checks. For production code, cryptographic work, protocol parsing, or hardware design, confirm the official specification, add test vectors at minimum and maximum values, and test zero, negative signed patterns, oversized shifts, and overflow boundaries.
Frequently asked questions
How do I add two binary numbers?
Choose Add, enter both base-2 operands, and use exact mode unless the question specifies a word size. For 1010 + 11, the exact result is 1101, which is decimal 13.
Can this calculator subtract binary numbers and show a negative result?
Yes. Exact mode displays a negative result with a leading minus sign, such as -10 for negative two. Fixed signed mode instead displays the stored two's-complement word and keeps the exact decimal result visible.
How does binary division handle a remainder?
Division returns an integer quotient truncated toward zero and reports the remainder separately. Dividing 10101 by 100 gives binary quotient 101 and binary remainder 1. Division by zero is rejected.
What is the difference between exact and fixed-width binary mode?
Exact mode preserves the complete mathematical integer. Fixed-width mode retains only the low selected number of bits, interprets that word as unsigned or signed, and reports whether the exact result overflowed the allowed range.
How do I convert a signed binary number from two's complement?
Choose Signed two's complement and the required word size. A leading 1 is negative only after the width is known. For example, 11111101 is -3 as signed 8-bit but 253 as unsigned 8-bit.
What happens when an 8-bit binary result overflows?
The calculator preserves the exact result, stores only the low eight bits, and marks overflow Yes when the exact value is outside the selected range. Unsigned 11111111 + 1 wraps to 00000000 while the exact result remains 256.
References
These sources support the method or guidance used for Binary Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Binary Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.