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There is no single maximum for a binary number: the answer depends on how many bits it has and whether those bits represent an unsigned or signed integer. With n bits, an unsigned integer tops out at 2n − 1; an n-bit two’s-complement signed integer tops out at 2n−1 − 1. An unlimited-length binary numeral has no finite maximum.
How bit width sets the maximum
A fixed-width value made of n bits has 2n possible bit patterns. For an unsigned integer, every pattern represents a nonnegative value, from 0 through 2n − 1. The maximum is the all-ones pattern.
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For example, the 8-bit pattern 11111111 represents 255 when read as unsigned: 28 − 1.
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Most modern signed integers use two’s-complement representation. Its highest-order bit contributes a negative weight, so an n-bit signed integer ranges from −2n−1 to 2n−1 − 1. The same width can therefore represent a smaller positive maximum when signed, in exchange for representing negative values. The GNU C Language Manual describes unsigned values as using the entire space for nonnegative numbers.
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| Width | Unsigned maximum | Signed two’s-complement maximum | Signed minimum |
|---|---|---|---|
| 8 bits | 255 | 127 | −128 |
| 16 bits | 65,535 | 32,767 | −32,768 |
| 32 bits | 4,294,967,295 | 2,147,483,647 | −2,147,483,648 |
| 64 bits | 18,446,744,073,709,551,615 | 9,223,372,036,854,775,807 | −9,223,372,036,854,775,808 |
For 8 bits, 01111111 is the largest signed pattern and equals 127. The all-ones pattern, 11111111, is −1 in two’s complement, even though it means 255 when interpreted as unsigned.
Why “binary number” needs a width and interpretation
A string of binary digits alone does not establish a maximum. Without a fixed length, another digit can always be added to make a larger nonnegative numeral. For a fixed-width integer, specify both the bit count and whether the value is signed or unsigned.
These formulas concern integer representations. Floating-point formats also use binary bits, but divide them among sign, exponent, and precision fields; their largest finite values are a separate question.
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A mathematical limit for a given width does not automatically tell you the width of a type named int, long, or short. In C, for example, the GNU manual notes that long int can be 32 or 64 bits depending on the system. Check the language rules, platform ABI, and documented limits for the actual type; fixed-width types make the intended width explicit when the language and implementation support them.
As one implementation reference, Microsoft Learn’s C range table, updated August 3, 2021, gives 4,294,967,295 as the maximum for its 32-bit unsigned long and 2,147,483,647 for its 32-bit signed long; it lists 65,535 as the upper bound for unsigned short. These are documented C implementation examples, not universal guarantees for every language or platform. See Microsoft’s range of integer values reference.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What happens at the maximum
Trying to represent a result beyond a type’s range raises an overflow issue. The precise behavior depends on the language and whether the type is signed or unsigned; do not assume that one language’s rules apply to another. The GNU C Language Manual discusses these representation and arithmetic distinctions at Integer Representations.
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