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The maximum value of an n-bit binary integer depends on whether it is unsigned or signed. An unsigned integer tops out at 2n − 1; an n-bit signed two’s-complement integer tops out at 2n−1 − 1.
How do you calculate the maximum value?
For an unsigned integer with n bits, the maximum is 2n − 1. Each bit can be 0 or 1, and the largest possible pattern sets every bit to 1. The bit positions have weights 1, 2, 4, and so on through 2n−1; adding those weights gives 2n − 1.
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The result is one less than the number of possible bit patterns because the values begin at zero. For example, 8 bits can encode 256 unsigned values, from 0 through 255.
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What changes for a signed binary integer?
For an n-bit signed integer using two’s-complement representation, the maximum is 2n−1 − 1. Its full range is −2n−1 through 2n−1 − 1, inclusive. One bit participates in the sign representation, leaving n − 1 bits for the positive side.
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This range is asymmetric: the minimum is one greater in magnitude than the maximum. For an 8-bit signed value, for instance, the range is −128 through 127, not −127 through 127.
Examples by bit width and signedness
| Integer type | Maximum value |
|---|---|
| 8-bit unsigned | 255 |
| 16-bit unsigned | 65,535 |
| 32-bit signed, two’s complement | 2,147,483,647 |
| 32-bit unsigned | 4,294,967,295 |
The two 32-bit examples show why “32-bit” alone does not determine a maximum: signedness changes the range.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How to find the limit for a programming type
Use the limit defined for the exact type rather than inferring width from a name. Language and platform conventions can differ; for example, GNU’s C manual notes that the width of long int depends on the system.
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One free scan finds every outdated or missing driver and matches the right update for your exact hardware.Free scan · exact hardware match- C: use limits macros such as
INT_MAXandUINT_MAXfrom the relevant limits header. GNU’s reference documents that integer type widths can vary by system: GNU C Library: Integer Types. - Rust: a fixed-width unsigned type such as
u32exposes its maximum asu32::MAX. See Rust documentation foru32::MAX. - Swift: fixed-width integer types provide a
maxproperty for the maximum representable value. See Apple Developer Documentation:max.
For exact-width integer limits, POSIX defines unsigned maxima as 2N − 1 and signed maxima as 2N−1 − 1: POSIX stdint.h definitions.
Quick Recap
What to check when comparing integer limits
- Bit width: the number of bits used to store the integer.
- Signedness and representation: unsigned values are nonnegative; the signed formula here assumes two’s complement.
- Actual language type: confirm its documented width and limit, especially for types whose width can vary by platform.
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