Disclaimer: this post is an English translation of my original article, first published at z0rr0.blog/unsigned-integers.
Why Do We Need Them?
This is a short discussion about unsigned integers in programming. From a mathematical point of view, they may look a bit strange - why do we even need such “truncated” numbers?
Nevertheless, there are solid reasons for their existence. Here are the main ones:
Extending the available numeric range using the same number of bits. You can store more values if negative numbers are not required. For example, an 8-bit signed integer can store values from −128 to 127, while an unsigned type can store values from 0 to 255.
Additional control over business logic. Sometimes application requirements explicitly forbid negative values (for example, quantity of goods, customer age, file size, etc.). Using unsigned types helps to clearly express these constraints and prevents errors related to improper use of negative numbers.
Performance optimizations. On some CPU architectures, operations on unsigned numbers may be faster than on signed ones. Additionally, this serves as a hint for the compiler or interpreter that certain optimizations are possible.
How Do They Work?
Let’s write a few small examples to clearly see how unsigned integers behave.
Here is an example in Go. The program iterates over values from 0 to 259 and prints their representation as signed and unsigned numbers, along with their binary and hexadecimal forms.
package main
import (
"fmt"
"unsafe"
)
func s2u(s int8) uint8 {
return *(*uint8)(unsafe.Pointer(&s))
}
func main() {
var (
signed int8
unsigned uint8
)
size := unsafe.Sizeof(signed)
fmt.Printf("Size of int8 and uint8: %d byte(s)\n\n", size)
println("+---------------------------------------------------------------------------------+")
println("| Compare signed and unsigned numbers |")
println("+-----+----------+----------+-----+--------+-----------+-----+--------------------+")
println("| # | unsigned | bin | hex | signed | bin | hex | signed as unsigned |")
println("|-----|----------|----------|-----|--------|-----------|-----|--------------------|")
for i := range 260 {
signed = int8(i)
unsigned = uint8(i)
signedAsUnsigned := s2u(signed)
fmt.Printf("| %-3[1]d | %-8[2]d | %-8[2]b | %-3[2]x | %-6[3]d | %-9[3]b | %-3[3]x | %-18[4]b |\n", i, unsigned, signed, signedAsUnsigned)
}
println("+-----+----------+----------+-----+--------+-----------+-----+--------------------+")
}
The output looks like this (heavily truncated for readability, with ... inserted):
Size of int8 and uint8: 1 byte(s)
+---------------------------------------------------------------------------------+
| Compare signed and unsigned numbers |
+-----+----------+----------+-----+--------+-----------+-----+--------------------+
| # | unsigned | bin | hex | signed | bin | hex | signed as unsigned |
|-----|----------|----------|-----|--------|-----------|-----|--------------------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 10 | 2 | 2 | 10 | 2 | 10 |
...
| 126 | 126 | 1111110 | 7e | 126 | 1111110 | 7e | 1111110 |
| 127 | 127 | 1111111 | 7f | 127 | 1111111 | 7f | 1111111 |
| 128 | 128 | 10000000 | 80 | -128 | -10000000 | -80 | 10000000 |
| 129 | 129 | 10000001 | 81 | -127 | -1111111 | -7f | 10000001 |
| 130 | 130 | 10000010 | 82 | -126 | -1111110 | -7e | 10000010 |
...
| 254 | 254 | 11111110 | fe | -2 | -10 | -2 | 11111110 |
| 255 | 255 | 11111111 | ff | -1 | -1 | -1 | 11111111 |
| 256 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 257 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 258 | 2 | 10 | 2 | 2 | 10 | 2 | 10 |
| 259 | 3 | 11 | 3 | 3 | 11 | 3 | 11 |
+-----+----------+----------+-----+--------+-----------+-----+--------------------+
And here is a similar example in Rust:
fn main() {
let size = std::mem::size_of::<u8>();
println!("Size of i8 and u8: {} byte(s)\n", size);
println!("+---------------------------------------------------------------------------------+");
println!("| Compare signed and unsigned numbers |");
println!("+-----+----------+----------+-----+--------+-----------+-----+--------------------+");
println!("| # | unsigned | bin | hex | signed | bin | hex | signed as unsigned |");
println!("|-----|----------|----------|-----|--------|-----------|-----|--------------------|");
for number in 0..260 {
let unsigned: u8 = number as u8;
let signed: i8 = unsigned as i8;
let s2u: u8 = signed as u8;
println!(
"| {0:<3} | {1:<8} | {1:<8b} | {1:<3x} | {2:<6} | {2:<9b} | {2:<3x} | {3:<18b} |",
number, unsigned, signed, s2u,
);
}
println!("+-----+----------+----------+-----+--------+-----------+-----+--------------------+");
}
The output is very similar, but with a few differences:
In Rust, negative numbers are printed without a minus sign in binary and hexadecimal representations.
In Go, the same values are printed using two’s complement with a leading minus sign.
It is important to understand that this is only a difference in representation. In memory, both languages store exactly the same data, and Rust also uses two’s complement internally.
Size of i8 and u8: 1 byte(s)
+---------------------------------------------------------------------------------+
| Compare signed and unsigned numbers |
+-----+----------+----------+-----+--------+-----------+-----+--------------------+
| # | unsigned | bin | hex | signed | bin | hex | signed as unsigned |
|-----|----------|----------|-----|--------|-----------|-----|--------------------|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 2 | 10 | 2 | 2 | 10 | 2 | 10 |
...
| 126 | 126 | 1111110 | 7e | 126 | 1111110 | 7e | 1111110 |
| 127 | 127 | 1111111 | 7f | 127 | 1111111 | 7f | 1111111 |
| 128 | 128 | 10000000 | 80 | -128 | 10000000 | 80 | 10000000 |
| 129 | 129 | 10000001 | 81 | -127 | 10000001 | 81 | 10000001 |
| 130 | 130 | 10000010 | 82 | -126 | 10000010 | 82 | 10000010 |
...
| 254 | 254 | 11111110 | fe | -2 | 11111110 | fe | 11111110 |
| 255 | 255 | 11111111 | ff | -1 | 11111111 | ff | 11111111 |
| 256 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 257 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 258 | 2 | 10 | 2 | 2 | 10 | 2 | 10 |
| 259 | 3 | 11 | 3 | 3 | 11 | 3 | 11 |
+-----+----------+----------+-----+--------+-----------+-----+--------------------+
A Short Digression on Two’s Complement. In a very simplified form, the algorithm for obtaining two’s complement (using -2 as an example) can be described as follows:
As shown in the Rust output, the binary representation of -2 is 11111110.
The most significant bit (the leftmost one) represents the sign: 0 for positive, 1 for negative.
If the most significant bit is 0, the two’s complement representation is identical to the direct binary representation.
Otherwise, to obtain the two’s complement of a negative number, invert all bits except the most significant one, resulting in 10000001, and then add 1, producing 10000010. The first bit is interpreted as the sign, and the remaining bits (0000010) represent the value 2. The final result is -2.
Looking at the outputs above, two natural questions arise:
How can the same binary representation be interpreted differently by the computer?
How do we correctly convert signed values to unsigned ones and vice versa?
For the first question, the answer is simple: there is no difference in memory. During compilation or interpretation, the analyzer tracks which type is used at each point. Depending on that type, the same sequence of bits is interpreted differently. Type information effectively lives in the program’s “metadata”, not in the data itself.
Consider this Go program program.go:
package main
func signedLess(a, b int8) bool {
return a < b
}
func unsignedLess(a, b uint8) bool {
return a < b
}
func main() {
var (
signedA int8 = -1
signedB int8 = 2
unsignedA uint8 = 255
unsignedB uint8 = 2
)
println(signedLess(signedA, signedB))
println(unsignedLess(unsignedA, unsignedB))
}
If we inspect its ARM64 assembly output:
go tool compile -S program.go
main.signedLess STEXT size=32 args=0x8 locals=0x0 funcid=0x0 align=0x0 leaf
0x0000 00000 (program.go:3) TEXT main.signedLess(SB), LEAF|NOFRAME|ABIInternal, $0-8
0x0000 00000 (program.go:3) FUNCDATA $0, gclocals·g5+hNtRBP6YXNjfog7aZjQ==(SB)
0x0000 00000 (program.go:3) FUNCDATA $1, gclocals·g5+hNtRBP6YXNjfog7aZjQ==(SB)
0x0000 00000 (program.go:3) FUNCDATA $5, main.signedLess.arginfo1(SB)
0x0000 00000 (program.go:3) FUNCDATA $6, main.signedLess.argliveinfo(SB)
0x0000 00000 (program.go:3) PCDATA $3, $1
0x0000 00000 (program.go:4) MOVB R0, R2
0x0004 00004 (program.go:4) MOVB R1, R1
0x0008 00008 (program.go:4) CMPW R2, R1
0x000c 00012 (program.go:4) CSET GT, R0
0x0010 00016 (program.go:4) RET (R30)
0x0000 02 1c 40 93 21 1c 40 93 3f 00 02 6b e0 d7 9f 9a ..@.!.@.?..k....
0x0010 c0 03 5f d6 00 00 00 00 00 00 00 00 00 00 00 00 .._.............
main.unsignedLess STEXT size=32 args=0x8 locals=0x0 funcid=0x0 align=0x0 leaf
0x0000 00000 (program.go:7) TEXT main.unsignedLess(SB), LEAF|NOFRAME|ABIInternal, $0-8
0x0000 00000 (program.go:7) FUNCDATA $0, gclocals·g5+hNtRBP6YXNjfog7aZjQ==(SB)
0x0000 00000 (program.go:7) FUNCDATA $1, gclocals·g5+hNtRBP6YXNjfog7aZjQ==(SB)
0x0000 00000 (program.go:7) FUNCDATA $5, main.unsignedLess.arginfo1(SB)
0x0000 00000 (program.go:7) FUNCDATA $6, main.unsignedLess.argliveinfo(SB)
0x0000 00000 (program.go:7) PCDATA $3, $1
0x0000 00000 (program.go:8) MOVBU R0, R2
0x0004 00004 (program.go:8) MOVBU R1, R1
0x0008 00008 (program.go:8) CMPW R2, R1
0x000c 00012 (program.go:8) CSET HI, R0
0x0010 00016 (program.go:8) RET (R30)
0x0000 02 1c 40 d3 21 1c 40 d3 3f 00 02 6b e0 97 9f 9a ..@.!.@.?..k....
0x0010 c0 03 5f d6 00 00 00 00 00 00 00 00 00 00 00 00 .._.............
As shown above, even though signedA and unsignedA are stored identically in memory, the load instructions (MOVB vs MOVBU) and the condition codes used to produce the result (CSET GT vs CSET HI) are different. This is because the compiler knows which type is used in each function and emits machine code appropriate for operating on that type. The CMPW R2, R1 instruction sets condition flags based on the most significant bit, which may be interpreted as a sign bit or as part of the magnitude depending on the comparison that follows. The comparison itself is selected based on the type: GT (greater than) is used for signed comparisons, while HI (higher) is used for unsigned comparisons.
Some languages are strict about signedness. Go and Rust do not allow implicit conversions between signed and unsigned integers, which is why the examples used unsafe.Pointer in Go and as casts in Rust.
var unsigned uint8 = -1
Even in Go, mistakes are possible. For example, this code produces an infinite loop:
package main
func main() {
var i uint
for i = 5; i >= 0; i-- {
println(i)
}
}
This is why the Google C++ Style Guide and the designers of Java were historically skeptical about using unsigned integers for arithmetic, recommending them mainly for bit masks.
In C and C++, implicit conversions are allowed and often happen without warnings:
uint8_t unsigned_number = -1;
A classic example is comparing signed and unsigned values, such as if (-1 > unsigned_number), which may unexpectedly evaluate to true.
Java historically had no unsigned primitives (except char, which is 16-bit), as the language designers considered them too error-prone.
Python goes even further: it has a single int type with arbitrary precision. When interpreting raw bytes, you must explicitly specify whether the value is signed or unsigned:
number = b"\xff"
unsigned = int.from_bytes(number, byteorder="big", signed=False)
signed = int.from_bytes(number, byteorder="big", signed=True)
Now let us move on to the second question: how to correctly convert a signed integer to an unsigned one and vice versa. In modern programming languages, this is typically done using two’s complement representation, which we discussed earlier — for example, the value 254 (binary 11111110) is interpreted as -2. In fact, no inverse algorithm is required here: we simply reinterpret the same sequence of bits, treating the most significant bit as part of the value rather than as an explicit - / + sign.
Another way to reason about two’s complement is the following:
If the number is non-negative, the signed and unsigned representations are identical.
If the number is negative, the most significant bit is 1, and the value can be computed as 254 − 256 = −2, where 256 is 2^8, i.e. the maximum representable value for an 8-bit number plus one.
Can We Look at Another Example?
I would like to share a few examples where incorrect handling of unsigned integers has led to bugs.
Consider a ClickHouse table:
CREATE TABLE test
(
timestamp DateTime,
sign Int8 DEFAULT 1,
value UInt8,
comment String
) ENGINE = MergeTree() ORDER BY timestamp;
This table stores append-only events. Logical deletions are represented by sign = -1, while value is stored as UInt8.
Now we aggregate by day using a materialized view:
CREATE MATERIALIZED VIEW mv_test
(
`day` Date,
`total` UInt64
)
ENGINE = SummingMergeTree() ORDER BY day
AS
SELECT toStartOfDay(timestamp) AS day,
sum(value * sign) AS total
FROM test
GROUP BY day;
We choose the UInt64 type for total because we expect it to store the sum of UInt8 value events; rollbacks should evaluate to 0 and therefore not affect the final result.
INSERT INTO test
VALUES ('2026-01-01 00:00:01', 1, 1, 'number 1'),
('2026-01-01 00:00:02', 1, 2, 'number 2'),
('2026-01-01 00:00:06', 1, 255, 'number 255, max unsigned 8-bit');
SELECT * FROM mv_test ORDER BY day FORMAT Vertical;
Row 1:
──────
day: 2026-01-01
total: 258
Everything appears to work correctly, but only while implicitly assuming that rollback timestamps always совпiding with the original events. Once this assumption is broken, either by mistake or by design — issues start to surface. For example, the "2nd event rollback" record falls on the day after the "2nd event":
INSERT INTO test
VALUES ('2026-01-02 23:59:55', 1, 1, '1st event'),
('2026-01-02 23:59:58', -1, 1, '1st event rollback'),
('2026-01-02 23:59:59', 1, 1, '2nd event'),
('2026-01-03 00:00:02', -1, 1, '2nd event rollback');
SELECT * FROM mv_test ORDER BY day FORMAT Vertical;
Row 1:
──────
day: 2026-01-01
total: 258
Row 2:
──────
day: 2026-01-02
total: 1
Row 3:
──────
day: 2026-01-03
total: 18446744073709551615
However, a query equivalent to the materialized view mv_test produces a different result:
SELECT toStartOfDay(timestamp) AS day,
sum(value * sign) AS total
FROM test
GROUP BY day
ORDER BY day
FORMAT Vertical;
Row 1:
──────
day: 2026-01-01 00:00:00
total: 258
Row 2:
──────
day: 2026-01-02 00:00:00
total: 1
Row 3:
──────
day: 2026-01-03 00:00:00
total: -1
The value 18446744073709551615 is the maximum representable value of UInt64, i.e. 2^64 − 1, which corresponds to -1 when interpreted as an unsigned number. At the same time, the second query returns the correct result of -1, because by default the database engine interprets total as a signed value.
We once encountered a very similar issue in a real production system and it involved money. At some point, the total amount of refunds for a single day unexpectedly exceeded the revenue accumulated over several years :)
However, since this is only a matter of how the result is interpreted, and the underlying bytes are physically identical, the fix can be straightforward. If the materialized view is based on a separate table, it is sufficient to change the column type using an ALTER TABLE operation. Otherwise, the materialized view must be recreated and fully repopulated.
What Conclusions Can We Draw?
Unsigned integers play an important role in programming, enabling efficient memory usage and self-documenting constraints.
Signed and unsigned integers are merely different interpretations of the same bit patterns.
Choosing between them requires careful consideration of language rules, implicit conversions, and edge cases.
What Else Is Worth Reading?
Wikipedia - Unsigned integer
Wikipedia - Two's complement
Blog posts by Julia Evans
