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{{Short description|Bit indicating the sign of a number}}
{{Refimprove|date=December 2009}}
In [[computer science]], the '''sign bit''' is a [[bit]] in a [[signed number representation]] that indicates the [[sign (mathematics)|sign]] of a number.  Although only [[signedness|signed]] numeric data types have a sign bit, it is invariably located in the [[most significant bit]] position,<ref name=":0">{{Cite web |title=Intel® 64 and IA-32 Architectures Software Developer's Manual Combined Volumes: 1, 2A, 2B, 2C, 2D, 3A, 3B, 3C, 3D, and 4 |url=https://www.intel.com/content/www/us/en/content-details/782158/intel-64-and-ia-32-architectures-software-developer-s-manual-combined-volumes-1-2a-2b-2c-2d-3a-3b-3c-3d-and-4.html |access-date=2024-03-13 |website=Intel}}</ref> so the term may be used interchangeably with "most significant bit" in some contexts.

Almost always, if the sign bit is 0, the number is non-negative (positive or zero).<ref name=":0" />  If the sign bit is 1 then the number is negative. Formats other than [[two's complement]] integers allow a [[signed zero]]: distinct "positive zero" and "negative zero" representations, the latter of which does not correspond to the mathematical concept of a [[negative number]].

When using a complement representation, to convert a signed number to a wider format the additional bits must be filled with copies of the sign bit in order to preserve its numerical value,<ref name="Bryant" />{{Rp|61–62}} a process called ''[[sign extension]]'' or ''sign propagation''.<ref>{{Cite web |url=http://www.adrc.com/glossary/s1.html |title=Data Dictionary (Glossary and Algorithms) |website=Adroit Data Recovery Centre Pte Ltd |access-date=2014-12-15 |archive-date=2017-04-19 |archive-url=https://web.archive.org/web/20170419101741/http://www.adrc.com/glossary/s1.html |url-status=dead }}</ref>

== Sign bit weight in Two's complement ==
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;" 
|+
!Bits
!Value using Two's complement
|-
| {{Mono|'''0'''000}}
| 0
|-
| {{Mono|'''0'''001}}
| 1
|-
| {{Mono|'''0'''111}}
| 7
|-
| {{Mono|'''1'''000}}
| −8
|-
| {{Mono|'''1'''001}}
| −7
|-
| {{Mono|'''1'''111}}
| −1
|}
[[Two's complement]] is by far the most common format for signed integers. In Two's complement, the sign bit has the weight {{Math|-2<sup>w-1</sup>}} where w is equal to the bits position in the number.<ref name=":0" /> With an 8-bit integer, the sign bit would have the value of {{Math|-2<sup>'''8'''-1</sup>}}, or -128. Due to this value being larger than all the other bits combined, having this bit set would ultimately make the number negative, thus changing the sign.

== Sign bit weight in Ones' complement ==
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;" 
|+
!Bits
!Value using One's complement
|-
| {{Mono|'''0'''000}}
| 0
|-
| {{Mono|'''0'''001}}
| 1
|-
| {{Mono|'''0'''111}}
| 7
|-
| {{Mono|'''1'''000}}
| −7
|-
| {{Mono|'''1'''001}}
| −6
|-
| {{Mono|'''1'''111}}
| −0
|}
[[Ones' complement]] is similar to Two's Complement, but the sign bit has the weight {{Math|-(2<sup>w-1</sup> +1)}} where w is equal to the bits position in the number.{{Fact|date=March 2024}} With an 8-bit integer, the sign bit would have a value of {{Math|-(2<sup>'''8'''-1</sup> +1)}}, or -127. This allows for [[Signed zero|two types of zero]]: positive and negative, which is not possible with Two's complement.

== Sign bit in sign magnitude integers ==
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;" 
|+
!Bits
!Value using sign magnitude
|-
| {{Mono|'''0'''000}}
| 0
|-
| {{Mono|'''0'''001}}
| 1
|-
| {{Mono|'''0'''111}}
| 7
|-
| {{Mono|'''1'''000}}
| −0
|-
| {{Mono|'''1'''001}}
| −1
|-
| {{Mono|'''1'''111}}
| −7
|}
Using [[sign magnitude]], the sign bit directly determines the sign. If the sign bit is 0, the number is positive; if the sign bit is 1, the number is negative.<ref name="Bryant" />{{Rp|52–54}} Similarly with Ones' Complement, this allows for both a positive and a negative zero.

== Sign bit in floating-point numbers ==
[[Floating-point arithmetic|Floating-point]] numbers, such as [[IEEE floating point|IEEE format]], [[IBM hexadecimal floating-point|IBM format]], [[VAX]] format, and even the format used by the Zuse [[Z1 (computer)|Z1]] and [[Z3 (computer)|Z3]] use a [[Sign and magnitude]] representation.{{Fact|date=April 2024}}

==References==
{{reflist|refs=
<ref name=Bryant>{{cite book |last1=Bryant |first1=Randal E. |last2=O'Hallaron |first2=David R. |title=Computer Systems: a Programmer's Perspective |year=2003 |publisher= Prentice Hall |location=Upper Saddle River, New Jersey |isbn=0-13-034074-X |chapter=Chapter 2: Representing and Manipulating Information}}</ref>
}}

{{DEFAULTSORT:Sign Bit}}
[[Category:Binary arithmetic]]
[[Category:Computer arithmetic]]
[[Category:Sign (mathematics)]]


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