Editing Excess-3 (section)

==Representation==
Biased codes are a way to represent values with a balanced number of positive and negative numbers using a pre-specified number ''N'' as a biasing value. Biased codes (and [[Gray code]]s) are non-weighted codes. In excess-3 code, numbers are represented as decimal digits, and each digit is represented by four [[bit]]s as the digit value plus 3 (the "excess" amount):
* The smallest binary number represents the smallest value ({{nobr|0 − excess}}).
* The greatest binary number represents the largest value ({{nobr|2<sup>''N''+1</sup> − excess − 1}}).

{| class="wikitable"
|+ Excess-3, and Stibitz code
|-
! Decimal
! Excess-3
! Stibitz
! [[BCD 8-4-2-1]]
! Binary
! 3-of-6 [[CCITT]] <br/>extension<ref name="CCITT_1959"/><ref name="Steinbuch_1962"/>
! 4-of-8 [[Hamming code|Hamming]] <br/>extension<ref name="Steinbuch_1962"/>
|- align="center"
| '''0'''
| style="background:#0FF" | '''0011'''
| '''0011'''
| 0000
| 0000
| …'''10'''
| …'''0011'''
|- align="center"
| '''1'''
| style="background:#0F0" | '''0100'''
| '''0100'''
| 0001
| 0001
| …'''11'''
| …'''1011'''
|- align="center"
| '''2'''
| style="background:#FF0" | '''0101'''
| '''0101'''
| 0010
| 0010
| …'''10'''
| …'''0101'''
|- align="center"
| '''3'''
| style="background:#FF8000" | '''0110'''
| '''0110'''
| 0011
| 0011
| …'''10'''
| …'''0110'''
|- align="center"
| '''4'''
| style="background:#F00;" | '''0111'''
| '''0111'''
| 0100
| 0100
| …'''00'''
| …'''1000'''
|- align="center"
| '''5'''
| style="background:#F00;" | '''1000'''
| '''1000'''
| 0101
| 0101
| …'''11'''
| …'''0111'''
|- align="center"
| '''6'''
| style="background:#FF8000" | '''1001'''
| '''1001'''
| 0110
| 0110
| …'''10'''
| …'''1001'''
|- align="center"
| '''7'''
| style="background:#FF0" | '''1010'''
| '''1010'''
| 0111
| 0111
| …'''10'''
| …'''1010'''
|- align="center"
| '''8'''
| style="background:#0F0" | '''1011'''
| '''1011'''
| 1000
| 1000
| …'''00'''
| …'''0100'''
|- align="center"
| '''9'''
| style="background:#0FF" | '''1100'''
| '''1100'''
| 1001
| 1001
| …'''10'''
| …'''1100'''
|}

To encode a number such as 127, one simply encodes each of the decimal digits as above, giving (0100, 0101, 1010).

Excess-3 arithmetic uses different [[algorithm]]s than normal non-biased BCD or binary [[positional system]] numbers. After adding two excess-3 digits, the raw sum is excess-6. For instance, after adding 1 (0100 in excess-3) and 2 (0101 in excess-3), the sum looks like 6 (1001 in excess-3) instead of 3 (0110 in excess-3). To correct this problem, after adding two digits, it is necessary to remove the extra bias by subtracting binary 0011 (decimal 3 in unbiased binary) if the resulting digit is less than decimal 10, or subtracting binary 1101 (decimal 13 in unbiased binary) if an [[integer overflow|overflow]] (carry) has occurred. (In 4-bit binary, subtracting binary 1101 is equivalent to adding 0011 and vice versa.)<ref>{{cite book |title=Computer Architecture and Organization  |last=Hayes |first=John P. |isbn=0-07-027363-4 |date=1978 |publisher=McGraw-Hill International Book Company |page=156}}</ref>