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<H2>2.6.5 Other Arithmetic Systems</H2>
<P><P CLASS="BodyAfterHead"><A NAME="pgfId=197759"></A>There are other schemes
for addition and multiplication that are useful in special circumstances.
Addition of numbers using <B>redundant binary encoding</B> avoids carry
propagation and is thus potentially very fast. Table 2.13 shows the
rules for addition using an intermediate carry and sum that are added without
the need for carry. For example,</P>
<P><SPAN CLASS="ComputerFirst"> <A NAME="pgfId=197760"></A> binary decimal redundant
CSD</SPAN>
<SPAN CLASS="Computer"> <A NAME="pgfId=197862"></A> binary vector</SPAN>
<SPAN CLASS="Computer"> <A NAME="pgfId=197887"></A> </SPAN> <SPAN CLASS="Computer">
<A NAME="pgfId=197546"></A> 1010111 87 <IMG SRC=
"CH02-81.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
<IMG SRC=
"CH02-82.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
addend
</SPAN> <SPAN CLASS="Computer"> <A NAME="pgfId=197547"></A> + 1100101 101 + <IMG SRC=
"CH02-83.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
+ <IMG SRC=
"CH02-84.gif" ALIGN="BASELINE" WIDTH="50" HEIGHT="14" NATURALSIZEFLAG="3">
augend
</SPAN> <SPAN CLASS="ComputerFirst"> <A NAME="pgfId=197580"></A>= 10111100 = 188 <IMG SRC=
"CH02-85.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
<IMG SRC=
"CH02-86.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
intermediate
sum</SPAN> <SPAN CLASS="Computer"> <A NAME="pgfId=197584"></A> <IMG SRC=
"CH02-87.gif" ALIGN="BASELINE" WIDTH="51" HEIGHT="16" NATURALSIZEFLAG="3">
<IMG SRC=
"CH02-88.gif" ALIGN="BASELINE" WIDTH="48" HEIGHT="16" NATURALSIZEFLAG="3">
intermediate
carry</SPAN> <SPAN CLASS="Computer"> <A NAME="pgfId=197591"></A> </SPAN>
<SPAN CLASS="ComputerLast"> <A NAME="pgfId=197548"></A> = <IMG SRC=
"CH02-89.gif" ALIGN="BASELINE" WIDTH="57" HEIGHT="16" NATURALSIZEFLAG="3">
= <IMG SRC=
"CH02-90.gif" ALIGN="BASELINE" WIDTH="57" HEIGHT="16" NATURALSIZEFLAG="3">
sum</SPAN>
<TABLE BORDER="0" CELLSPACING="2" CELLPADDING="0">
<TR>
<TD COLSPAN="6"><P CLASS="TableTitle"><A NAME="pgfId=198900"></A>TABLE 2.13 Redundant
binary addition.</TD></TR>
<TR>
<TD><P CLASS="TableFirst"><A NAME="pgfId=198912"></A>A[<SPAN CLASS="EquationVariables">
i</SPAN> <SPAN CLASS="White"> </SPAN>]</TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=198914"></A>B[<SPAN CLASS="EquationVariables">
i</SPAN> <SPAN CLASS="White"> </SPAN>]</TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=198916"></A>A[<SPAN CLASS="EquationVariables">
i</SPAN> <SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]</TD>
<TD><P CLASS="TableFirst"><A NAME="pgfId=198918"></A>B[<SPAN CLASS="EquationVariables">
i</SPAN> <SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]</TD>
<TD><P><P CLASS="TableFirst"><A NAME="pgfId=198920"></A>Intermediate</P>
<P><P CLASS="TableFirst"><A NAME="pgfId=198921"></A>sum</TD>
<TD><P><P CLASS="TableFirst"><A NAME="pgfId=198923"></A>Intermediate</P>
<P><P CLASS="TableFirst"><A NAME="pgfId=198924"></A>carry</TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=198929"></A><CODE><IMG SRC="CH02-91.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=198934"></A><CODE><IMG SRC="CH02-92.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=198936"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198938"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198940"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198945"></A><CODE><IMG SRC="CH02-93.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=198950"></A><CODE><IMG SRC="CH02-94.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=198952"></A><CODE>0</CODE></TD>
<TD COLSPAN="2"><P CLASS="Table"><A NAME="pgfId=198954"></A><CODE>A[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=0/1
and B[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=0/1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198961"></A><CODE><IMG SRC="CH02-95.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=198963"></A><CODE>0</CODE></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=198965"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198970"></A><CODE><IMG SRC="CH02-96.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD COLSPAN="2"><P CLASS="Table"><A NAME="pgfId=198978"></A><CODE>A[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=<IMG SRC=
"CH02-97.gif" ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3">
or B[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=<IMG SRC=
"CH02-98.gif" ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3">
</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198982"></A><CODE>1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198987"></A><CODE><IMG SRC="CH02-99.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=198992"></A><CODE><IMG SRC="CH02-100.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=198994"></A><CODE>1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198996"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=198998"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199000"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199002"></A><CODE>0</CODE></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=199004"></A><CODE>1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199009"></A><CODE><IMG SRC="CH02-101.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=199011"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199013"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199015"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199017"></A><CODE>0</CODE></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=199019"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199021"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199023"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199025"></A><CODE>x</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199027"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199029"></A><CODE>0</CODE></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=199031"></A><CODE>0</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199033"></A><CODE>1</CODE></TD>
<TD COLSPAN="2"><P CLASS="Table"><A NAME="pgfId=199035"></A><CODE>A[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=0/1
and B[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=0/1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199042"></A><CODE><IMG SRC="CH02-102.gif"
ALIGN="BASELINE" WIDTH="10" HEIGHT="16" NATURALSIZEFLAG="3"> </CODE> </TD>
<TD><P CLASS="Table"><A NAME="pgfId=199044"></A><CODE>1</CODE></TD></TR>
<TR>
<TD><P CLASS="Table"><A NAME="pgfId=199046"></A><CODE>1</CODE></TD>
<TD><P CLASS="Table"><A NAME="pgfId=199048"></A><CODE>0</CODE></TD>
<TD COLSPAN="2"><P CLASS="Table"><A NAME="pgfId=199056"></A><CODE>A[i<SPAN CLASS="White"> </SPAN><SPAN CLASS="White"> </SPAN>1]=<IMG SRC=
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