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In fact, the signed-magnitude and 1s complement data types
both require unnecessarily cumbersome hardware to do addition. Because com-
puter designers knew what it would take to design a circuit to add two integers,
they chose representations that simplied the circuit. The result is the 2s comple-
ment data type, also shown in Figure 2.1. It is used on just about every computer
manufactured today. | [
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] |
1. It is used on just about every computer
manufactured today.
2.3 2s Complement Integers
We see in Figure 2.1 the representations of the integers from 16 to +15 for the
2s complement data type. Why were those representations chosen?
The positive integers, we saw, are represented in the straightforward posi-
tional scheme. With ve bits, we use exactly half of the 25 codes to represent 0
and the positive integers from 1 to 24 1. | [
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] |
Why were those representations chosen?
The positive integers, we saw, are represented in the straightforward posi-
tional scheme. With ve bits, we use exactly half of the 25 codes to represent 0
and the positive integers from 1 to 24 1.
The choice of representations for the negative integers was based, as we said
previously, on the wish to keep the logic circuits as simple as possible. | [
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] |
With ve bits, we use exactly half of the 25 codes to represent 0
and the positive integers from 1 to 24 1.
The choice of representations for the negative integers was based, as we said
previously, on the wish to keep the logic circuits as simple as possible. Almost
all computers use the same basic mechanism to perform addition. It is called an
arithmetic and logic unit, usually known by its acronym ALU. | [
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] |
Almost
all computers use the same basic mechanism to perform addition. It is called an
arithmetic and logic unit, usually known by its acronym ALU. We will get into
the actual structure of the ALU in Chapters 3 and 4. What is relevant right now
is that an ALU has two inputs and one output. It performs addition by adding the
binary bit patterns at its inputs, producing a bit pattern at its output that is the
sum of the two input bit patterns. | [
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] |
What is relevant right now
is that an ALU has two inputs and one output. It performs addition by adding the
binary bit patterns at its inputs, producing a bit pattern at its output that is the
sum of the two input bit patterns.
For example, if the ALU processe
were 00110 and 00101, the result (o
addition is as follows:
00110
00101
01011
The addition of two binary strings is performed in the same way the addi-
tion of two decimal strings is performed, from right to left, column by column. | [
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] |
It performs addition by adding the
binary bit patterns at its inputs, producing a bit pattern at its output that is the
sum of the two input bit patterns.
For example, if the ALU processe
were 00110 and 00101, the result (o
addition is as follows:
00110
00101
01011
The addition of two binary strings is performed in the same way the addi-
tion of two decimal strings is performed, from right to left, column by column.
If the addition in a column generates a carry, the carry is added to the column
immediately to its left. | [
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] |
For example, if the ALU processe
were 00110 and 00101, the result (o
addition is as follows:
00110
00101
01011
The addition of two binary strings is performed in the same way the addi-
tion of two decimal strings is performed, from right to left, column by column.
If the addition in a column generates a carry, the carry is added to the column
immediately to its left.
What is particularly relevant is that the binary ALU does not know (and does
not care) what the two patterns it is adding represent. | [
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] |
If the addition in a column generates a carry, the carry is added to the column
immediately to its left.
What is particularly relevant is that the binary ALU does not know (and does
not care) what the two patterns it is adding represent. It simply adds the two binary
patterns. Since the binary ALU only ADDs and does not CARE, it would be nice
if our assignment of codes to the integers resulted in the ALU producing correct
results when it added two integers. | [
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It simply adds the two binary
patterns. Since the binary ALU only ADDs and does not CARE, it would be nice
if our assignment of codes to the integers resulted in the ALU producing correct
results when it added two integers.
For starters, it would be nice if, when the ALU adds the representation for an
arbitrary integer to the representation of the integer having the same magnitude
but opposite sign, the sum would be 0. | [
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] |
Since the binary ALU only ADDs and does not CARE, it would be nice
if our assignment of codes to the integers resulted in the ALU producing correct
results when it added two integers.
For starters, it would be nice if, when the ALU adds the representation for an
arbitrary integer to the representation of the integer having the same magnitude
but opposite sign, the sum would be 0. That is, if the inputs to the ALU are the
representations of non-zero integers A and A, the output of the ALU should
be 00000. | [
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] |
For starters, it would be nice if, when the ALU adds the representation for an
arbitrary integer to the representation of the integer having the same magnitude
but opposite sign, the sum would be 0. That is, if the inputs to the ALU are the
representations of non-zero integers A and A, the output of the ALU should
be 00000. | [
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] |
To accomplish that, the 2s complement data type species the representation
for each negative integer so that when the ALU adds it to the representation of
the positive integer of the same magnitude, the result will be the representation
for 0. For example, since 00101 is the representation of +5, 11011 is chosen as
the representation for 5.
Moreover, and actually more importantly, as we sequence through represen-
tations of 15 to +15, the ALU is adding 00001 to each successive representation. | [
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] |
For example, since 00101 is the representation of +5, 11011 is chosen as
the representation for 5.
Moreover, and actually more importantly, as we sequence through represen-
tations of 15 to +15, the ALU is adding 00001 to each successive representation.
We can express this mathematically as:
REPRESENTATION(value + 1) =
REPRESENTATION(value) + REPRESENTATION(1).
This is sucient to guarantee (as long as we do not get a result larger than
+15 or smaller than 15) that the binary ALU will perform addition correctly. | [
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] |
We can express this mathematically as:
REPRESENTATION(value + 1) =
REPRESENTATION(value) + REPRESENTATION(1).
This is sucient to guarantee (as long as we do not get a result larger than
+15 or smaller than 15) that the binary ALU will perform addition correctly.
Note in particular the representations for 1 and 0, that is, 11111 and 00000.
When we add 00001 to the representation for 1, we do get 00000, but we also
generate a carry. | [
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] |
Note in particular the representations for 1 and 0, that is, 11111 and 00000.
When we add 00001 to the representation for 1, we do get 00000, but we also
generate a carry. That carry, however, does not inuence the result. That is, the
correct result of adding 00001 to the representation for 1 is 0, not 100000.
Therefore, the carry is ignored. In fact, because the carry obtained by adding
00001 to 11111 is ignored, the carry can always be ignored when dealing with
2s complement arithmetic. | [
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0.0038851862773299217,
0.0036457348614931107,
-0.00497452262789011,
-0.04527321457862854
] |
Therefore, the carry is ignored. In fact, because the carry obtained by adding
00001 to 11111 is ignored, the carry can always be ignored when dealing with
2s complement arithmetic.
Note: If we know the representation for A, a shortcut for guring out the
representation for A(A 0) is as follows: Flip all the bits of A (the ocial term
for ip is complement), and add 1 to the complement of A. | [
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] |
In fact, because the carry obtained by adding
00001 to 11111 is ignored, the carry can always be ignored when dealing with
2s complement arithmetic.
Note: If we know the representation for A, a shortcut for guring out the
representation for A(A 0) is as follows: Flip all the bits of A (the ocial term
for ip is complement), and add 1 to the complement of A. The sum of A and
Exa
10011
00000
You may have noticed that the addition of 01101 and 10011, in addition to
producing 00000, also produces a carry out of the ve-bit ALU. | [
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] |
Note: If we know the representation for A, a shortcut for guring out the
representation for A(A 0) is as follows: Flip all the bits of A (the ocial term
for ip is complement), and add 1 to the complement of A. The sum of A and
Exa
10011
00000
You may have noticed that the addition of 01101 and 10011, in addition to
producing 00000, also produces a carry out of the ve-bit ALU. That is, the binary
addition of 01101 and 10011 is really 100000. | [
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] |
The sum of A and
Exa
10011
00000
You may have noticed that the addition of 01101 and 10011, in addition to
producing 00000, also produces a carry out of the ve-bit ALU. That is, the binary
addition of 01101 and 10011 is really 100000. However, as we saw previously,
this carry out can be ignored in the case of the 2s complement data type.
At this point, we have identied in our ve-bit scheme 15 positive inte-
gers. | [
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] |
However, as we saw previously,
this carry out can be ignored in the case of the 2s complement data type.
At this point, we have identied in our ve-bit scheme 15 positive inte-
gers. We have constructed 15 negative integers. We also have a representation | [
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] |
for 0. With k = 5, we can uniquely identify 32 distinct quantities, and we have
accounted for only 31 (15+15+1). The remaining representation is 10000. What
value shall we assign to it?
We note that 1 is 11111, 2 is 11110, 3 is 11101, and so on. If we continue
this, we note that 15 is 10001. Note that, as in the case of the positive represen-
tations, as we sequence backwards from representations of 1 to 15, the ALU
is subtracting 00001 from each successive representation. | [
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] |
If we continue
this, we note that 15 is 10001. Note that, as in the case of the positive represen-
tations, as we sequence backwards from representations of 1 to 15, the ALU
is subtracting 00001 from each successive representation. Thus, it is convenient
to assign to 10000 the value 16; that is the value one gets by subtracting 00001
from 10001 (the representation for 15).
In Chapter 5 we will specify a computer that we aectionately have named
the LC-3 (for Little Computer 3). | [
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] |
Thus, it is convenient
to assign to 10000 the value 16; that is the value one gets by subtracting 00001
from 10001 (the representation for 15).
In Chapter 5 we will specify a computer that we aectionately have named
the LC-3 (for Little Computer 3). The LC-3 operates on 16-bit values. Therefore,
the 2s complement integers that can be represented in the LC-3 are the integers
from 32,768 to +32,767. | [
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0.010399837046861649,
-0.0046491376124322414,
-0.008011979050934315,
-0.006174902897328138,
-0.038014139980077744
] |
The LC-3 operates on 16-bit values. Therefore,
the 2s complement integers that can be represented in the LC-3 are the integers
from 32,768 to +32,767.
2.4 Conversion Between Binary and
Decimal
It is often useful to convert numbers between the 2s complement data type
the computer likes and the decimal representation that you have used all
your life.
2.4.1 Binary to Decimal Conversion
We convert a 2s complement representation of an integer to a decimal represen-
tation as follows: For purposes of illustration, we will assume our number can be
represented in eight bits, corresponding to decimal integer values from 128 to
+127. | [
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0.00395039189606905,
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0.04551612585783005,
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0.0070855701342225075,
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0.02540750801563263,
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0.008213282562792301,
0.020067855715751648,
0.017649387940764427,
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0.002282599685713649,
0.005692912265658379,
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-0.023383060470223427
] |
4.1 Binary to Decimal Conversion
We convert a 2s complement representation of an integer to a decimal represen-
tation as follows: For purposes of illustration, we will assume our number can be
represented in eight bits, corresponding to decimal integer values from 128 to
+127.
Recall that an eight-bit 2s complement integer takes the form
b7 b6 b5 b4 b3 b2 b1 b0
where each of the bits bi is either 0 or 1. | [
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] |
1 Binary to Decimal Conversion
We convert a 2s complement representation of an integer to a decimal represen-
tation as follows: For purposes of illustration, we will assume our number can be
represented in eight bits, corresponding to decimal integer values from 128 to
+127.
Recall that an eight-bit 2s complement integer takes the form
b7 b6 b5 b4 b3 b2 b1 b0
where each of the bits bi is either 0 or 1.
1. | [
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] |
Recall that an eight-bit 2s complement integer takes the form
b7 b6 b5 b4 b3 b2 b1 b0
where each of the bits bi is either 0 or 1.
1. Examine the leading bit b7. If it is a 0, the integer is positive, and we can
begin evaluating its magnitude. If it is a 1, the integer is negative. In that
case, we need to rst obtain the 2s complement representation of the
positive number having the same magnitude. | [
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] |
If it is a 1, the integer is negative. In that
case, we need to rst obtain the 2s complement representation of the
positive number having the same magnitude. We do this by ipping all the
bits and adding 1.
2. The magnitude is simply
b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
In either case, we obtain the decimal magnitude by simply adding the
powers of 2 that have coecients of 1. | [
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] |
2. The magnitude is simply
b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
In either case, we obtain the decimal magnitude by simply adding the
powers of 2 that have coecients of 1.
3. Finally, if the original number is negative, we ax a minus sign in front.
Done! | [
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] |
Exa
.
3. The decimal integer value corresponding to 11000111 is 57.
2.4.2 Decimal to Binary Conversion
Converting from decimal to 2s complement is a little more complicated. The
crux of the method is to note that a positive binary number is odd if the rightmost
digit is 1 and even if the rightmost digit is 0.
Consider again our generic eight-bit representation:
b7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
We can illustrate the conversion best by rst working through an example. | [
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0.004626563750207424,
-0.021550917997956276,
-0.022001000121235847,
-0.003140636021271348,
0.026395903900265694,
-0.0021726316772401333,
0.04469035565853119,
-0.011622670106589794,
0.03102908656001091,
0.01325090229511261,
0.037462592124938965,
0.016930973157286644,
-0.016692696139216423,
0.026872459799051285,
0.004653039388358593,
-0.0471525639295578,
0.014786471612751484,
-0.011046831496059895,
0.007532231509685516,
-0.005705433432012796,
0.006142276339232922,
0.010418042540550232,
-0.01648089289665222,
0.03637710213661194,
-0.00820735190063715,
-0.0013105288380756974,
0.028063850477337837,
0.015541018918156624,
0.0016356789274141192,
-0.007214527577161789,
0.0355563685297966,
0.02096846140921116,
-0.01241031102836132,
0.004646420478820801,
-0.0030430082697421312,
0.00005279553079162724,
-0.010054006241261959,
0.010477611795067787,
-0.020518381148576736,
-0.015077700838446617,
-0.024410253390669823,
0.0253236535936594,
-0.017738470807671547,
0.03431202843785286,
-0.017619332298636436,
0.02058456838130951,
-0.01768551953136921,
0.014640857465565205,
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-0.013926023617386818,
-0.011516768485307693,
0.018506255000829697,
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0.008286777883768082,
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0.008425774052739143,
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0.016096999868750572,
-0.02108759991824627,
0.0012220019707456231,
0.01573958434164524,
0.002392707858234644,
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0.020862560719251633,
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0.01172857079654932,
0.0189298614859581,
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0.033173587173223495,
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0.032141052186489105,
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0.001487582572735846,
-0.0022901161573827267,
0.004798653535544872,
0.0010176454670727253,
0.005083263386040926,
-0.028434503823518753,
0.00972968339920044,
0.008465486578643322,
-0.004136770498007536,
0.0015752821927890182,
0.009961343370378017,
0.0625876784324646,
0.062428828328847885,
-0.011880804784595966,
0.03301473706960678,
0.008730240166187286,
0.013555369339883327,
0.020902272313833237,
0.012754490599036217,
0.0026127840392291546,
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0.025058899074792862,
0.004431308247148991,
-0.0021709769498556852,
-0.012099226005375385,
0.004020940512418747,
0.023920461535453796,
-0.017937036231160164,
0.013211189769208431,
-0.019565269351005554,
-0.003964680712670088,
-0.0062117744237184525,
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0.025032425299286842,
0.0004645592998713255,
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-0.010001055896282196,
0.007922742515802383,
-0.009431836195290089,
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-0.021842148154973984,
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-0.022795259952545166,
-0.03875988349318504,
-0.01331047248095274,
0.00016443661297671497,
-0.01600433699786663,
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-0.03412669897079468,
0.05676310881972313,
0.01335680391639471,
0.00198564981110394,
0.010100338608026505,
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0.020743420347571373,
-0.004371738992631435,
0.0206772331148386,
0.010801934637129307,
-0.02218632586300373
] |
The
crux of the method is to note that a positive binary number is odd if the rightmost
digit is 1 and even if the rightmost digit is 0.
Consider again our generic eight-bit representation:
b7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
We can illustrate the conversion best by rst working through an example.
Suppose we wish to convert the value +105 to a 2s complement binary code. | [
-0.019782057031989098,
0.014494187198579311,
0.014846712350845337,
-0.011484169401228428,
-0.0015100936871021986,
0.012379039078950882,
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-0.034845709800720215,
-0.02703593112528324,
-0.01883295178413391,
-0.00931478664278984,
0.0382082499563694,
-0.02425641007721424,
-0.0051658423617482185,
-0.0007995361229404807,
0.015253471210598946,
0.009707987308502197,
0.009958822280168533,
0.004484520759433508,
0.006796268746256828,
-0.03113063983619213,
0.018358400091528893,
-0.016758481040596962,
-0.01253496389836073,
-0.004772641696035862,
-0.022886985912919044,
-0.0029201924335211515,
-0.019199036061763763,
0.006270871497690678,
-0.007104727905243635,
0.011396038345992565,
-0.005701408721506596,
-0.039292942732572556,
-0.011104526929557323,
0.007416576612740755,
-0.0061725713312625885,
-0.015999196097254753,
0.011124865151941776,
0.018954981118440628,
-0.009165641851723194,
0.039699699729681015,
0.01721947453916073,
-0.01392472442239523,
0.016934743151068687,
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0.006111557595431805,
0.020744720473885536,
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0.0004277327680028975,
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-0.00606071250513196,
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-0.019280387088656425,
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-0.020961659029126167,
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0.03042559139430523,
0.006623396184295416,
-0.0039455643855035305,
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0.0222361721098423,
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] |
Consider again our generic eight-bit representation:
b7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
We can illustrate the conversion best by rst working through an example.
Suppose we wish to convert the value +105 to a 2s complement binary code.
We note that +105 is positive. We rst nd values for bi, representing the mag-
nitude 105. Since the value is positive, we will then obtain the 2s complement
result by simply appending b7, which we know is 0. | [
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We rst nd values for bi, representing the mag-
nitude 105. Since the value is positive, we will then obtain the 2s complement
result by simply appending b7, which we know is 0.
Our rst step is to nd values for bi that satisfy the following:
105 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
Since 105 is odd, we know that b0 is 1. We subtract 1 from both sides of the
equation, yielding
104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21
We next divide both sides of the equation by 2, yielding
52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20
Since 52 is even, b1, the only coecient not multiplied by a power of 2, must be
equal to 0. | [
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Our rst step is to nd values for bi that satisfy the following:
105 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
Since 105 is odd, we know that b0 is 1. We subtract 1 from both sides of the
equation, yielding
104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21
We next divide both sides of the equation by 2, yielding
52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20
Since 52 is even, b1, the only coecient not multiplied by a power of 2, must be
equal to 0.
We iterate this process, each time subtracting the rightmost digit from both
sides of the equation, then dividing both sides by 2, and nally noting whether
the new decimal number on the left side is odd or even. | [
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We subtract 1 from both sides of the
equation, yielding
104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21
We next divide both sides of the equation by 2, yielding
52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20
Since 52 is even, b1, the only coecient not multiplied by a power of 2, must be
equal to 0.
We iterate this process, each time subtracting the rightmost digit from both
sides of the equation, then dividing both sides by 2, and nally noting whether
the new decimal number on the left side is odd or even. Continuing where we left
o, with
52 = b6 25 + j5 24 + b4 23 + b3 22 + b2 21
the process produces, in turn:
26 = b6 24 + b5 23 + b4 22 + b3 21 + b2 20. | [
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] |
We iterate this process, each time subtracting the rightmost digit from both
sides of the equation, then dividing both sides by 2, and nally noting whether
the new decimal number on the left side is odd or even. Continuing where we left
o, with
52 = b6 25 + j5 24 + b4 23 + b3 22 + b2 21
the process produces, in turn:
26 = b6 24 + b5 23 + b4 22 + b3 21 + b2 20 | [
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] |
Therefore, b2 = 0.
13 = b6 23 + b5 22 + b4 21 + b3 20
Therefore, b3 = 1.
6 = b6 22 + b5 21 + b4 20
Therefore, b4 = 0.
3 = b6 21 + b5 20
Therefore, b5 = 1.
1 = b6 20
Therefore, b6 = 1, and we are done. The binary representation is 01101001.
Lets summarize the process. If we are given a decimal integer value N, we
construct the 2s complement representation as follows:
1. We rst obtain the binary representation of the magnitude of N by forming
the equation
N = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
and repeating the following, until the left side of the equation is 0:
a. | [
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] |
If we are given a decimal integer value N, we
construct the 2s complement representation as follows:
1. We rst obtain the binary representation of the magnitude of N by forming
the equation
N = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20
and repeating the following, until the left side of the equation is 0:
a. If N is odd, the rightmost bit is 1. If N is even, the rightmost bit is 0. | [
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If N is odd, the rightmost bit is 1. If N is even, the rightmost bit is 0.
b. Subtract 1 or 0 (according to whether N is odd or even) from N, remove
the least signicant term from the right side, and divide both sides of
the equation by 2.
Each iteration produces the value of one coecient bi.
2. If the original decimal number is positive, append a leading 0 sign bit, and
you are done. | [
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] |
2. If the original decimal number is positive, append a leading 0 sign bit, and
you are done.
3. If the original decimal number is negative, append a leading 0 and then
form the negative of this 2s complement representation, and then you
are done.
2.4.3 Extending Conversion to Numbers with Fractional Parts
What if the number we wish to convert is not an integer, but instead has a
fractional part. | [
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] |
4.3 Extending Conversion to Numbers with Fractional Parts
What if the number we wish to convert is not an integer, but instead has a
fractional part. How do we handle that wrinkle?
Binary to decimal
The binary to decimal case is straightforward. In a positional
notation system, the number
0.b1b2b3b4
shows four bits to the right of the binary point, representing (when the cor-
responding bi = 1) the values 0. | [
0.011358310468494892,
0.020204557105898857,
0.018205709755420685,
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0.003697190899401903,
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] |
In a positional
notation system, the number
0.b1b2b3b4
shows four bits to the right of the binary point, representing (when the cor-
responding bi = 1) the values 0.5, 0.25, 0.125, and 0.0625. To complete the | [
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] |
conversion to decimal, we simply add those values where the corresponding bi =
1. For example, if the fractional part of the binary representation is
. 1 0 1 1
we would add 0.5 plus 0.125 plus 0.0625, or 0.6875.
Decimal to binary
The decimal to binary case requires a little more work. Sup-
pose we wanted to convert 0.421 to binary. As we did for integer conversion, we
rst form the equation
0. | [
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] |
421 to binary. As we did for integer conversion, we
rst form the equation
0.421 = b1 21 + b2 22 + b3 23 + b4 24 + ...
In the case of converting a decimal integer value to binary, we divided by 2 and
assigned a 1 or 0 to the coecient of 20 depending on whether the number on the
left of the equal sign is odd or even. Here (i.e., in the case of converting a decimal
fraction to binary), we multiply both sides of the equation by 2 and assign a 1 or
a 0 to the coecient of 20 depending on whether the left side of the equation is
greater than or equal to 1 or whether the left side is less than 1. | [
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] |
e., in the case of converting a decimal
fraction to binary), we multiply both sides of the equation by 2 and assign a 1 or
a 0 to the coecient of 20 depending on whether the left side of the equation is
greater than or equal to 1 or whether the left side is less than 1. Do you see why?
Since
0.842 = b1 20 + b2 21 + b3 22 + b4 23 + ...
we assign b1 = 0. Continuing,
1.684 = b2 20 + b3 21 + b4 22 +. | [
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Continuing,
1.684 = b2 20 + b3 21 + b4 22 + ...
so we assign b2 = 1 and subtract 1 from both sides of the equation, yielding
0.684 = b3 21 + b4 22 + ...
Multiplying by 2, we get
1.368 = b3 20 + b4 21 + ...
so we assign b3 = 1 and subtract 1 from both sides of the equation, yielding
0.368 = b4 20 + ...
which assigns 0 to b4. We can continue this process indenitely, until we are
simply too tired to go on, or until the left side = 0, in which case all bits to
the right of where we stop are 0s. | [
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] |
which assigns 0 to b4. We can continue this process indenitely, until we are
simply too tired to go on, or until the left side = 0, in which case all bits to
the right of where we stop are 0s. In our case, stopping with four bits, we have
converted 0.421 decimal to 0.0110 in binary.
2.5 Operations on Bits
Part I: Arithmetic
2.5.1 Addition and Subtraction
Arithmetic on 2s complement numbers is very much like the arithmetic on
decimal numbers that you have been doing for a long time. | [
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0.009075259789824486,
0.009102677926421165,
0.0050174398347735405,
-0.031831953674554825
] |
5.1 Addition and Subtraction
Arithmetic on 2s complement numbers is very much like the arithmetic on
decimal numbers that you have been doing for a long time. | [
0.0009934501722455025,
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] |
Addition still proceeds from right to left, one digit at a time. At each point,
we generate a sum digit and a carry. Instead of generating a carry after 9 (since
9 is the largest decimal digit), we generate a carry after 1 (since 1 is the largest
binary digit).
Example 2.3
Using our ve-bit notation, what is 11 + 3?
The decimal value 11 is represented as 01011
The decimal value 3 is represented as
00011
The sum, which is the value 14, is
01110
Subtraction is simply addition, preceded by determining the negative of the
number to be subtracted. | [
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] |
Example 2.3
Using our ve-bit notation, what is 11 + 3?
The decimal value 11 is represented as 01011
The decimal value 3 is represented as
00011
The sum, which is the value 14, is
01110
Subtraction is simply addition, preceded by determining the negative of the
number to be subtracted. That is, A B is simply A + (B).
Example 2.4
What is 14 9?
The decimal value 14 is represented as
01110
The decimal value 9 is represented as
01001
First we form the negative, that is, -9: 10111
Adding 14 to -9, we get
01110
10111
which results in the value 5. | [
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] |
Example 2.4
What is 14 9?
The decimal value 14 is represented as
01110
The decimal value 9 is represented as
01001
First we form the negative, that is, -9: 10111
Adding 14 to -9, we get
01110
10111
which results in the value 5.
00101
Note again that the carry out is ignored.
Exam
What happens when we add a number to itself (e.g., x + x)?
Lets assume for this example eight-bit codes, which would allow us to represent
integers from 128 to 127. | [
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g., x + x)?
Lets assume for this example eight-bit codes, which would allow us to represent
integers from 128 to 127. Consider a value for x, the integer 59, represented as
00111011. If we add 59 to itself, we get the code 01110110. Note that the bits have
all shifted to the left by one position. Is that a curiosity, or will that happen all the
time as long as the sum x + x is not too large to represent with the available number
of bits?
Using our positional notation, the number 59 is
0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20
The sum 59 + 59 is 2 59, which, in our representation, is
2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)
But that is nothing more than
0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21
which shifts each digit one position to the left. | [
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] |
Note that the bits have
all shifted to the left by one position. Is that a curiosity, or will that happen all the
time as long as the sum x + x is not too large to represent with the available number
of bits?
Using our positional notation, the number 59 is
0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20
The sum 59 + 59 is 2 59, which, in our representation, is
2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)
But that is nothing more than
0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21
which shifts each digit one position to the left. Thus, adding a number to itself
(provided there are enough bits to represent the result) is equivalent to shifting the
representation one bit position to the left. | [
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] |
Is that a curiosity, or will that happen all the
time as long as the sum x + x is not too large to represent with the available number
of bits?
Using our positional notation, the number 59 is
0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20
The sum 59 + 59 is 2 59, which, in our representation, is
2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)
But that is nothing more than
0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21
which shifts each digit one position to the left. Thus, adding a number to itself
(provided there are enough bits to represent the result) is equivalent to shifting the
representation one bit position to the left.. | [
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0.007895577698946,
0.003906949423253536,
0.0049721719697117805,
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] |
Thus, adding a number to itself
(provided there are enough bits to represent the result) is equivalent to shifting the
representation one bit position to the left. | [
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] |
2.5.2 Sign-Extension
It is often useful to represent a small number with fewer bits. For example, rather
than represent the value 5 as 0000000000000101, there are times when it makes
nse to use only six bits to represent the value 5: 000101. There is little
n, since we are all used to adding leading zeros without aecting the
a number. A check for $456.78 and a check for $0000456.78 are checks
he same value. | [
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] |
78 and a check for $0000456.78 are checks
he same value.
at about negative representations? We obtained the negative representa-
its positive counterpart by complementing the positive representation
ing 1. Thus, the representation for 5, given that 5 is represented as
, is 111011. If 5 is represented as 0000000000000101, then the represen-
tation for 5 is 1111111111111011. In the same way that leading 0s do not aect
the value of a positive number, leading 1s do not aect the value of a negative
number. | [
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] |
If 5 is represented as 0000000000000101, then the represen-
tation for 5 is 1111111111111011. In the same way that leading 0s do not aect
the value of a positive number, leading 1s do not aect the value of a negative
number.
In order to add representations of dierent lengths, it is rst necessary to
represent them with the same number of bits. For example, suppose we wish to
add the number 13 to 5, where 13 is represented as 0000000000001101 and 5
is represented as 111011. | [
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In order to add representations of dierent lengths, it is rst necessary to
represent them with the same number of bits. For example, suppose we wish to
add the number 13 to 5, where 13 is represented as 0000000000001101 and 5
is represented as 111011. If we do not represent the two values with the same
number of bits, we have
0000000000001101
+ 111011
When we attempt to perform the addition, what shall we do with the missing bits
in the representation for 5? If we take the absence of a bit to be a 0, then we are
no longer adding 5 to 13. | [
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For example, suppose we wish to
add the number 13 to 5, where 13 is represented as 0000000000001101 and 5
is represented as 111011. If we do not represent the two values with the same
number of bits, we have
0000000000001101
+ 111011
When we attempt to perform the addition, what shall we do with the missing bits
in the representation for 5? If we take the absence of a bit to be a 0, then we are
no longer adding 5 to 13. On the contrary, if we take the absence of bits to be 0s,
we have changed the 5 to the number represented as 0000000000111011, that
is, +59. | [
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] |
If we do not represent the two values with the same
number of bits, we have
0000000000001101
+ 111011
When we attempt to perform the addition, what shall we do with the missing bits
in the representation for 5? If we take the absence of a bit to be a 0, then we are
no longer adding 5 to 13. On the contrary, if we take the absence of bits to be 0s,
we have changed the 5 to the number represented as 0000000000111011, that
is, +59. Not surprisingly, then, our result turns out to be the representation for 72. | [
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] |
On the contrary, if we take the absence of bits to be 0s,
we have changed the 5 to the number represented as 0000000000111011, that
is, +59. Not surprisingly, then, our result turns out to be the representation for 72.
However, if we understand that a 6-bit 5 and a 16-bit 5 dier only in the
number of meaningless leading 1s, then we rst extend the value of 5 to 16 bits
before we perform the addition. | [
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] |
Not surprisingly, then, our result turns out to be the representation for 72.
However, if we understand that a 6-bit 5 and a 16-bit 5 dier only in the
number of meaningless leading 1s, then we rst extend the value of 5 to 16 bits
before we perform the addition. Thus, we have
0000000000001101
+ 1111111111111011
0000000000001000
result is +8, as we should expect.
e value of a positive number does not change if we extend the sign bit
any bit positions to the left as desired. | [
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0.010302744805812836,
0.017133550718426704,
-0.03724684938788414
] |
Thus, we have
0000000000001101
+ 1111111111111011
0000000000001000
result is +8, as we should expect.
e value of a positive number does not change if we extend the sign bit
any bit positions to the left as desired. Similarly, the value of a negative
r does not change by extending the sign bit 1 as many bit positions to the
desired. Since in both cases it is the sign bit that is extended, we refer
to the operation as Sign-EXTension, often abbreviated SEXT. | [
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] |
Similarly, the value of a negative
r does not change by extending the sign bit 1 as many bit positions to the
desired. Since in both cases it is the sign bit that is extended, we refer
to the operation as Sign-EXTension, often abbreviated SEXT. Sign-extension is
performed in order to be able to operate on representations of dierent lengths.
It does not aect the values of the numbers being represented. | [
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Sign-extension is
performed in order to be able to operate on representations of dierent lengths.
It does not aect the values of the numbers being represented.
2.5.3 Overow
Up to now, we have always insisted that the sum of two integers be small enough
to be represented by the available bits. What happens if such is not the case? | [
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You are undoubtedly familiar with the odometer on the front dashboard of
your automobile. It keeps track of how many miles your car has been drivenbut
only up to a point. In the old days, when the odometer registered 99992 and you
drove it 100 miles, its new reading became 00092. A brand new car! The problem,
as you know, is that the largest value the odometer could store was 99999, so the
value 100092 showed up as 00092. | [
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In the old days, when the odometer registered 99992 and you
drove it 100 miles, its new reading became 00092. A brand new car! The problem,
as you know, is that the largest value the odometer could store was 99999, so the
value 100092 showed up as 00092. The carry out of the ten-thousands digit was
lost. (Of course, if you grew up in Boston, the carry out was not lost at allit
was in full display in the rusted chrome all over the car. | [
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] |
The carry out of the ten-thousands digit was
lost. (Of course, if you grew up in Boston, the carry out was not lost at allit
was in full display in the rusted chrome all over the car.)
We say the odometer overowed. Representing 100092 as 00092 is unaccept-
able. As more and more cars lasted more than 100,000 miles, car makers felt the
pressure to add a digit to the odometer. Today, practically all cars overow at
1,000,000 miles, rather than 100,000 miles. | [
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As more and more cars lasted more than 100,000 miles, car makers felt the
pressure to add a digit to the odometer. Today, practically all cars overow at
1,000,000 miles, rather than 100,000 miles.
The odometer provides an example of unsigned arithmetic. The miles you
add are always positive miles. The odometer reads 000129 and you drive 50 miles.
The odometer now reads 000179. Overow is a carry out of the leading digit. | [
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] |
The odometer now reads 000179. Overow is a carry out of the leading digit.
In the case of signed arithmetic, or more particularly, 2s complement
arithmetic, overow is a little more subtle.
Lets return to our ve-bit 2s complement data type, which allowed us to
represent integers from 16 to +15. Suppose we wish to add +9 and +11. Our
arithmetic takes the following form:
01001
01011
10100
Note that the sum is larger than +15, and therefore too large to represent with
our 2s complement scheme. | [
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] |
Suppose we wish to add +9 and +11. Our
arithmetic takes the following form:
01001
01011
10100
Note that the sum is larger than +15, and therefore too large to represent with
our 2s complement scheme. The fact that the number is too large means that the
number is larger than 01111, the largest positive number we can represent with
a ve-bit 2s complement data type. Note that because our positive result was
larger than +15, it generated a carry into the leading bit position. | [
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] |
The fact that the number is too large means that the
number is larger than 01111, the largest positive number we can represent with
a ve-bit 2s complement data type. Note that because our positive result was
larger than +15, it generated a carry into the leading bit position. But this bit
position is used to indicate the sign of a value. Thus, detecting that the result is
too large is an easy matter. | [
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] |
But this bit
position is used to indicate the sign of a value. Thus, detecting that the result is
too large is an easy matter. Since we are adding two positive numbers, the result
must be positive. Since the ALU has produced a negative result, something must
be wrong. The thing that is wrong is that the sum of the two positive numbers
is too large to be represented with the available bits. | [
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] |
Since the ALU has produced a negative result, something must
be wrong. The thing that is wrong is that the sum of the two positive numbers
is too large to be represented with the available bits. We say that the result has
overowed the capacity of the representation.
Suppose instead, we had started with negative numbers, for example, 12
and 6. In this case, our arithmetic takes the following form:
10100
11010
01110
Here, too, the result has overowed the capacity of the machine, since 12 + 6
equals 18, which is more negative than 16, the negative number with the
largest allowable magnitude. | [
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Suppose instead, we had started with negative numbers, for example, 12
and 6. In this case, our arithmetic takes the following form:
10100
11010
01110
Here, too, the result has overowed the capacity of the machine, since 12 + 6
equals 18, which is more negative than 16, the negative number with the
largest allowable magnitude. The ALU obliges by producing a positive result.
Again, this is easy to detect since the sum of two negative numbers cannot be
positive. | [
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] |
The ALU obliges by producing a positive result.
Again, this is easy to detect since the sum of two negative numbers cannot be
positive. | [
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] |
Note that the sum of a negative number and a positive number never presents
a problem. Why is that? See Exercise 2.25.
2.6 Operations on Bits
Part II: Logical Operations
We have seen that it is possible to perform arithmetic (e.g., add, subtract) on val-
ues represented as binary patterns. Another class of operations useful to perform
on binary patterns is the set of logical operations. | [
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, add, subtract) on val-
ues represented as binary patterns. Another class of operations useful to perform
on binary patterns is the set of logical operations.
2.6.1 A Logical Variable
Logical operations operate on logical variables. A logical variable can have one
of two values, 0 or 1. The name logical is a historical one; it comes from the fact
that the two values 0 and 1 can represent the two logical values false and true,
but the use of logical operations has traveled far from this original meaning. | [
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] |
A logical variable can have one
of two values, 0 or 1. The name logical is a historical one; it comes from the fact
that the two values 0 and 1 can represent the two logical values false and true,
but the use of logical operations has traveled far from this original meaning.
There are several basic logic functions, and most ALUs perform all of them.
2.6.2 The AND Function
AND is a binary logical function. | [
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6.2 The AND Function
AND is a binary logical function. This means it requires two pieces of input data.
Said another way, AND requires two source operands. Each source is a logical
variable, taking the value 0 or 1. The output of AND is 1 only if both sources have
the value 1. Otherwise, the output is 0. We can think of the AND operation as the
ALL operation; that is, the output is 1 only if ALL two inputs are 1. | [
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] |
Otherwise, the output is 0. We can think of the AND operation as the
ALL operation; that is, the output is 1 only if ALL two inputs are 1. Otherwise,
the output is 0.
A convenient mechanism for representing the behavior of a logical operation
is the truth table. A truth table consists of n + 1 columns and 2n rows. The rst
n columns correspond to the n source operands. Since each source operand is a
logical variable and can have one of two values, there are 2n unique values that
these source operands can have. | [
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] |
The rst
n columns correspond to the n source operands. Since each source operand is a
logical variable and can have one of two values, there are 2n unique values that
these source operands can have. Each such set of values (sometimes called an
input combination) is represented as one row of the truth table. The nal column
in the truth table shows the output for each input combination. | [
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] |
Each such set of values (sometimes called an
input combination) is represented as one row of the truth table. The nal column
in the truth table shows the output for each input combination.
In the case of a two-input AND function, the truth table has two columns for
source operands, and four (22) rows for unique input combinations.
A
B
AND
0
0
0
0
1
0
1
0
0
1
1
1
We can apply the logical operation AND to two bit patterns of m bits each. | [
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] |
In the case of a two-input AND function, the truth table has two columns for
source operands, and four (22) rows for unique input combinations.
A
B
AND
0
0
0
0
1
0
1
0
0
1
1
1
We can apply the logical operation AND to two bit patterns of m bits each. This
involves applying the operation individually and independently to each pair of | [
-0.005007173400372267,
0.012643774971365929,
-0.011100558564066887,
-0.010325639508664608,
-0.03473229706287384,
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0.022995907813310623,
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0.00023036474885884672,
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0.02164476551115513,
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0.007689587771892548,
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0.004212383646517992,
0.00043920392636209726,
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0.0019141178345307708,
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0.006278836168348789,
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0.0029291301034390926,
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] |
bits in the two source operands. For example, if a and b in Example 2.6 are 16-
bit patterns, then c is the AND of a and b. This operation is often called a bit-
wise AND because the operation is applied to each pair of bits individually and
independently.
Example 2.6
If c is the AND of a and b, where a=0011101001101001 and b=0101100100100001,
what is c?
We form the AND of a and b by bit-wise ANDing the two values. | [
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0.004759406670928001,
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0.01603873074054718,
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-0.009023459628224373,
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-0.0189439095556736,
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-0.020282702520489693,
0.010074411518871784,
-0.009485343471169472,
-0.003050772938877344,
-0.01950620301067829,
0.05703255161643028,
0.013608823530375957,
-0.011928639374673367,
0.009304606355726719,
-0.0023797033354640007,
0.006874698214232922,
0.0399763360619545,
0.001354690408334136,
0.0022993755992501974,
0.011493531987071037,
-0.029962170869112015
] |
Example 2.6
If c is the AND of a and b, where a=0011101001101001 and b=0101100100100001,
what is c?
We form the AND of a and b by bit-wise ANDing the two values.
That means individually ANDing each pair of bits ai and bi to form ci. For
example, since a0=1 and b0=1, c0 is the AND of a0 and b0, which is 1.
Since a6=1 and b6=0, c6 is the AND of a6 and b6, which is 0.
The complete solution for c is
a: 0011101001101001
b: 0101100100100001
c: 0001100000100001
Exam
Suppose we have an eight-bit patternlets call it Ain which the rightmost two
bits have particular signicance. | [
-0.019354086369276047,
0.005058757960796356,
0.01545657403767109,
-0.0066638123244047165,
-0.02475988306105137,
0.020488636568188667,
-0.01773902215063572,
-0.019420824944972992,
-0.020542027428746223,
-0.015429878607392311,
-0.0015700168441981077,
0.042418818920850754,
0.002764631062746048,
-0.012673589400947094,
0.00396758783608675,
0.012760349549353123,
0.0012346572475507855,
0.02000812068581581,
0.021916834637522697,
0.01493601594120264,
-0.02318486012518406,
0.03176739811897278,
0.005626032594591379,
-0.026361601427197456,
0.005115485284477472,
0.004371353890746832,
0.0006285907002165914,
-0.015336445532739162,
0.004548210185021162,
-0.017832454293966293,
0.008322257548570633,
0.02729593589901924,
-0.05293676257133484,
-0.009343352168798447,
-0.003530452260747552,
0.0010419506579637527,
-0.01137219462543726,
-0.015269706957042217,
0.017538806423544884,
0.012052924372255802,
0.031954266130924225,
0.012713632546365261,
-0.019527606666088104,
0.0011779298074543476,
-0.008862837217748165,
-0.018059365451335907,
-0.004581579472869635,
-0.025947824120521545,
-0.0013990001752972603,
-0.025934476405382156,
-0.01373472809791565,
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] |
Since a6=1 and b6=0, c6 is the AND of a6 and b6, which is 0.
The complete solution for c is
a: 0011101001101001
b: 0101100100100001
c: 0001100000100001
Exam
Suppose we have an eight-bit patternlets call it Ain which the rightmost two
bits have particular signicance. The computer could be asked to do one of four tasks
depending on the value stored in the two rightmost bits of A. Can we isolate those
two bits?
Yes, we can, using a bit mask. | [
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] |
The computer could be asked to do one of four tasks
depending on the value stored in the two rightmost bits of A. Can we isolate those
two bits?
Yes, we can, using a bit mask. A bit mask is a binary pattern that enables the bits
of A to be separated into two partsgenerally the part you care about and the part
you wish to ignore. In this case, the bit mask 00000011 ANDed with A produces 0 in
bit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions
1 and 0. | [
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A bit mask is a binary pattern that enables the bits
of A to be separated into two partsgenerally the part you care about and the part
you wish to ignore. In this case, the bit mask 00000011 ANDed with A produces 0 in
bit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions
1 and 0. The bit mask is said to mask out the values in bit positions 7 through 2. | [
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] |
In this case, the bit mask 00000011 ANDed with A produces 0 in
bit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions
1 and 0. The bit mask is said to mask out the values in bit positions 7 through 2.
If A is 01010110, the AND of A and the bit mask 00000011 is 00000010. If A is
11111100, the AND of A and the bit mask 00000011 is 00000000.
That is, the result of ANDing any eight-bit pattern with the mask 00000011 is
one of the four patterns: 00000000, 00000001, 00000010, or 00000011. | [
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] |
If A is
11111100, the AND of A and the bit mask 00000011 is 00000000.
That is, the result of ANDing any eight-bit pattern with the mask 00000011 is
one of the four patterns: 00000000, 00000001, 00000010, or 00000011. The result
of ANDing with the mask is to highlight the two bits that are relevant.
2.6.3 The OR Function
OR is also a binary logical function. It requires two source operands, both of
which are logical variables. | [
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] |
3 The OR Function
OR is also a binary logical function. It requires two source operands, both of
which are logical variables. The output of OR is 1 if any source has the value 1.
Only if both sources are 0 is the output 0. We can think of the OR operation as
the ANY operation; that is, the output is 1 if ANY of the two inputs are 1.
The truth table for a two-input OR function is
A
B
OR
0
0
0
0
1
1
1
0
1
1
1
1
In the same way that we applied the logical operation AND to two m-bit patterns,
we can apply the OR operation bit-wise to two m-bit patterns. | [
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We can think of the OR operation as
the ANY operation; that is, the output is 1 if ANY of the two inputs are 1.
The truth table for a two-input OR function is
A
B
OR
0
0
0
0
1
1
1
0
1
1
1
1
In the same way that we applied the logical operation AND to two m-bit patterns,
we can apply the OR operation bit-wise to two m-bit patterns. | [
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Example 2.8
If c is the OR of a and b, where a=0011101001101001 and b=0101100100100001,
as before, what is c?
We form the OR of a and b by bit-wise ORing the two values. That means
individually ORing each pair of bits ai and bi to form ci. For example, since a0=1
and b0=1, c0 is the OR of a0 and b0, which is 1. Since a6=1 and b6=0, c6 is the OR
of a6 and b6, which is also 1.
The complete solution for c is
a: 0011101001101001
b: 0101100100100001
c: 0111101101101001
Sometimes this OR operation is referred to as the inclusive-OR in order to distinguish
it from the exclusive-OR function, which we will discuss momentarily. | [
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] |
Since a6=1 and b6=0, c6 is the OR
of a6 and b6, which is also 1.
The complete solution for c is
a: 0011101001101001
b: 0101100100100001
c: 0111101101101001
Sometimes this OR operation is referred to as the inclusive-OR in order to distinguish
it from the exclusive-OR function, which we will discuss momentarily.
2.6.4 The NOT Function
NOT is a unary logical function. This means it operates on only one source
operand. | [
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] |
4 The NOT Function
NOT is a unary logical function. This means it operates on only one source
operand. It is also known as the complement operation. The output is formed by
complementing the input. We sometimes say the output is formed by inverting
the input. A 1 input results in a 0 output. A 0 input results in a 1 output.
The truth table for the NOT function is
A
NOT
0
1
1
0
In the same way that we applied the logical operation AND and OR to two m-bit
patterns, we can apply the NOT operation bit-wise to one m-bit pattern. | [
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] |
A 0 input results in a 1 output.
The truth table for the NOT function is
A
NOT
0
1
1
0
In the same way that we applied the logical operation AND and OR to two m-bit
patterns, we can apply the NOT operation bit-wise to one m-bit pattern. If a is as
before, then c is the NOT of a.
a: 0011101001101001
c: 1100010110010110
2.6.5 The Exclusive-OR Function
Exclusive-OR, often abbreviated XOR, is a binary logical function. | [
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] |
6.5 The Exclusive-OR Function
Exclusive-OR, often abbreviated XOR, is a binary logical function. It, too,
requires two source operands, both of which are logical variables. The output
of XOR is 1 if one (but not both) of the two sources is 1. The output of XOR is 0
if both sources are 1 or if neither source is 1. In other words, the output of XOR is
1 if the two sources are dierent. The output is 0 if the two sources are the same. | [
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