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} | {
"context": "In fact, the signed-magnitude and 1s complement data types\nboth require unnecessarily cumbersome hardware to do addition. Because com-\nputer designers knew what it would take to design a circuit to add two integers,\nthey chose representations that simplied the circuit. The result is the 2s comple-\nment data type, also shown in Figure 2.1. It is used on just about every computer\nmanufactured today."
} |
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] | {
"indices": [
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"values": [
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]
} | {
"context": "1. It is used on just about every computer\nmanufactured today.\n2.3 2s Complement Integers\nWe see in Figure 2.1 the representations of the integers from 16 to +15 for the\n2s complement data type. Why were those representations chosen?\nThe positive integers, we saw, are represented in the straightforward posi-\ntional scheme. With ve bits, we use exactly half of the 25 codes to represent 0\nand the positive integers from 1 to 24 1."
} |
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} | {
"context": "Why were those representations chosen?\nThe positive integers, we saw, are represented in the straightforward posi-\ntional scheme. With ve bits, we use exactly half of the 25 codes to represent 0\nand the positive integers from 1 to 24 1.\nThe choice of representations for the negative integers was based, as we said\npreviously, on the wish to keep the logic circuits as simple as possible."
} |
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"indices": [
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],
"values": [
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} | {
"context": "With ve bits, we use exactly half of the 25 codes to represent 0\nand the positive integers from 1 to 24 1.\nThe choice of representations for the negative integers was based, as we said\npreviously, on the wish to keep the logic circuits as simple as possible. Almost\nall computers use the same basic mechanism to perform addition. It is called an\narithmetic and logic unit, usually known by its acronym ALU."
} |
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} | {
"context": "Almost\nall computers use the same basic mechanism to perform addition. It is called an\narithmetic and logic unit, usually known by its acronym ALU. We will get into\nthe actual structure of the ALU in Chapters 3 and 4. What is relevant right now\nis that an ALU has two inputs and one output. It performs addition by adding the\nbinary bit patterns at its inputs, producing a bit pattern at its output that is the\nsum of the two input bit patterns."
} |
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"values": [
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} | {
"context": "What is relevant right now\nis that an ALU has two inputs and one output. It performs addition by adding the\nbinary bit patterns at its inputs, producing a bit pattern at its output that is the\nsum of the two input bit patterns.\nFor example, if the ALU processe\nwere 00110 and 00101, the result (o\naddition is as follows:\n00110\n00101\n01011\nThe addition of two binary strings is performed in the same way the addi-\ntion of two decimal strings is performed, from right to left, column by column."
} |
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"indices": [
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} | {
"context": "It performs addition by adding the\nbinary bit patterns at its inputs, producing a bit pattern at its output that is the\nsum of the two input bit patterns.\nFor example, if the ALU processe\nwere 00110 and 00101, the result (o\naddition is as follows:\n00110\n00101\n01011\nThe addition of two binary strings is performed in the same way the addi-\ntion of two decimal strings is performed, from right to left, column by column.\nIf the addition in a column generates a carry, the carry is added to the column\nimmediately to its left."
} |
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"indices": [
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"values": [
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} | {
"context": "For example, if the ALU processe\nwere 00110 and 00101, the result (o\naddition is as follows:\n00110\n00101\n01011\nThe addition of two binary strings is performed in the same way the addi-\ntion of two decimal strings is performed, from right to left, column by column.\nIf the addition in a column generates a carry, the carry is added to the column\nimmediately to its left.\nWhat is particularly relevant is that the binary ALU does not know (and does\nnot care) what the two patterns it is adding represent."
} |
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} | {
"context": "If the addition in a column generates a carry, the carry is added to the column\nimmediately to its left.\nWhat is particularly relevant is that the binary ALU does not know (and does\nnot care) what the two patterns it is adding represent. It simply adds the two binary\npatterns. Since the binary ALU only ADDs and does not CARE, it would be nice\nif our assignment of codes to the integers resulted in the ALU producing correct\nresults when it added two integers."
} |
720406 | [
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} | {
"context": "It simply adds the two binary\npatterns. Since the binary ALU only ADDs and does not CARE, it would be nice\nif our assignment of codes to the integers resulted in the ALU producing correct\nresults when it added two integers.\nFor starters, it would be nice if, when the ALU adds the representation for an\narbitrary integer to the representation of the integer having the same magnitude\nbut opposite sign, the sum would be 0."
} |
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] | {
"indices": [
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"values": [
0.45886343419922815,
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]
} | {
"context": "Since the binary ALU only ADDs and does not CARE, it would be nice\nif our assignment of codes to the integers resulted in the ALU producing correct\nresults when it added two integers.\nFor starters, it would be nice if, when the ALU adds the representation for an\narbitrary integer to the representation of the integer having the same magnitude\nbut opposite sign, the sum would be 0. That is, if the inputs to the ALU are the\nrepresentations of non-zero integers A and A, the output of the ALU should\nbe 00000."
} |
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] | {
"indices": [
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],
"values": [
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]
} | {
"context": "\nFor starters, it would be nice if, when the ALU adds the representation for an\narbitrary integer to the representation of the integer having the same magnitude\nbut opposite sign, the sum would be 0. That is, if the inputs to the ALU are the\nrepresentations of non-zero integers A and A, the output of the ALU should\nbe 00000."
} |
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"values": [
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} | {
"context": "To accomplish that, the 2s complement data type species the representation\nfor each negative integer so that when the ALU adds it to the representation of\nthe positive integer of the same magnitude, the result will be the representation\nfor 0. For example, since 00101 is the representation of +5, 11011 is chosen as\nthe representation for 5.\nMoreover, and actually more importantly, as we sequence through represen-\ntations of 15 to +15, the ALU is adding 00001 to each successive representation."
} |
408434 | [
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"values": [
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} | {
"context": "For example, since 00101 is the representation of +5, 11011 is chosen as\nthe representation for 5.\nMoreover, and actually more importantly, as we sequence through represen-\ntations of 15 to +15, the ALU is adding 00001 to each successive representation.\nWe can express this mathematically as:\nREPRESENTATION(value + 1) =\nREPRESENTATION(value) + REPRESENTATION(1).\nThis is sucient to guarantee (as long as we do not get a result larger than\n+15 or smaller than 15) that the binary ALU will perform addition correctly."
} |
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"indices": [
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} | {
"context": "We can express this mathematically as:\nREPRESENTATION(value + 1) =\nREPRESENTATION(value) + REPRESENTATION(1).\nThis is sucient to guarantee (as long as we do not get a result larger than\n+15 or smaller than 15) that the binary ALU will perform addition correctly.\nNote in particular the representations for 1 and 0, that is, 11111 and 00000.\nWhen we add 00001 to the representation for 1, we do get 00000, but we also\ngenerate a carry."
} |
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"indices": [
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} | {
"context": "Note in particular the representations for 1 and 0, that is, 11111 and 00000.\nWhen we add 00001 to the representation for 1, we do get 00000, but we also\ngenerate a carry. That carry, however, does not inuence the result. That is, the\ncorrect result of adding 00001 to the representation for 1 is 0, not 100000.\nTherefore, the carry is ignored. In fact, because the carry obtained by adding\n00001 to 11111 is ignored, the carry can always be ignored when dealing with\n2s complement arithmetic."
} |
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0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.7477917007047616,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.6640519892753799,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196,
0.49706424497402196
]
} | {
"context": "Therefore, the carry is ignored. In fact, because the carry obtained by adding\n00001 to 11111 is ignored, the carry can always be ignored when dealing with\n2s complement arithmetic.\nNote: If we know the representation for A, a shortcut for guring out the\nrepresentation for A(A 0) is as follows: Flip all the bits of A (the ocial term\nfor ip is complement), and add 1 to the complement of A."
} |
368479 | [
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} | {
"context": "In fact, because the carry obtained by adding\n00001 to 11111 is ignored, the carry can always be ignored when dealing with\n2s complement arithmetic.\nNote: If we know the representation for A, a shortcut for guring out the\nrepresentation for A(A 0) is as follows: Flip all the bits of A (the ocial term\nfor ip is complement), and add 1 to the complement of A. The sum of A and\nExa\n10011\n00000\nYou may have noticed that the addition of 01101 and 10011, in addition to\nproducing 00000, also produces a carry out of the ve-bit ALU."
} |
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] | {
"indices": [
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"values": [
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]
} | {
"context": "Note: If we know the representation for A, a shortcut for guring out the\nrepresentation for A(A 0) is as follows: Flip all the bits of A (the ocial term\nfor ip is complement), and add 1 to the complement of A. The sum of A and\nExa\n10011\n00000\nYou may have noticed that the addition of 01101 and 10011, in addition to\nproducing 00000, also produces a carry out of the ve-bit ALU. That is, the binary\naddition of 01101 and 10011 is really 100000."
} |
366488 | [
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0.4633143131816905,
0.4633143131816905,
0.4633143131816905,
0.4633143131816905
]
} | {
"context": "The sum of A and\nExa\n10011\n00000\nYou may have noticed that the addition of 01101 and 10011, in addition to\nproducing 00000, also produces a carry out of the ve-bit ALU. That is, the binary\naddition of 01101 and 10011 is really 100000. However, as we saw previously,\nthis carry out can be ignored in the case of the 2s complement data type.\nAt this point, we have identied in our ve-bit scheme 15 positive inte-\ngers."
} |
300962 | [
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"indices": [
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"values": [
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} | {
"context": " However, as we saw previously,\nthis carry out can be ignored in the case of the 2s complement data type.\nAt this point, we have identied in our ve-bit scheme 15 positive inte-\ngers. We have constructed 15 negative integers. We also have a representation"
} |
947421 | [
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} | {
"context": "for 0. With k = 5, we can uniquely identify 32 distinct quantities, and we have\naccounted for only 31 (15+15+1). The remaining representation is 10000. What\nvalue shall we assign to it?\nWe note that 1 is 11111, 2 is 11110, 3 is 11101, and so on. If we continue\nthis, we note that 15 is 10001. Note that, as in the case of the positive represen-\ntations, as we sequence backwards from representations of 1 to 15, the ALU\nis subtracting 00001 from each successive representation."
} |
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} | {
"context": "If we continue\nthis, we note that 15 is 10001. Note that, as in the case of the positive represen-\ntations, as we sequence backwards from representations of 1 to 15, the ALU\nis subtracting 00001 from each successive representation. Thus, it is convenient\nto assign to 10000 the value 16; that is the value one gets by subtracting 00001\nfrom 10001 (the representation for 15).\nIn Chapter 5 we will specify a computer that we aectionately have named\nthe LC-3 (for Little Computer 3)."
} |
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"values": [
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} | {
"context": "Thus, it is convenient\nto assign to 10000 the value 16; that is the value one gets by subtracting 00001\nfrom 10001 (the representation for 15).\nIn Chapter 5 we will specify a computer that we aectionately have named\nthe LC-3 (for Little Computer 3). The LC-3 operates on 16-bit values. Therefore,\nthe 2s complement integers that can be represented in the LC-3 are the integers\nfrom 32,768 to +32,767."
} |
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"indices": [
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"values": [
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} | {
"context": "The LC-3 operates on 16-bit values. Therefore,\nthe 2s complement integers that can be represented in the LC-3 are the integers\nfrom 32,768 to +32,767.\n2.4 Conversion Between Binary and\nDecimal\nIt is often useful to convert numbers between the 2s complement data type\nthe computer likes and the decimal representation that you have used all\nyour life.\n2.4.1 Binary to Decimal Conversion\nWe convert a 2s complement representation of an integer to a decimal represen-\ntation as follows: For purposes of illustration, we will assume our number can be\nrepresented in eight bits, corresponding to decimal integer values from 128 to\n+127."
} |
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] | {
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"values": [
0.4418834306199022,
0.4418834306199022,
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]
} | {
"context": "4.1 Binary to Decimal Conversion\nWe convert a 2s complement representation of an integer to a decimal represen-\ntation as follows: For purposes of illustration, we will assume our number can be\nrepresented in eight bits, corresponding to decimal integer values from 128 to\n+127.\nRecall that an eight-bit 2s complement integer takes the form\nb7 b6 b5 b4 b3 b2 b1 b0\nwhere each of the bits bi is either 0 or 1."
} |
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} | {
"context": "1 Binary to Decimal Conversion\nWe convert a 2s complement representation of an integer to a decimal represen-\ntation as follows: For purposes of illustration, we will assume our number can be\nrepresented in eight bits, corresponding to decimal integer values from 128 to\n+127.\nRecall that an eight-bit 2s complement integer takes the form\nb7 b6 b5 b4 b3 b2 b1 b0\nwhere each of the bits bi is either 0 or 1.\n1."
} |
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] | {
"indices": [
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],
"values": [
0.45021338715555703,
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0.6208926095194913,
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0.7107036060130572,
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]
} | {
"context": "Recall that an eight-bit 2s complement integer takes the form\nb7 b6 b5 b4 b3 b2 b1 b0\nwhere each of the bits bi is either 0 or 1.\n1. Examine the leading bit b7. If it is a 0, the integer is positive, and we can\nbegin evaluating its magnitude. If it is a 1, the integer is negative. In that\ncase, we need to rst obtain the 2s complement representation of the\npositive number having the same magnitude."
} |
146279 | [
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] | {
"indices": [
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"values": [
0.7071961376506675,
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]
} | {
"context": "If it is a 1, the integer is negative. In that\ncase, we need to rst obtain the 2s complement representation of the\npositive number having the same magnitude. We do this by ipping all the\nbits and adding 1.\n2. The magnitude is simply\nb6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nIn either case, we obtain the decimal magnitude by simply adding the\npowers of 2 that have coecients of 1."
} |
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] | {
"indices": [
19522071,
4077204723,
963753705,
1321343407,
494070171,
3546764081,
1246280156,
4268804430,
2219907560,
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385678680,
3982917068,
3690722467,
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1117261085
],
"values": [
0.6460876948075268,
0.6460876948075268,
0.6460876948075268,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
0.47720054860988836,
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0.47720054860988836,
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0.47720054860988836,
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]
} | {
"context": "\n2. The magnitude is simply\nb6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nIn either case, we obtain the decimal magnitude by simply adding the\npowers of 2 that have coecients of 1.\n3. Finally, if the original number is negative, we ax a minus sign in front.\nDone!"
} |
559524 | [
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} | {
"context": "Exa\n.\n3. The decimal integer value corresponding to 11000111 is 57.\n2.4.2 Decimal to Binary Conversion\nConverting from decimal to 2s complement is a little more complicated. The\ncrux of the method is to note that a positive binary number is odd if the rightmost\ndigit is 1 and even if the rightmost digit is 0.\nConsider again our generic eight-bit representation:\nb7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nWe can illustrate the conversion best by rst working through an example."
} |
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"context": "The\ncrux of the method is to note that a positive binary number is odd if the rightmost\ndigit is 1 and even if the rightmost digit is 0.\nConsider again our generic eight-bit representation:\nb7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nWe can illustrate the conversion best by rst working through an example.\nSuppose we wish to convert the value +105 to a 2s complement binary code."
} |
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} | {
"context": "Consider again our generic eight-bit representation:\nb7 27 + b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nWe can illustrate the conversion best by rst working through an example.\nSuppose we wish to convert the value +105 to a 2s complement binary code.\nWe note that +105 is positive. We rst nd values for bi, representing the mag-\nnitude 105. Since the value is positive, we will then obtain the 2s complement\nresult by simply appending b7, which we know is 0."
} |
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} | {
"context": "We rst nd values for bi, representing the mag-\nnitude 105. Since the value is positive, we will then obtain the 2s complement\nresult by simply appending b7, which we know is 0.\nOur rst step is to nd values for bi that satisfy the following:\n105 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nSince 105 is odd, we know that b0 is 1. We subtract 1 from both sides of the\nequation, yielding\n104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21\nWe next divide both sides of the equation by 2, yielding\n52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20\nSince 52 is even, b1, the only coecient not multiplied by a power of 2, must be\nequal to 0."
} |
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} | {
"context": "Our rst step is to nd values for bi that satisfy the following:\n105 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nSince 105 is odd, we know that b0 is 1. We subtract 1 from both sides of the\nequation, yielding\n104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21\nWe next divide both sides of the equation by 2, yielding\n52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20\nSince 52 is even, b1, the only coecient not multiplied by a power of 2, must be\nequal to 0.\nWe iterate this process, each time subtracting the rightmost digit from both\nsides of the equation, then dividing both sides by 2, and nally noting whether\nthe new decimal number on the left side is odd or even."
} |
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} | {
"context": "We subtract 1 from both sides of the\nequation, yielding\n104 = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21\nWe next divide both sides of the equation by 2, yielding\n52 = b6 25 + b5 24 + b4 23 + b3 22 + b2 21 + b1 20\nSince 52 is even, b1, the only coecient not multiplied by a power of 2, must be\nequal to 0.\nWe iterate this process, each time subtracting the rightmost digit from both\nsides of the equation, then dividing both sides by 2, and nally noting whether\nthe new decimal number on the left side is odd or even. Continuing where we left\no, with\n52 = b6 25 + j5 24 + b4 23 + b3 22 + b2 21\nthe process produces, in turn:\n26 = b6 24 + b5 23 + b4 22 + b3 21 + b2 20."
} |
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] | {
"indices": [
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],
"values": [
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} | {
"context": "\nWe iterate this process, each time subtracting the rightmost digit from both\nsides of the equation, then dividing both sides by 2, and nally noting whether\nthe new decimal number on the left side is odd or even. Continuing where we left\no, with\n52 = b6 25 + j5 24 + b4 23 + b3 22 + b2 21\nthe process produces, in turn:\n26 = b6 24 + b5 23 + b4 22 + b3 21 + b2 20"
} |
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] | {
"indices": [
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"values": [
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0.48774936983707934,
0.5881814813230422,
0.3225320989183732,
0.7406980729092179,
0.48774936983707934,
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0.3225320989183732,
0.3225320989183732,
0.48774936983707934,
0.3225320989183732,
0.3225320989183732,
0.3225320989183732,
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]
} | {
"context": "Therefore, b2 = 0.\n13 = b6 23 + b5 22 + b4 21 + b3 20\nTherefore, b3 = 1.\n6 = b6 22 + b5 21 + b4 20\nTherefore, b4 = 0.\n3 = b6 21 + b5 20\nTherefore, b5 = 1.\n1 = b6 20\nTherefore, b6 = 1, and we are done. The binary representation is 01101001.\nLets summarize the process. If we are given a decimal integer value N, we\nconstruct the 2s complement representation as follows:\n1. We rst obtain the binary representation of the magnitude of N by forming\nthe equation\nN = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nand repeating the following, until the left side of the equation is 0:\na."
} |
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] | {
"indices": [
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],
"values": [
0.4261152563889012,
0.4261152563889012,
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]
} | {
"context": "If we are given a decimal integer value N, we\nconstruct the 2s complement representation as follows:\n1. We rst obtain the binary representation of the magnitude of N by forming\nthe equation\nN = b6 26 + b5 25 + b4 24 + b3 23 + b2 22 + b1 21 + b0 20\nand repeating the following, until the left side of the equation is 0:\na. If N is odd, the rightmost bit is 1. If N is even, the rightmost bit is 0."
} |
530200 | [
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} | {
"context": "If N is odd, the rightmost bit is 1. If N is even, the rightmost bit is 0.\nb. Subtract 1 or 0 (according to whether N is odd or even) from N, remove\nthe least signicant term from the right side, and divide both sides of\nthe equation by 2.\nEach iteration produces the value of one coecient bi.\n2. If the original decimal number is positive, append a leading 0 sign bit, and\nyou are done."
} |
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] | {
"indices": [
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],
"values": [
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} | {
"context": "2. If the original decimal number is positive, append a leading 0 sign bit, and\nyou are done.\n3. If the original decimal number is negative, append a leading 0 and then\nform the negative of this 2s complement representation, and then you\nare done.\n2.4.3 Extending Conversion to Numbers with Fractional Parts\nWhat if the number we wish to convert is not an integer, but instead has a\nfractional part."
} |
475752 | [
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] | {
"indices": [
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"values": [
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]
} | {
"context": "4.3 Extending Conversion to Numbers with Fractional Parts\nWhat if the number we wish to convert is not an integer, but instead has a\nfractional part. How do we handle that wrinkle?\nBinary to decimal\nThe binary to decimal case is straightforward. In a positional\nnotation system, the number\n0.b1b2b3b4\nshows four bits to the right of the binary point, representing (when the cor-\nresponding bi = 1) the values 0."
} |
318855 | [
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] | {
"indices": [
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"values": [
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} | {
"context": " In a positional\nnotation system, the number\n0.b1b2b3b4\nshows four bits to the right of the binary point, representing (when the cor-\nresponding bi = 1) the values 0.5, 0.25, 0.125, and 0.0625. To complete the"
} |
742549 | [
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]
] | {
"indices": [
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],
"values": [
0.616883240004695,
0.7071961376506675,
0.4460095183914834,
0.616883240004695,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.763052303025794,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.763052303025794,
0.4460095183914834,
0.616883240004695,
0.4460095183914834,
0.4460095183914834,
0.616883240004695,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
0.4460095183914834,
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]
} | {
"context": "conversion to decimal, we simply add those values where the corresponding bi =\n1. For example, if the fractional part of the binary representation is\n. 1 0 1 1\nwe would add 0.5 plus 0.125 plus 0.0625, or 0.6875.\nDecimal to binary\nThe decimal to binary case requires a little more work. Sup-\npose we wanted to convert 0.421 to binary. As we did for integer conversion, we\nrst form the equation\n0."
} |
784498 | [
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] | {
"indices": [
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"values": [
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]
} | {
"context": "421 to binary. As we did for integer conversion, we\nrst form the equation\n0.421 = b1 21 + b2 22 + b3 23 + b4 24 + ...\nIn the case of converting a decimal integer value to binary, we divided by 2 and\nassigned a 1 or 0 to the coecient of 20 depending on whether the number on the\nleft of the equal sign is odd or even. Here (i.e., in the case of converting a decimal\nfraction to binary), we multiply both sides of the equation by 2 and assign a 1 or\na 0 to the coecient of 20 depending on whether the left side of the equation is\ngreater than or equal to 1 or whether the left side is less than 1."
} |
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} | {
"context": "e., in the case of converting a decimal\nfraction to binary), we multiply both sides of the equation by 2 and assign a 1 or\na 0 to the coecient of 20 depending on whether the left side of the equation is\ngreater than or equal to 1 or whether the left side is less than 1. Do you see why?\nSince\n0.842 = b1 20 + b2 21 + b3 22 + b4 23 + ...\nwe assign b1 = 0. Continuing,\n1.684 = b2 20 + b3 21 + b4 22 +."
} |
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] | {
"indices": [
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"values": [
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0.5591160146868198,
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]
} | {
"context": "Continuing,\n1.684 = b2 20 + b3 21 + b4 22 + ...\nso we assign b2 = 1 and subtract 1 from both sides of the equation, yielding\n0.684 = b3 21 + b4 22 + ...\nMultiplying by 2, we get\n1.368 = b3 20 + b4 21 + ...\nso we assign b3 = 1 and subtract 1 from both sides of the equation, yielding\n0.368 = b4 20 + ...\nwhich assigns 0 to b4. We can continue this process indenitely, until we are\nsimply too tired to go on, or until the left side = 0, in which case all bits to\nthe right of where we stop are 0s."
} |
924429 | [
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} | {
"context": "which assigns 0 to b4. We can continue this process indenitely, until we are\nsimply too tired to go on, or until the left side = 0, in which case all bits to\nthe right of where we stop are 0s. In our case, stopping with four bits, we have\nconverted 0.421 decimal to 0.0110 in binary.\n2.5 Operations on Bits\nPart I: Arithmetic\n2.5.1 Addition and Subtraction\nArithmetic on 2s complement numbers is very much like the arithmetic on\ndecimal numbers that you have been doing for a long time."
} |
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] | {
"indices": [
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],
"values": [
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]
} | {
"context": "5.1 Addition and Subtraction\nArithmetic on 2s complement numbers is very much like the arithmetic on\ndecimal numbers that you have been doing for a long time."
} |
934501 | [
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0.6614917680604173,
0.5657382420345549,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.5657382420345549,
0.5657382420345549,
0.6614917680604173,
0.5657382420345549,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.5657382420345549,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288,
0.3944456016432288
]
} | {
"context": "Addition still proceeds from right to left, one digit at a time. At each point,\nwe generate a sum digit and a carry. Instead of generating a carry after 9 (since\n9 is the largest decimal digit), we generate a carry after 1 (since 1 is the largest\nbinary digit).\nExample 2.3\nUsing our ve-bit notation, what is 11 + 3?\nThe decimal value 11 is represented as 01011\nThe decimal value 3 is represented as\n00011\nThe sum, which is the value 14, is\n01110\nSubtraction is simply addition, preceded by determining the negative of the\nnumber to be subtracted."
} |
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} | {
"context": "Example 2.3\nUsing our ve-bit notation, what is 11 + 3?\nThe decimal value 11 is represented as 01011\nThe decimal value 3 is represented as\n00011\nThe sum, which is the value 14, is\n01110\nSubtraction is simply addition, preceded by determining the negative of the\nnumber to be subtracted. That is, A B is simply A + (B).\nExample 2.4\nWhat is 14 9?\nThe decimal value 14 is represented as\n01110\nThe decimal value 9 is represented as\n01001\nFirst we form the negative, that is, -9: 10111\nAdding 14 to -9, we get\n01110\n10111\nwhich results in the value 5."
} |
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} | {
"context": "Example 2.4\nWhat is 14 9?\nThe decimal value 14 is represented as\n01110\nThe decimal value 9 is represented as\n01001\nFirst we form the negative, that is, -9: 10111\nAdding 14 to -9, we get\n01110\n10111\nwhich results in the value 5.\n00101\nNote again that the carry out is ignored.\nExam\nWhat happens when we add a number to itself (e.g., x + x)?\nLets assume for this example eight-bit codes, which would allow us to represent\nintegers from 128 to 127."
} |
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} | {
"context": "g., x + x)?\nLets assume for this example eight-bit codes, which would allow us to represent\nintegers from 128 to 127. Consider a value for x, the integer 59, represented as\n00111011. If we add 59 to itself, we get the code 01110110. Note that the bits have\nall shifted to the left by one position. Is that a curiosity, or will that happen all the\ntime as long as the sum x + x is not too large to represent with the available number\nof bits?\nUsing our positional notation, the number 59 is\n0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20\nThe sum 59 + 59 is 2 59, which, in our representation, is\n2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)\nBut that is nothing more than\n0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21\nwhich shifts each digit one position to the left."
} |
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] | {
"indices": [
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"values": [
0.3079750566011035,
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} | {
"context": "Note that the bits have\nall shifted to the left by one position. Is that a curiosity, or will that happen all the\ntime as long as the sum x + x is not too large to represent with the available number\nof bits?\nUsing our positional notation, the number 59 is\n0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20\nThe sum 59 + 59 is 2 59, which, in our representation, is\n2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)\nBut that is nothing more than\n0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21\nwhich shifts each digit one position to the left. Thus, adding a number to itself\n(provided there are enough bits to represent the result) is equivalent to shifting the\nrepresentation one bit position to the left."
} |
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} | {
"context": "Is that a curiosity, or will that happen all the\ntime as long as the sum x + x is not too large to represent with the available number\nof bits?\nUsing our positional notation, the number 59 is\n0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20\nThe sum 59 + 59 is 2 59, which, in our representation, is\n2 (0 26 + 1 25 + 1 24 + 1 23 + 0 22 + 1 21 + 1 20)\nBut that is nothing more than\n0 27 + 1 26 + 1 25 + 1 24 + 0 23 + 1 22 + 1 21\nwhich shifts each digit one position to the left. Thus, adding a number to itself\n(provided there are enough bits to represent the result) is equivalent to shifting the\nrepresentation one bit position to the left.."
} |
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] | {
"indices": [
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],
"values": [
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]
} | {
"context": " Thus, adding a number to itself\n(provided there are enough bits to represent the result) is equivalent to shifting the\nrepresentation one bit position to the left."
} |
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} | {
"context": "2.5.2 Sign-Extension\nIt is often useful to represent a small number with fewer bits. For example, rather\nthan represent the value 5 as 0000000000000101, there are times when it makes\nnse to use only six bits to represent the value 5: 000101. There is little\nn, since we are all used to adding leading zeros without aecting the\na number. A check for $456.78 and a check for $0000456.78 are checks\nhe same value."
} |
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"indices": [
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"values": [
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} | {
"context": "78 and a check for $0000456.78 are checks\nhe same value.\nat about negative representations? We obtained the negative representa-\nits positive counterpart by complementing the positive representation\ning 1. Thus, the representation for 5, given that 5 is represented as\n, is 111011. If 5 is represented as 0000000000000101, then the represen-\ntation for 5 is 1111111111111011. In the same way that leading 0s do not aect\nthe value of a positive number, leading 1s do not aect the value of a negative\nnumber."
} |
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"context": "If 5 is represented as 0000000000000101, then the represen-\ntation for 5 is 1111111111111011. In the same way that leading 0s do not aect\nthe value of a positive number, leading 1s do not aect the value of a negative\nnumber.\nIn order to add representations of dierent lengths, it is rst necessary to\nrepresent them with the same number of bits. For example, suppose we wish to\nadd the number 13 to 5, where 13 is represented as 0000000000001101 and 5\nis represented as 111011."
} |
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} | {
"context": "In order to add representations of dierent lengths, it is rst necessary to\nrepresent them with the same number of bits. For example, suppose we wish to\nadd the number 13 to 5, where 13 is represented as 0000000000001101 and 5\nis represented as 111011. If we do not represent the two values with the same\nnumber of bits, we have\n0000000000001101\n+ 111011\nWhen we attempt to perform the addition, what shall we do with the missing bits\nin the representation for 5? If we take the absence of a bit to be a 0, then we are\nno longer adding 5 to 13."
} |
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"indices": [
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} | {
"context": "For example, suppose we wish to\nadd the number 13 to 5, where 13 is represented as 0000000000001101 and 5\nis represented as 111011. If we do not represent the two values with the same\nnumber of bits, we have\n0000000000001101\n+ 111011\nWhen we attempt to perform the addition, what shall we do with the missing bits\nin the representation for 5? If we take the absence of a bit to be a 0, then we are\nno longer adding 5 to 13. On the contrary, if we take the absence of bits to be 0s,\nwe have changed the 5 to the number represented as 0000000000111011, that\nis, +59."
} |
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"indices": [
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"values": [
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} | {
"context": "If we do not represent the two values with the same\nnumber of bits, we have\n0000000000001101\n+ 111011\nWhen we attempt to perform the addition, what shall we do with the missing bits\nin the representation for 5? If we take the absence of a bit to be a 0, then we are\nno longer adding 5 to 13. On the contrary, if we take the absence of bits to be 0s,\nwe have changed the 5 to the number represented as 0000000000111011, that\nis, +59. Not surprisingly, then, our result turns out to be the representation for 72."
} |
721685 | [
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"indices": [
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} | {
"context": "On the contrary, if we take the absence of bits to be 0s,\nwe have changed the 5 to the number represented as 0000000000111011, that\nis, +59. Not surprisingly, then, our result turns out to be the representation for 72.\nHowever, if we understand that a 6-bit 5 and a 16-bit 5 dier only in the\nnumber of meaningless leading 1s, then we rst extend the value of 5 to 16 bits\nbefore we perform the addition."
} |
247613 | [
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"indices": [
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"values": [
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]
} | {
"context": "Not surprisingly, then, our result turns out to be the representation for 72.\nHowever, if we understand that a 6-bit 5 and a 16-bit 5 dier only in the\nnumber of meaningless leading 1s, then we rst extend the value of 5 to 16 bits\nbefore we perform the addition. Thus, we have\n0000000000001101\n+ 1111111111111011\n0000000000001000\nresult is +8, as we should expect.\ne value of a positive number does not change if we extend the sign bit\nany bit positions to the left as desired."
} |
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] | {
"indices": [
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"values": [
0.45449725721703593,
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]
} | {
"context": "Thus, we have\n0000000000001101\n+ 1111111111111011\n0000000000001000\nresult is +8, as we should expect.\ne value of a positive number does not change if we extend the sign bit\nany bit positions to the left as desired. Similarly, the value of a negative\nr does not change by extending the sign bit 1 as many bit positions to the\ndesired. Since in both cases it is the sign bit that is extended, we refer\nto the operation as Sign-EXTension, often abbreviated SEXT."
} |
170509 | [
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} | {
"context": "Similarly, the value of a negative\nr does not change by extending the sign bit 1 as many bit positions to the\ndesired. Since in both cases it is the sign bit that is extended, we refer\nto the operation as Sign-EXTension, often abbreviated SEXT. Sign-extension is\nperformed in order to be able to operate on representations of dierent lengths.\nIt does not aect the values of the numbers being represented."
} |
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"indices": [
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"values": [
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} | {
"context": " Sign-extension is\nperformed in order to be able to operate on representations of dierent lengths.\nIt does not aect the values of the numbers being represented.\n2.5.3 Overow\nUp to now, we have always insisted that the sum of two integers be small enough\nto be represented by the available bits. What happens if such is not the case?"
} |
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} | {
"context": "You are undoubtedly familiar with the odometer on the front dashboard of\nyour automobile. It keeps track of how many miles your car has been drivenbut\nonly up to a point. In the old days, when the odometer registered 99992 and you\ndrove it 100 miles, its new reading became 00092. A brand new car! The problem,\nas you know, is that the largest value the odometer could store was 99999, so the\nvalue 100092 showed up as 00092."
} |
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"context": "In the old days, when the odometer registered 99992 and you\ndrove it 100 miles, its new reading became 00092. A brand new car! The problem,\nas you know, is that the largest value the odometer could store was 99999, so the\nvalue 100092 showed up as 00092. The carry out of the ten-thousands digit was\nlost. (Of course, if you grew up in Boston, the carry out was not lost at allit\nwas in full display in the rusted chrome all over the car."
} |
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"context": "The carry out of the ten-thousands digit was\nlost. (Of course, if you grew up in Boston, the carry out was not lost at allit\nwas in full display in the rusted chrome all over the car.)\nWe say the odometer overowed. Representing 100092 as 00092 is unaccept-\nable. As more and more cars lasted more than 100,000 miles, car makers felt the\npressure to add a digit to the odometer. Today, practically all cars overow at\n1,000,000 miles, rather than 100,000 miles."
} |
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} | {
"context": "As more and more cars lasted more than 100,000 miles, car makers felt the\npressure to add a digit to the odometer. Today, practically all cars overow at\n1,000,000 miles, rather than 100,000 miles.\nThe odometer provides an example of unsigned arithmetic. The miles you\nadd are always positive miles. The odometer reads 000129 and you drive 50 miles.\nThe odometer now reads 000179. Overow is a carry out of the leading digit."
} |
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} | {
"context": "The odometer now reads 000179. Overow is a carry out of the leading digit.\nIn the case of signed arithmetic, or more particularly, 2s complement\narithmetic, overow is a little more subtle.\nLets return to our ve-bit 2s complement data type, which allowed us to\nrepresent integers from 16 to +15. Suppose we wish to add +9 and +11. Our\narithmetic takes the following form:\n01001\n01011\n10100\nNote that the sum is larger than +15, and therefore too large to represent with\nour 2s complement scheme."
} |
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"indices": [
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"values": [
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} | {
"context": "Suppose we wish to add +9 and +11. Our\narithmetic takes the following form:\n01001\n01011\n10100\nNote that the sum is larger than +15, and therefore too large to represent with\nour 2s complement scheme. The fact that the number is too large means that the\nnumber is larger than 01111, the largest positive number we can represent with\na ve-bit 2s complement data type. Note that because our positive result was\nlarger than +15, it generated a carry into the leading bit position."
} |
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"indices": [
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"values": [
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} | {
"context": "The fact that the number is too large means that the\nnumber is larger than 01111, the largest positive number we can represent with\na ve-bit 2s complement data type. Note that because our positive result was\nlarger than +15, it generated a carry into the leading bit position. But this bit\nposition is used to indicate the sign of a value. Thus, detecting that the result is\ntoo large is an easy matter."
} |
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} | {
"context": "But this bit\nposition is used to indicate the sign of a value. Thus, detecting that the result is\ntoo large is an easy matter. Since we are adding two positive numbers, the result\nmust be positive. Since the ALU has produced a negative result, something must\nbe wrong. The thing that is wrong is that the sum of the two positive numbers\nis too large to be represented with the available bits."
} |
246720 | [
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} | {
"context": "Since the ALU has produced a negative result, something must\nbe wrong. The thing that is wrong is that the sum of the two positive numbers\nis too large to be represented with the available bits. We say that the result has\noverowed the capacity of the representation.\nSuppose instead, we had started with negative numbers, for example, 12\nand 6. In this case, our arithmetic takes the following form:\n10100\n11010\n01110\nHere, too, the result has overowed the capacity of the machine, since 12 + 6\nequals 18, which is more negative than 16, the negative number with the\nlargest allowable magnitude."
} |
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} | {
"context": "Suppose instead, we had started with negative numbers, for example, 12\nand 6. In this case, our arithmetic takes the following form:\n10100\n11010\n01110\nHere, too, the result has overowed the capacity of the machine, since 12 + 6\nequals 18, which is more negative than 16, the negative number with the\nlargest allowable magnitude. The ALU obliges by producing a positive result.\nAgain, this is easy to detect since the sum of two negative numbers cannot be\npositive."
} |
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] | {
"indices": [
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],
"values": [
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]
} | {
"context": " The ALU obliges by producing a positive result.\nAgain, this is easy to detect since the sum of two negative numbers cannot be\npositive."
} |
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} | {
"context": "Note that the sum of a negative number and a positive number never presents\na problem. Why is that? See Exercise 2.25.\n2.6 Operations on Bits\nPart II: Logical Operations\nWe have seen that it is possible to perform arithmetic (e.g., add, subtract) on val-\nues represented as binary patterns. Another class of operations useful to perform\non binary patterns is the set of logical operations."
} |
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"context": ", add, subtract) on val-\nues represented as binary patterns. Another class of operations useful to perform\non binary patterns is the set of logical operations.\n2.6.1 A Logical Variable\nLogical operations operate on logical variables. A logical variable can have one\nof two values, 0 or 1. The name logical is a historical one; it comes from the fact\nthat the two values 0 and 1 can represent the two logical values false and true,\nbut the use of logical operations has traveled far from this original meaning."
} |
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"indices": [
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"values": [
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} | {
"context": "A logical variable can have one\nof two values, 0 or 1. The name logical is a historical one; it comes from the fact\nthat the two values 0 and 1 can represent the two logical values false and true,\nbut the use of logical operations has traveled far from this original meaning.\nThere are several basic logic functions, and most ALUs perform all of them.\n2.6.2 The AND Function\nAND is a binary logical function."
} |
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} | {
"context": "6.2 The AND Function\nAND is a binary logical function. This means it requires two pieces of input data.\nSaid another way, AND requires two source operands. Each source is a logical\nvariable, taking the value 0 or 1. The output of AND is 1 only if both sources have\nthe value 1. Otherwise, the output is 0. We can think of the AND operation as the\nALL operation; that is, the output is 1 only if ALL two inputs are 1."
} |
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} | {
"context": "Otherwise, the output is 0. We can think of the AND operation as the\nALL operation; that is, the output is 1 only if ALL two inputs are 1. Otherwise,\nthe output is 0.\nA convenient mechanism for representing the behavior of a logical operation\nis the truth table. A truth table consists of n + 1 columns and 2n rows. The rst\nn columns correspond to the n source operands. Since each source operand is a\nlogical variable and can have one of two values, there are 2n unique values that\nthese source operands can have."
} |
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] | {
"indices": [
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"values": [
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} | {
"context": "The rst\nn columns correspond to the n source operands. Since each source operand is a\nlogical variable and can have one of two values, there are 2n unique values that\nthese source operands can have. Each such set of values (sometimes called an\ninput combination) is represented as one row of the truth table. The nal column\nin the truth table shows the output for each input combination."
} |
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] | {
"indices": [
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"values": [
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} | {
"context": "Each such set of values (sometimes called an\ninput combination) is represented as one row of the truth table. The nal column\nin the truth table shows the output for each input combination.\nIn the case of a two-input AND function, the truth table has two columns for\nsource operands, and four (22) rows for unique input combinations.\nA\nB\nAND\n0\n0\n0\n0\n1\n0\n1\n0\n0\n1\n1\n1\nWe can apply the logical operation AND to two bit patterns of m bits each."
} |
146400 | [
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] | {
"indices": [
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"values": [
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} | {
"context": "\nIn the case of a two-input AND function, the truth table has two columns for\nsource operands, and four (22) rows for unique input combinations.\nA\nB\nAND\n0\n0\n0\n0\n1\n0\n1\n0\n0\n1\n1\n1\nWe can apply the logical operation AND to two bit patterns of m bits each. This\ninvolves applying the operation individually and independently to each pair of"
} |
143957 | [
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"indices": [
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"values": [
0.7362650941286006,
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0.7362650941286006,
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]
} | {
"context": "bits in the two source operands. For example, if a and b in Example 2.6 are 16-\nbit patterns, then c is the AND of a and b. This operation is often called a bit-\nwise AND because the operation is applied to each pair of bits individually and\nindependently.\nExample 2.6\nIf c is the AND of a and b, where a=0011101001101001 and b=0101100100100001,\nwhat is c?\nWe form the AND of a and b by bit-wise ANDing the two values."
} |
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} | {
"context": "Example 2.6\nIf c is the AND of a and b, where a=0011101001101001 and b=0101100100100001,\nwhat is c?\nWe form the AND of a and b by bit-wise ANDing the two values.\nThat means individually ANDing each pair of bits ai and bi to form ci. For\nexample, since a0=1 and b0=1, c0 is the AND of a0 and b0, which is 1.\nSince a6=1 and b6=0, c6 is the AND of a6 and b6, which is 0.\nThe complete solution for c is\na: 0011101001101001\nb: 0101100100100001\nc: 0001100000100001\nExam\nSuppose we have an eight-bit patternlets call it Ain which the rightmost two\nbits have particular signicance."
} |
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} | {
"context": "Since a6=1 and b6=0, c6 is the AND of a6 and b6, which is 0.\nThe complete solution for c is\na: 0011101001101001\nb: 0101100100100001\nc: 0001100000100001\nExam\nSuppose we have an eight-bit patternlets call it Ain which the rightmost two\nbits have particular signicance. The computer could be asked to do one of four tasks\ndepending on the value stored in the two rightmost bits of A. Can we isolate those\ntwo bits?\nYes, we can, using a bit mask."
} |
320929 | [
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] | {
"indices": [
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],
"values": [
0.4150083828869232,
0.4150083828869232,
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]
} | {
"context": "The computer could be asked to do one of four tasks\ndepending on the value stored in the two rightmost bits of A. Can we isolate those\ntwo bits?\nYes, we can, using a bit mask. A bit mask is a binary pattern that enables the bits\nof A to be separated into two partsgenerally the part you care about and the part\nyou wish to ignore. In this case, the bit mask 00000011 ANDed with A produces 0 in\nbit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions\n1 and 0."
} |
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} | {
"context": "A bit mask is a binary pattern that enables the bits\nof A to be separated into two partsgenerally the part you care about and the part\nyou wish to ignore. In this case, the bit mask 00000011 ANDed with A produces 0 in\nbit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions\n1 and 0. The bit mask is said to mask out the values in bit positions 7 through 2."
} |
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] | {
"indices": [
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],
"values": [
0.41864577566799605,
0.8520923031101176,
0.8120558655058222,
0.7826374421676355,
0.5902048035505816,
0.41864577566799605,
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]
} | {
"context": "In this case, the bit mask 00000011 ANDed with A produces 0 in\nbit positions 7 through 2, and the original values of bits 1 and 0 of A in bit positions\n1 and 0. The bit mask is said to mask out the values in bit positions 7 through 2.\nIf A is 01010110, the AND of A and the bit mask 00000011 is 00000010. If A is\n11111100, the AND of A and the bit mask 00000011 is 00000000.\nThat is, the result of ANDing any eight-bit pattern with the mask 00000011 is\none of the four patterns: 00000000, 00000001, 00000010, or 00000011."
} |
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] | {
"indices": [
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"values": [
0.4820161344390456,
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0.7362650941286006,
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0.6504870267441352,
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]
} | {
"context": "If A is\n11111100, the AND of A and the bit mask 00000011 is 00000000.\nThat is, the result of ANDing any eight-bit pattern with the mask 00000011 is\none of the four patterns: 00000000, 00000001, 00000010, or 00000011. The result\nof ANDing with the mask is to highlight the two bits that are relevant.\n2.6.3 The OR Function\nOR is also a binary logical function. It requires two source operands, both of\nwhich are logical variables."
} |
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} | {
"context": "3 The OR Function\nOR is also a binary logical function. It requires two source operands, both of\nwhich are logical variables. The output of OR is 1 if any source has the value 1.\nOnly if both sources are 0 is the output 0. We can think of the OR operation as\nthe ANY operation; that is, the output is 1 if ANY of the two inputs are 1.\nThe truth table for a two-input OR function is\nA\nB\nOR\n0\n0\n0\n0\n1\n1\n1\n0\n1\n1\n1\n1\nIn the same way that we applied the logical operation AND to two m-bit patterns,\nwe can apply the OR operation bit-wise to two m-bit patterns."
} |
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"indices": [
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"values": [
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} | {
"context": " We can think of the OR operation as\nthe ANY operation; that is, the output is 1 if ANY of the two inputs are 1.\nThe truth table for a two-input OR function is\nA\nB\nOR\n0\n0\n0\n0\n1\n1\n1\n0\n1\n1\n1\n1\nIn the same way that we applied the logical operation AND to two m-bit patterns,\nwe can apply the OR operation bit-wise to two m-bit patterns."
} |
115729 | [
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} | {
"context": "Example 2.8\nIf c is the OR of a and b, where a=0011101001101001 and b=0101100100100001,\nas before, what is c?\nWe form the OR of a and b by bit-wise ORing the two values. That means\nindividually ORing each pair of bits ai and bi to form ci. For example, since a0=1\nand b0=1, c0 is the OR of a0 and b0, which is 1. Since a6=1 and b6=0, c6 is the OR\nof a6 and b6, which is also 1.\nThe complete solution for c is\na: 0011101001101001\nb: 0101100100100001\nc: 0111101101101001\nSometimes this OR operation is referred to as the inclusive-OR in order to distinguish\nit from the exclusive-OR function, which we will discuss momentarily."
} |
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} | {
"context": "Since a6=1 and b6=0, c6 is the OR\nof a6 and b6, which is also 1.\nThe complete solution for c is\na: 0011101001101001\nb: 0101100100100001\nc: 0111101101101001\nSometimes this OR operation is referred to as the inclusive-OR in order to distinguish\nit from the exclusive-OR function, which we will discuss momentarily.\n2.6.4 The NOT Function\nNOT is a unary logical function. This means it operates on only one source\noperand."
} |
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} | {
"context": "4 The NOT Function\nNOT is a unary logical function. This means it operates on only one source\noperand. It is also known as the complement operation. The output is formed by\ncomplementing the input. We sometimes say the output is formed by inverting\nthe input. A 1 input results in a 0 output. A 0 input results in a 1 output.\nThe truth table for the NOT function is\nA\nNOT\n0\n1\n1\n0\nIn the same way that we applied the logical operation AND and OR to two m-bit\npatterns, we can apply the NOT operation bit-wise to one m-bit pattern."
} |
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0.7250884400239858,
0.46785238308125726,
0.46785238308125726,
0.7250884400239858,
0.46785238308125726,
0.46785238308125726,
0.46785238308125726,
0.7250884400239858,
0.46785238308125726,
0.6374651681242771,
0.6374651681242771,
0.6374651681242771,
0.46785238308125726,
0.6374651681242771,
0.6374651681242771,
0.46785238308125726,
0.46785238308125726,
0.6374651681242771,
0.46785238308125726,
0.46785238308125726,
0.46785238308125726,
0.46785238308125726,
0.6374651681242771,
0.46785238308125726,
0.46785238308125726,
0.46785238308125726,
0.46785238308125726
]
} | {
"context": "A 0 input results in a 1 output.\nThe truth table for the NOT function is\nA\nNOT\n0\n1\n1\n0\nIn the same way that we applied the logical operation AND and OR to two m-bit\npatterns, we can apply the NOT operation bit-wise to one m-bit pattern. If a is as\nbefore, then c is the NOT of a.\na: 0011101001101001\nc: 1100010110010110\n2.6.5 The Exclusive-OR Function\nExclusive-OR, often abbreviated XOR, is a binary logical function."
} |
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} | {
"context": "6.5 The Exclusive-OR Function\nExclusive-OR, often abbreviated XOR, is a binary logical function. It, too,\nrequires two source operands, both of which are logical variables. The output\nof XOR is 1 if one (but not both) of the two sources is 1. The output of XOR is 0\nif both sources are 1 or if neither source is 1. In other words, the output of XOR is\n1 if the two sources are dierent. The output is 0 if the two sources are the same."
} |