4-10 A NOTE ON INTEGERS
************************
Signed and unsigned integers
----------------------------
FORTRAN integers are SIGNED INTEGERS, half the possible bit patterns
are used to represent positive values (and 0), the other half is used
for the negative values.
A table showing the bit patterns that can be constructed with 4 bits,
and the integers that are associated with them:
Bit Signed Unsigned
Pattern integer integer
------------ -------- --------
1111 -1 15
1110 -2 14
1101 -3 13
1100 -4 12
1011 -5 11
1010 -6 10
1001 -7 9
1000 -8 8
0111 7 7
0110 6 6
0101 5 5
0100 4 4
0011 3 3
0010 2 2
0001 1 1
0000 0 0
Of course no FORTRAN integer type is really made of half a byte,
but that is enough to illustrate the general principles.
CPU hardware adds integers bit by bit, from the Least Significant
Bit (LSB) to the Most Significant Bit (MSB), taking into account
the value carried from the previous binary bit addition.
We will call the correspondence between the possible bit patterns
and unsigned integers the natural representation.
Natural binary representation of positive integers
--------------------------------------------------
The binary representation of a positive integer N is:
N = A0*1 + A1*2 + A2*4 + A3*8 + A4*16 + ... Ai = {0,1}
The Ai coefficients are the reminders on repeated divisions
by 2:
A0 = N - (2 * (N / 2)) = MOD(N, 2)
N = N / 2
A1 = N - (2 * (N / 2)) = MOD(N, 2)
......................
The resulting binary number can be written:
... A4 A3 A2 A1 A0
Other binary representations of positive integers
-------------------------------------------------
Two more representation methods were invented for positive integers,
offset binary and the binary coded decimal family (BCD).
These methods are better understood in the context of extending the
representation to include negative integers, and so will be discussed
in the next paragraph.
Number systems that include negative & positive integers
--------------------------------------------------------
Extending the 'natural' binary representation of positive integers
to negative integers can be done in at least 3 different schemes:
sign-magnitude, one's complement and two's complement.
SIGN-MAGNITUDE The most significant bit (MSB) is reserved to
the sign, 0 is positive, 1 is negative.
All other bits are used to store the
magnitude in the natural representation.
Addition and subtraction are complicated.
There are two representations for zero!
ONE'S COMPLEMENT Positive integers are like in the natural
representation, negative numbers are obtained
by complementing each bit of the corresponding
positive number (i.e. the absolute value)
There are two representations for zero!
Bitwise addition of N and -N gives -0.
Positive integers still have MSB = 0, and
negative integers have MSB=1.
TWO'S COMPLEMENT Like one's complement, but negative numbers
are having 1 added after complementation.
Bitwise addition of N and -N gives 0
if you ignore the carry out of the MSB.
Positive integers still have MSB = 0, and
negative integers have MSB=1.
Only one representation for zero.
Negating an integer is always done using the operation of bitwise
complementation, i.e. reversing the value of each bit:
0 ---> 1
1 ---> 0
The only integer representation used in modern computers is two's
complement (One's complement was used in the classic CDC machines).
Two's complement allows the same CPU circuitry to perform addition
and subtraction, subtraction is just addition of the negated number.
See Appendix A for a method to diagnose the type of integer arithmetic
on your computer.
Two other number systems that are not extensions of the 'natural'
binary representation of positive integers, i.e. they give other
values to the positive integers are offset binary and BCD.
OFFSET BINARY Rarely used method, the value assigned to a bit
pattern is the 'natural value' minus the bit
pattern 100...0.
Forms an 'ascending' series from the negative to
the positive numbers.
Equals two's complement but with MSB complemented.
Only one representation for zero.
BCD A family of coding systems, all of them generated
by taking the decimal representation and coding
each decimal digit (and the sign) with 4 bits.
The following table illustrates the representation methods with
4 bits bit patterns:
Sign- One's two's Offset
Integer magnitude complement complement binary
------- --------- ---------- ---------- ------
+7 0111 0111 0111 1111
+6 0110 0110 0110 1110
+5 0101 0101 0101 1101
+4 0100 0100 0100 1100
+3 0011 0011 0011 1011
+2 0010 0010 0010 1010
+1 0001 0001 0001 1001
0 0000 0000 0000 1000
-1 1001 1110 1111 0111
-2 1010 1101 1110 0110
-3 1011 1100 1101 0101
-4 1100 1011 1100 0100
-5 1101 1010 1011 0011
-6 1110 1001 1010 0010
-7 1111 1000 1001 0001
-8 ---- ---- 1000 0000
(-0) 1000 1111 ---- ----
The following table shows the three common BCD coding systems
(BCD 8421 is known as just BCD):
Pattern BCD 8421 Excess-3 BCD 2421
-------- -------- -------- --------
1111 9
1110 8
1101 7
1100 9 6
1011 8 5
1010 7
1001 9 6
1000 8 5
0111 7 4
0110 6 3
0101 5 2
0100 4 1 4
0011 3 0 3
0010 2 2
0001 1 1
0000 0 0
BCD arithmetic on strings is really a kind of multiple precision
arithmetic (bignums). Old computers implemented BCD in hardware,
but it is inefficient compared with integers and reals.
Excess-3 and BCD 2421 were devised so that bitwise complementation
will give the 9's complement (the radix 2 analog of 1's complement)
of the decimal number, making hardware implementation require more
simple circuitry.
Using integers in place of floating-point numbers
-------------------------------------------------
Integers are sometimes called fixed-point numbers because they can
be viewed as floats with radix point after the least significant
digit, and zero fractional digits:
1 1.000...
10 10.000...
100 100.000...
This strange representation hints at a possible use of integers as
a replacement for floats. Instead of a float X with p digits decimal
mantissa, we can use an integer N:
N = INT(X * (10**p)) (Float to integer transformation)
X = N / (10**p) (Recovering the float from the integer)
Addition is simple, you just add the integers.
X1 + X2 = N1 / (10**p) + N2 / (10**p) = (N1 + N2) / (10**p)
Multiplication is more tricky:
X1 * X2 = [N1 / (10**p)] * [N2 / (10**p)]
= (N1 * N2) / [(10**p)**2]
= [(N1 * N2) / (10**p)] / (10**p)
Division by an extra (10**p) factor is needed to get the right answer.
A common INTEGER type is the INTEGER*4 which has range:
[-2,147,483,648 : 2,147,483,647]
that means that with a 3 digits decimal mantissa you will get only
the range:
[-2,147,483.648 : 2,147,483.647]
The advantage of using integers instead of floats is that roundoff
errors are eliminated on addition and subtraction (problematic
operations with floats). Radix conversions between the internal
binary representation and external decimal are also errorless.
Integer operations takes about the same CPU time as the corresponding
float operations.
Unsigned integers
-----------------
If you work with quantities that are always positive, and even
intermediary and temporary results are positive, half of the integer
range is 'wasted'. However, unsigned integers are awkward to emulate
with signed integer arithmetic(?).
You may get the output of a measuring device in a file containing
unsigned integers, or get a file created by a program written in
a language that supports unsigned integers, e.g. C.
Comparing unsigned integers
---------------------------
Unsigned integers can be compared with a small routine that uses the
ordinary FORTRAN comparison operators:
LOGICAL FUNCTION UGE(I, J)
C ------------------------------------------------------------------
INTEGER I, J
C ------------------------------------------------------------------
IF (I .GE. 0) THEN
IF (J .LT. 0) THEN
UGE = .FALSE.
RETURN
ENDIF
ELSE
IF (J .GE. 0) THEN
UGE = .TRUE.
RETURN
ENDIF
ENDIF
C ------------------------------------------------------------------
IF (I .GE. J) THEN
UGE = .TRUE.
ELSE
UGE = .FALSE.
ENDIF
C ------------------------------------------------------------------
RETURN
END
This little monster can be further optimized ...
A much more important factor in the social movement than those already mentioned was the ever-increasing influence of women. This probably stood at the lowest point to which it has ever fallen, during the classic age of Greek life and thought. In the history of Thucydides, so far as it forms a connected series of events, four times only during a period of nearly seventy years does a woman cross the scene. In each instance her apparition only lasts for a moment. In three of the four instances she is a queen or a princess, and belongs either to the half-barbarous kingdoms of northern Hellas or to wholly barbarous Thrace. In the one remaining instance208— that of the woman who helps some of the trapped Thebans to make their escape from Plataea—while her deed of mercy will live for ever, her name is for ever lost.319 But no sooner did philosophy abandon physics for ethics and religion than the importance of those subjects to women was perceived, first by Socrates, and after him by Xenophon and Plato. Women are said to have attended Plato’s lectures disguised as men. Women formed part of the circle which gathered round Epicurus in his suburban retreat. Others aspired not only to learn but to teach. Arêtê, the daughter of Aristippus, handed on the Cyrenaic doctrine to her son, the younger Aristippus. Hipparchia, the wife of Crates the Cynic, earned a place among the representatives of his school. But all these were exceptions; some of them belonged to the class of Hetaerae; and philosophy, although it might address itself to them, remained unaffected by their influence. The case was widely different in Rome, where women were far more highly honoured than in Greece;320 and even if the prominent part assigned to them in the legendary history of the city be a proof, among others, of its untrustworthiness, still that such stories should be thought worth inventing and preserving is an indirect proof of the extent to which feminine influence prevailed. With the loss of political liberty, their importance, as always happens at such a conjuncture, was considerably increased. Under a personal government there is far more scope for intrigue than where law is king; and as intriguers women are at least the209 equals of men. Moreover, they profited fully by the levelling tendencies of the age. One great service of the imperial jurisconsults was to remove some of the disabilities under which women formerly suffered. According to the old law, they were placed under male guardianship through their whole life, but this restraint was first reduced to a legal fiction by compelling the guardian to do what they wished, and at last it was entirely abolished. Their powers both of inheritance and bequest were extended; they frequently possessed immense wealth; and their wealth was sometimes expended for purposes of public munificence. Their social freedom seems to have been unlimited, and they formed combinations among themselves which probably served to increase their general influence.321 The old religions of Greece and Italy were essentially oracular. While inculcating the existence of supernatural beings, and prescribing the modes according to which such beings were to be worshipped, they paid most attention to the interpretation of the signs by which either future events in general, or the consequences of particular actions, were supposed to be divinely revealed. Of these intimations, some were given to the whole world, so that he who ran might read, others were reserved for certain favoured localities, and only communicated through the appointed ministers of the god. The Delphic oracle in particular enjoyed an enormous reputation both among Greeks and barbarians for guidance afforded under the latter conditions; and during a considerable period it may even be said to have directed the course of Hellenic civilisation. It was also under this form that supernatural religion suffered most injury from the great intellectual movement which followed the Persian wars. Men who had learned to study the constant sequences of Nature for themselves, and to shape their conduct according to fixed principles of prudence or of justice, either thought it irreverent to trouble the god about questions on which they were competent to form an opinion for themselves, or did not choose to place a well-considered scheme at the mercy of his possibly interested responses. That such a revolution occurred about the middle of the fifth century B.C., seems proved by the great change of tone in reference to this subject which one perceives on passing from Aeschylus to Sophocles. That anyone should question the veracity of an oracle is a supposition which never crosses the mind of the elder dramatist. A knowledge of augury counts among the greatest benefits222 conferred by Prometheus on mankind, and the Titan brings Zeus himself to terms by his acquaintance with the secrets of destiny. Sophocles, on the other hand, evidently has to deal with a sceptical generation, despising prophecies and needing to be warned of the fearful consequences brought about by neglecting their injunctions. The stranger had a pleasant, round face, with eyes that twinkled in spite of the creases around them that showed worry. No wonder he was worried, Sandy thought: having deserted the craft they had foiled in its attempt to get the gems, the man had returned from some short foray to discover his craft replaced by another. “Thanks,” Dick retorted, without smiling. When they reached him, in the dying glow of the flashlight Dick trained on a body lying in a heap, they identified the man who had been warned by his gypsy fortune teller to “look out for a hidden enemy.” He was lying at full length in the mould and leaves. "But that is sport," she answered carelessly. On the retirement of Townshend, Walpole reigned supreme and without a rival in the Cabinet. Henry Pelham was made Secretary at War; Compton Earl of Wilmington Privy Seal. He left foreign affairs chiefly to Stanhope, now Lord Harrington, and to the Duke of Newcastle, impressing on them by all means to avoid quarrels with foreign Powers, and maintain the blessings of peace. With all the faults of Walpole, this was the praise of his political system, which system, on the meeting of Parliament in the spring of 1731, was violently attacked by Wyndham and Pulteney, on the plea that we were making ruinous treaties, and sacrificing British interests, in order to benefit Hanover, the eternal millstone round the neck of England. Pulteney and Bolingbroke carried the same attack into the pages of The Craftsman, but they failed to move Walpole, or to shake his power. The English Government, instead of treating Wilkes with a dignified indifference, was weak enough to show how deeply it was touched by him, dismissed him from his commission of Colonel of the Buckinghamshire Militia, and treated Lord Temple as an abettor of his, by depriving him of the Lord-Lieutenancy of the same county, and striking his name from the list of Privy Councillors, giving the Lord-Lieutenancy to Dashwood, now Lord Le Despencer. "I tell you what I'll do," said the Deacon, after a little consideration. "I feel as if both Si and you kin stand a little more'n you had yesterday. I'll cook two to-day. We'll send a big cupful over to Capt. McGillicuddy. That'll leave us two for to-morrer. After that we'll have to trust to Providence." "Indeed you won't," said the Surgeon decisively. "You'll go straight home, and stay there until you are well. You won't be fit for duty for at least a month yet, if then. If you went out into camp now you would have a relapse, and be dead inside of a week. The country between here and Chattanooga is dotted with the graves of men who have been sent back to the front too soon." "Adone do wud that—though you sound more as if you wur in a black temper wud me than as if you pitied me." "Wot about this gal he's married?" "Don't come any further." "Davy, it 'ud be cruel of us to go and leave him." "Insolent priest!" interrupted De Boteler, "do you dare to justify what you have done? Now, by my faith, if you had with proper humility acknowledged your fault and sued for pardon—pardon you should have had. But now, you leave this castle instantly. I will teach you that De Boteler will yet be master of his own house, and his own vassals. And here I swear (and the baron of Sudley uttered an imprecation) that, for your meddling knavery, no priest or monk shall ever again abide here. If the varlets want to shrieve, they can go to the Abbey; and if they want to hear mass, a priest can come from Winchcombe. But never shall another of your meddling fraternity abide at Sudley while Roland de Boteler is its lord." "My lord," said Edith, in her defence, "this woman has sworn falsely. The medicine I gave was a sovereign remedy, if given as I ordered. Ten drops would have saved the child's life; but the contents of the phial destroyed it. The words I uttered were prayers for the life of the child. My children, and all who know me, can bear witness that I have a custom of asking His blessing upon all I take in hand. I raised my eyes towards heaven, and muttered words; but, my lord, they were words of prayer—and I looked up as I prayed, to the footstool of the Lord. But it is in vain to contend: the malice of the wicked will triumph, and Edith Holgrave, who even in thought never harmed one of God's creatures, must be sacrificed to cover the guilt, or hide the thoughtlessness of another." "Aye, Sir Treasurer, thou hast reason to sink thy head! Thy odious poll-tax has mingled vengeance—nay, blood—with the cry of the bond." HoME古一级毛片免费观看
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