What are some of the most used properties of real numbers.

You have asked too many questions at the same time. I have
edited your question as you are allowed to ask only one
question.

Some of the properties of real numbers
are:

Commutative properties: This applies to addition and
multiplication and states that for two real numbers a and b, a + b = b + a and a*b =
b*a

Associative properties: This states that for three real
numbers a, b and c, (a + b) + c = a + (b + c) and (a*b)*c =
(a*(b*c)

Distributive property: This states that
for three real numbers a, b and c, a*(b + c) = a*b +
a*c

Density property: This states that for any
two real numbers it is possible to find another real number that lies between the two.
Put another way, there are an infinite number of real numbers.

Identity properties: For any number a, a + 0 = a, a is
called the additive identity. Similarly for multiplication a*1 = a, 1 is the
multiplicative identity.

What is a specific detail about the setting in Gathering Blue?

You might want to consider the description we are given of
the community of which Kira is a part, and the way that it features elements that are
both pre- and post-Ruin. This helps to present us with the post-apocalyptic setting that
the novel presents us with. Although the precise nature of the Ruin is never specified,
it is clear that we can link it to some kind of nuclear or environmental disaster that
returned civilisation to its primitive roots. Note how the following quote describes
Kira's community:

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Very little had survived the Ruin, but somehow
the structure called the Council Edifice had remained standing and firm. It was
immeasurably old. Several windows still contained patterned glass of deep reds and
golds, amazing things, for knowledge of the way of making such remarkable glass had been
lost. Some remaining windows, ones in which the colored glass had shattered, were now
paned in a thick, ordinary glass that distorted the view through bubbles and ripples.
Other windows were simply boarded over, and parts of the building's interior were darkly
shadowed. Still, the Edifice was magnificent in comparison to the ordinary sheds and
cottages of the village.

Note
the way that this description makes clear the way that so much knowledge and technology
has been lost. The old church building that is now the Council Edifice boasts forgotten
knowledge through its stained glass windows, and stands in contrast to the "ordinary
sheds and cottages" that the majority of the people in the village live in. This is yet
another detail that the author gives us of the setting that points to the cataclysmic
nature of the Ruin and the way that we are presented with some grim vision of the
future.

In The Miracle Worker, please explain what Keller, Aunt Ev and Kate are talking about when they speak of writing in Act One.

Let us remember that as the play opens we are introduced
to a family that has tried every possible contact to look for somebody who can help
Helen with her condition and also assist them in supporting Helen. Now, it is clear from
this opening scene that Keller has reached a point where he has had too many
disappointments, and has decided that it is wiser to accept what he thinks is the
inevitable, and that Helen can never be helped. This of course brings him into conflict
with Kate and Aunt Ev, who still maintain hope that somebody could be found who could
help Helen. Thus when they talk of a famous oculist in Baltimore, and Aunt Ev asks
Keller why he doesn't write to him, note how Keller
responds:

I've
stopped believing in wonders... The child's been to specialists all over Alabama and
Tennessee, if I thought it would do good I'd have her to every fool doctor in the
country.

Then, after this, he
says to his wife a very revealing comment:

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Katie. How many times can you let them break your
heart?

Keller, it is clear,
has reached the limit of what he can cope with in terms of having yet another false hope
that is doomed to be crushed. Thus the context of the opening scene refers to writing to
another specialist who they hope can help Helen.

Describe the main character in "Lamb to the Slaughter."

Roald Dahl shows his great creative talent in making Mary
Maloney the wife of a policeman. She has learned a great deal about how  police think
and operate, and this knowledge helps her commit the perfect crime. Furthermore, the
fact that the police regard Patrick Maloney as "one of ours" brings many more
investigators to the scene of the crime than would have otherwise been the case. They
stay longer, too, because they are intent on solving this crime. And with many men
working late in her house, Mary has no trouble disposing of the murder weapon by feeding
it to the investigators. Ordinarily these policemen would not have sat down to a meal in
a victim's home, but they decide it is okay in this case because Mary, as a cop's wife,
is also "one of ours."

Mary is described as a very simple,
domestic woman who is devoted to her husband.

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For her, this was always a blissful time of day.
She knew he didn't want to speak much until the first drink was finished, and she, on
her side, was content to sit quietly, enjoying his company after the long hours alone in
the house. She loved to luxuriate in the presence of this man, and to feel--almost as a
sunbather feels the sun--that warm male glow that came out of him to her when they were
alone together. She loved him for the way he sat loosely in a chair, for the way he came
in a door, or moved slowly across the room with long strides. She loved the intent, far
look in his eyes when they rested on her, the funny shape of the mouth, and especially
the way he remained silent about his tiredness, sitting still with himself until the
whisky had taken some of it
away.

Her loving devotion
will stand her in good stead after the murder because no one will suspect that such a
loving, devoted wife could commit such a crime. This excessive love, however, is the
reason Patrick wants a divorce. She is too loving, too possessive, too concerned about
his comfort. She is a nurturing, motherly type of woman, but her mothering is
suffocating her husband. She is six months pregnant, and no doubt she would have
transferred some of her mothering to her baby and given her husband some breathing room.
Perhaps Patrick would have been wise to put up with her over-attentiveness for just a
few more months. It is not at all uncommon for women to transfer their affections to
their babies, which is exactly why nature has given most of them these nurturing
instincts.

Mary thinks that by waiting on her husband like
a slave and by taking such a strong interest in everything he says and does, she will be
able to retain his affection. But she is doing just the opposite of what she intends.
Dahl deliberately refrains from quoting exactly what Patrick tells his wife when he
informs her that he wants a divorce, but the main reason must be that he is sick and
tired of their suffocating relationship.

Dahl did not want
to indicate, for example, that Patrick was having an affair with another woman. That
could be found out and would suggest a strong motive for Mary to have killed him. Dahl
characterizes Mary through her dialogue and actions, while at the same time illustrating
what it is about her that makes Patrick want to escape from her. She says such things
as:

"Tired,
darling?

"I'll get it!" she cried, jumping
up.

"Darling, shall I get your
slippers?"

"Darling," she said. "Would you like me to get
you some cheese?"

"But, darling, you must
eat!"

We like Mary better
after she stops being a doormat and plans the perfect crime.

when walking, why do cats walk diagonally across one's path both on a staircase and on level ground?

I don't believe anyone has done a scientific study on this
matter, but as the owner of three cats, I certainly do recognize the phenomenon you are
asking about!

Horses, deer, and other grazing animals
typically have eyes located on opposite sides of their skulls, an arrangement which
means that, while they have virtually no depth perception, they do have a very wide
field of view and can see almost all the way around themselves without having to move
their heads or roll their eyes to do so.

By contrast, cats
and other predator animals have their eyes located in the front of their skull like
humans do. This allows for good depth perception, which is essential for pouncing upon
or manipulating things. The trade off for good depth perception is a narrower field of
vision. Humans have a field of vision of about 180 degrees. Cats, because their eyes sit
farther forward in the sockets, can see a bit further to the sides, so their field of
vision is around 200 degrees. This means that there is a 160 degree field behind them
that they cannot see without moving their heads. Cats, like most hunters, tend to be
very attuned to what is going on around them. I believe they walk diagonally so they can
more easily keep track of what's happening behind them. Most house cats have learned the
hard way to watch out for human feet.

In "Dusk," what type of character does Norman Gortsby have?

The article on Saki in Wikipedia states: "Politically,
Munro was a Tory and somewhat reactionary in his views." It gives as a reference an
essay by Dominic Hibberd in the Oxford Dictionary of National
Biography. This helps clarify the meaning of the story "Dusk" and the
characterization of Norman Gortsby. The young con man who sits beside Gortsby on the
park bench and the elderly gentleman who "lost" the soap are "flat" characters. Only
Norman is a "round" character, and the story is about his learning experience.

Gortsby is not an educated gentleman but a naive young
clerical type who has probably been working all day in some office and is relaxing on a
park bench before going home. He is observing the "defeated" people who only come out in
the dusk. Gortsby thinks of them as

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Men and women, who had fought and lost, who hid
their fallen fortunes and dead hopes as far as possible from the scrutiiny of the
curious. . .

When the young
man sits beside him and tells his complicated story about losing his hotel, Gortsby
listens skeptically. It would appear that he has no intention of giving the stranger
money regardless of whether or not he is telling the truth. He says
bluntly:

"Of
course, the weak point of your story is that you can't produce the
soap."

The young man departs
muttering angrily, "I must have lost it." There is nothing sympathetic about this con
man. When he sits down

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As if to emphasize the fact that the world went
badly with him the new-comer unburdened himself of an angry and very audible expletive
as he flung himself into the
seat.

He is angry when he
sits down and angry when he leaves. And when Gortsby chases him down to give him a
sovereign and the cake of soap:

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He turned round sharply with an air of defensive
hostility when he found Gortsby hailing
him.

This young man wouldn't
give anybody a sovereign, or even a shilling. That is part of the
message.

Gortsby is portrayed as a young urbanite who
thinks himself sophisticated. He has no sympathy for any of the "defeated" people he
observes and none for the young man who sits down beside him. But then he finds the cake
of soap on the ground and he  feels ashamed of himself. He was in a bad mood and
naturally was regarding the people around him with a jaundiced attitude. He feels shame,
guilt, and remorse as he hurries to catch up with the young man. He is lending him the
money he needs because he wants to feel better about himself. He
wants to change, to atone, to be more humane, to be more
Christian.

Then when he passes by the bench where he had
been sitting and encounters the elderly gentleman looking for the cake of soap, Gortsby
realizes he has been a sucker. That is the point of Saki's story. Saki is a Tory and a
reactionary. As such, he feels that people should take care of themselves and that if
they can't, then they should perish. This is a dog-eat-dog world in which the species is
improved through survival of the fittest. That is the lesson Saki is teaching through
the painful learning experience of his viewpoint character Norman
Gortsby.

It seems possible that even the elderly gentleman
is a con man who left the cake of soap by the bench intending to come back and use it as
an excuse to start a conversatioin with Gortsby--but Gortsby had found the soap and had
chased after the other con man. This would suggest that Saki's thesis is that you can't
feel sorry for anybody and you can't trust anybody. Gortsby's experience dramatizes the
reactionary message.

Describe the fall of the Communist Bloc in Eastern Europe and especially in the Soviet Union.

The "communist bloc in eastern Europe" came into existence
after World War II as the Soviet Union expanded its sphere of influence into the
countries of Bulgaria, Czechoslovakia, East Germany, Hungary, Poland and Romania.  This
expansion of communist governmental and economic philosophies was viewed as
self-protection by the Soviets and as unwarranted aggression by western Europe and its
allies. Over the years, incidents of intense conflict (the construction of the Berlin
Wall, the Cuban missile crisis) occurred, as did periods of comparatively peaceful
coexistence (partial nuclear test ban treaty in 1963, SALT I agreement banning
construction of new ballistic armaments).

As the 1980's
drew to a close, the Soviet Union was facing serious economic challenges within its own
country and was unable to continue sustaining the level of involvement and support it
had been providing to bloc allies. Soviet Prime Minister Mikhail Gorbachev began reform
movements ("perestroika," or reconstruction and "glastnost," or openness) "emphasizing
global interdependence and cooperation and the avoidance of force in the conduct of
foreign policy," with the goal of reviving the Soviet economy and helping the countries
of eastern Europe to find more support from the western world. He negotiated the end to
many of the Soviet Union's former efforts to test, build and maintain weapons arsenals
and put an end to the "Brezhnev Doctrine," which had been used to justify the use of
troops to keep communist governments in power in communist bloc countries. Once this
practice ended, communist bloc countries began to overthrow their communist governments
and broke away from the partnership with the Soviet Union. The varied republics within
the Soviet Union continued to rebellion from the centralized control of the past and
ended the USSR union in 1991, declaring themselves separate and independent
countries.