find square roots of -1+2i

We have to find the square root of `-1+2i` i.e. `\sqrt{-1+2i}`

We will find the square roots of the complex number of the form x+yi , where x and y are real numbers, by the following method:

Let `z^2=(x+yi)^2=-1+2i`

i.e. `(x^2-y^2)+2xyi=-1+2i`

Comparing real and imaginary terms we get,

`x^2-y^2=-1 -------> (1)`

`2xy=2` implies `xy=1 ------>(2)`  

So from (2) we get,  y=1/x . Substituting this in (1) we have,

`x^2-\frac{1}{x^2}=-1`

i.e. `x^4+x^2-1=0`

implies `x^2=\frac{-1\pm\sqrt{5}}{2}`

               `=0.62, -1.62`

Therefore, `x=\pm\sqrt{0.62}=\pm 0.79` ``

`x^2=-1.62` is discarded since it gives imaginary value.

hence,

When x=0.79,  y= 1.27

         x=-0.79 , y= -1.27

i.e we have,  `\sqrt{-1+2i}=0.79+1.27i or -0.79-1.27i`

                                `=\pm (0.79+1.27i)`

Hence the square roots of -1+2i are:  `\pm` (0.79+1.27i)  

Why do people from different races look different?

Great question! Actually, it's not so much that people from different "races" look different, but that people who look different were categorized into races. I think it's really important for people to understand the historical and social context of the idea of race in order to fully make sense of how it impacts our lives today. Because I am most familiar with racial categories in the United States, I will use this as my central frame of reference. 

Race, as it is thought of in the Western world, describes categories of physical attributes like color of eyes, skin, and hair. In some parts of the world, race is quite different. Historically, Japan has been quite ethnically and phenotypically (the way people look) homogenous, so there wasn't really much of a concept of race until Europeans and Mainland Asians began to visit. In Brazil, racial categories are far more flexible and may take into account factors like socioeconomic status and who you hang out with-- not just appearance! Comparing understandings of race from around the world should tip us off to the fact that race is a cultural phenomenon rather than a physical one.

The truth is that there is no biological reality of race. There is no one gene, or even a collection of genes, which determine that a person is White or Black or Asian. Even trying to lump people into these categories leaves a lot out of the picture. Within one "racial category" and even within one nation or city, there's a lot of variation in how people look. Let's imagine you have a bag full of jellybeans of every possible flavor and color, and I asked you to sort them into distinct categories. You could try to sort them into categories like blue, red, green, and so on-- but what about the purple jellybeans, the light blue, or the red with orange spots? You'd probably do your best to sort these with the jellybeans they look most similar to, right? Our understanding of race is the same way-- we try to sort people into groups based on how they look in comparison with stereotypes of race.

Though race is not a biological reality in the sense that there is one gene or one phenotype for how people of a certain race look, our stereotypes of race are based on actual physical characteristics. Color of skin, hair, and eyes, facial bone structure, and the texture of hair all may play a part in how we intentionally or unintentionally categorize someone. That physical categorization on its own isn't necessarily a bad thing-- humans love to categorize things, even each other! When we treat these categorizations as being fixed realities, creating a sense of "either-or," it can cause a lot of conflict about physical appearance. Even worse, historically and to this day, many people assign value to certain characteristics or associate a physical appearance with social behavior.

This is why race is a damaging social construct. The idea of race and racial categories takes the full spectrum of human appearance, our mixed bag of jellybeans, and tries to push them into a handful of boxes that leave out a lot of information. Prior to European colonialism, the idea of race wasn't very fixed. In fact, prior to and during the 16th century, most people used the term "race" to refer to people of differing nationalities. With increasing exploration and colonization of the world, race came to refer to the many different ways people looked in addition to nationality. With the international slave trade, the idea that people from a particular place look a certain way transformed into a very rigid hierarchy of phenotype as justification for human cruelty.

In the 17th and 18th centuries, explorers and scientists tried to devise a system of neat categories of race. The most popular and persistent of these systems was that devised by John Friedman Blumenbach, who argued that there was a biologically real hierarchy of humans categorized as Caucasian, Mongoloid, Malay, Negroid, and (Indigenous) American. This hierarchy was primarily based on skin color, with the lightest (European) skin being favored, and the darkest (African) skin being considered lowest. Though it was outwardly based on appearance, this hierarchy took on all sorts of connotations of physical and mental capabilities. Much of the racial oppression and human injustice caused by the international slave trade was "justified" because light-skinned Europeans genuinely believed that people of darker skin tones were not as smart and possibly couldn't even feel pain. Some people went as far as believing that it was their duty as a light-skinned person to rule over, exploit, and "shepherd" darker-skinned people. 

As for why people from different parts of the world have different appearances, it's all about adaptation! Our species, Homo sapiens, first arose in Southeast Africa around 200,000 years ago. Since then, our ancestors have been busy spreading out around the globe. Though this journey was most likely motivated by a search for food, people eventually settled in every possible habitable environment. With exposure to these many variable environments, the general appearance of a population began to shift in response to environmental pressure. Our first Homo sapiens ancestors were likely dark-skinned, but those who migrated out of the high-sun environment of Sub-Saharan Africa did not require as much melanin to be present in the skin. Skeletal features like nose shape, height, or jaw shape may be a response to the air quality and diet of a particular region. 

Today, we know that the social connotations of race aren't true. Especially as genetic evidence expands, we know that there is no real, fixed, biological basis for racial identities. However, beliefs about race are so deeply ingrained in society that it takes active and critical thinking to undo the things we have been taught (implicitly and explicitly) about race. 

What does Bryon mean when he says "that was then, this is now"?

In Chapter 4, Bryon and Mark are reminiscing about playing as children and getting into trouble with their friends when they were young. Mark then asks Bryon if he ever gets the feeling that things are coming to an end because something new is about to begin. Bryon tells Mark that he understands how he is feeling. Mark then mentions how they used to be very close with their other friends, but recently things have begun to change. They don't get to see their friends anymore because everyone is in a relationship or doing their own thing. Bryon agrees with Mark that things have changed and says that it is kind of a good thing. Mark then sighs and comments, "but there's a difference. I wonder what the difference is?" (Hinton 69). Bryon responds by saying,

"The difference is...that was then, this is now" (Hinton 69).

Bryon essentially means that as time progresses, people change and grow apart from one another. Bryon realizes that their childhood has come to an end and that the present stage of their lives is not the same as it once was. He is saying that the past is the past, and one must accept the ever-changing course of life.

In Shakespeare's Macbeth, what are the obstacles to Lady Macbeth's plan?

One major obstacle to Lady Macbeth's plan is Macbeth himself.  She initially fears that he is "too full o' th' milk of human kindness" to consider killing the king in order to take the throne (1.5.17).  She turns out to be wrong -- he is willing to consider it -- but after Macbeth has had time to consider all the reasons he has not to kill Duncan, he tells her, "We will proceed no further in this business" (1.7.34).  Lady Macbeth must then convince him to overcome his scruples.

Lady Macbeth's own scruples and feelings present another obstacle to her plan.  She actually prays to be "unsex[ed]" because she wants to rid herself of any "remorse" she might have; likewise, she wishes to squash any compassion or womanly feelings that might prevent her from acting brutally (1.5.48, 1.5.51).  Then, when she goes to kill Duncan, she cannot.  Alone on stage, she says, "Had he not resembled / My father as he slept, I had done 't" (2.2.16-17).  So, it seems that Lady Macbeth is not quite as hard-hearted as she'd like to be, and her compassion -- however small -- presents another obstacle to her plan.

What are the conflicts in the story "A & P" by John Updike?

There are three main conflicts in John Updike's "A & P": Man vs. Society, Man vs. Man, and Man vs. Self.

• Man vs. Society

As Sammy, who narrates, checks out an older woman's groceries, he notices three girls who enter the store wearing "nothing but bathing suits." Because he becomes so distracted by these scantily-clad girls, Sammy rings up a box of HiHo crackers twice and the "witch about fifty with rouge on her cheekbones and no eyebrows" is able to catch him making an error. She complains to Sammy.
Further in the narrative, Sammy is critical of some of the customers, calling them names, such as "houseslaves in curlers." Then, he begins to feel defensive of the girls.

• Man vs. Man 

Sammy comes into conflict with the store manager, Mr. Lengel, because he wants to defend the three girls that Lengel has accosted and told, "We want you decently dressed when you come in here." Lengel adds that they must have their shoulders covered, as it is "policy." 
All the time that Lengel talks with the girls, Sammy watches them until he becomes infatuated enough--"scrunchy inside"--to feel that he must defend them. He tells Mr. Lengel that he did not have to embarrass the girls, and in an act of bravado, he quits.

"Sammy, you don't want to do this to your Mom and Dad," he [Lengel] tells me. It's true, I don't. But it seems to me that once you begin a gesture it's fatal not to go through with it....

"You'll feel this for the rest of your life," Lengel says, and I know it's true.

• Man vs. Self

Having acted against his better judgment, Sammy narrates that he keeps thinking of how Lengel made the pretty girl blush. After he has quit, Sammy searches for the girls, but they are gone. He looks through the window of the store, and realizes that he has defended the girls of his generation against the older generation only to find himself in some kind of limbo between the two. For, he does not know if he has acted as a fool or as a hero. Nevertheless, Sammy knows life will be harder: "I felt how hard the world was going to be to me hereafter."

What are examples of figurative language used in Act III, Scene 2 of King Lear?

Act III, Scene 2 of Shakespeare's King Lear is absolutely central to the play, and it accordingly employs some of the story's most lyrical language. The most extensive example of figurative language in this scene is personification, as Lear personifies the storm and raging elements by giving them human qualities. 

The first example of personification comes in the first line of the scene, when Lear urges the storm to "crack your cheeks; rage, blow" (1). The stormy elements of nature do not actually have cheeks, nor do they literally "rage." As such, we can see Lear is giving the elements human qualities and thus heightening our ability to imagine the intensity of the storm. Later in the scene, Lear advances his personification of the storm by calling the elements "servile ministers" (21), and so he directly imagines the storm having a human profession. 

It's also worth mentioning that this scene employs an apostrophe. In terms of figurative language, an apostrophe is an address to an absent character, or to a thing or abstract idea. In this scene, Lear very clearly addresses the storm as if it were a thinking being, saying "I tax not you, you elements, with unkindness" (16). Since the weather is not a thinking being or an actual character, Lear's direct address of the storm as "you" is an apostrophe.  

What are the names of important characters in Heidi by Johanna Spyri?

The protagonist is Heidi, a Swiss girl who lives with her aunt, Dete (also called "Detie"), in the mountains. Her aunt regards Heidi as an imposition, as she has to raise Heidi after her sister and brother-in-law die. Dete takes Heidi to live with Heidi's grandfather, Alm-Uncle, who is the father of Heidi's father. Her grandfather lives in Dorfli, where he maintains his distance from others and grazes livestock ("Alm" refers to the mountain ranges where he grazes his flock). Alm-Uncle grows to love Heidi, and she also gets to know Peter, a goatherd, and Peter's blind grandmother, Bridget (also called "Grannie"). 

Heidi later goes to Frankfurt to work as a companion to Clara Sesemann (spelled "Klara" in German), a sick girl who can't walk and who lives with her father, Mr. Sesemann. Heidi charms Clara with her friendliness and country manners. Fraulein Rottenmeier, Clara's housekeeper, is a strict woman who dislikes Heidi. Doctor Classen diagnoses Heidi with homesickness and sends her home to the mountains.