New York City with her nuclear family. Especially for holidays, she visits her extended family in the city. Although she acquiesces to her family’s wishes, she does not appreciate the serious side or deeper importance of commemorating Jewish traditions such as the Passover Seder. She ultimately learns these things through the frightening experience of living during the Holocaust, and symbolically sacrificing “her” life as Chaya. Once she returns to New York, understanding why she has received this Hebrew name, she shares Rivka’s role of storyteller and conscience.

Chaya, a Polish farm girl, but moves from Lublin to live with relatives after a cholera epidemic leaves her an orphan. The Nazis raid a town where the family is attending a wedding and take them to a concentration camp. Their lives become unspeakable as they not only do manual labor, but are forced to sort the dead prisoners’ possessions. At a crucial moment, Chaya takes her friend Rivka’s place in entering the gas chamber so that Rivka can survive.

In The Devil's Arithmetic, Hannah is celebrating with her family in modern-day New York when she opens the door and is transported back to the 1940s. At this moment in the story, Chaya and Hannah become the same person. The logistics of Hannah's missing family is explained by the fact that Chaya has been orphaned due to the cholera epidemic, and Chaya's own recovery from illness also accounts for the confusion as Hannah makes sense of what is happening. Because Hannah becomes Chaya, everything they experience and all of their feelings and personality traits are one and the same.

At the onset of the novel, Hannah is portrayed as a bit self-centered, and she is clearly apathetic about her family's experiences during the Holocaust. However, as she evolves through her experiences as Chaya, she learns the true meaning of sacrifice and loss. As her friends and family are executed at the camp, Hannah/Chaya becomes broken-hearted and begins to gain a deeper understanding of what Grandpa Will and Aunt Eva went through and why Grandpa Will is so resentful. The reader sees the evolution and understanding that Hannah/Chaya has gained when she sacrifices herself to save Rivka's life, an act that is unimaginable of the Hannah that is introduced at the beginning of the novel.

In many cultures in the United States, people have two names. For example, they may have a Chinese name and then an American name.  Some are changed because they are difficult for Americans to pronounce, some are changed because they want to blend in with American culture, and some are changed because there is an American equivalent.  In Ireland, John is Sean.  In Mexico, Antonio is Anthony. In Germany, Johann is John.  In Isreal, Chaya is Hannah.  Chaya is Hannah's Isreali name.

We know this because in chapter 4, she is with Schmuel and Yitzchak.  They call her Chaya.  She thinks,

"Chaya.  but that's my Hebrew name.....The one I was given to honor Aunt Eva's dead friend. Weird." (pg 24)

So the answer to your question is that Hannah, the American, has turned into Chaya, the Isreali.  They are the same person having a different experience.