Tough equation, easy solution

Problem: Solve the equation

\(x^{2}+2 a x+\displaystyle\frac{1}{16}=-a+\sqrt{a^2+x-\displaystyle\frac{1}{16}}\)

for \(x\) real and \(0alt : plots

There is an easy way to attack the solution of this equation and a hard way. The hard way is to add [latex "size="-2"]a\) to both sides and square, generating a fourth degree equation in \(x\). No thanks. The easy way, however, is to note that the right-hand side is a solution to the equation

\(x=y^{2}+2 a y+\displaystyle\frac{1}{16}\)

Furthermore, the solution to this equation gives

\(y=x^{2}+2 a x+\displaystyle\frac{1}{16}\).

That is, we have an equation for points where a function equal to its inverse function. The points in such a case lie along the line \(y=x\) (as can be verified in the plot), and we simply have a quadratic to solve:

\(x=x^{2}+2 a x+\displaystyle\frac{1}{16}\),

the solution to which is

\(x=\displaystyle\frac{1}{2}-a \pm \sqrt{\displaystyle\frac{3}{4}-4 a (1-a)}\).

Jew Who?

The Finkler Question, by Howard Jacobson

In graduate school at the University of Rochester, we of the class that entered in 1992 were a tight bunch. We studied together, met at The Distillery or MacGregor’s after a long day of school or in the lab, and generally formed each others’ complete social circle. It was kind of difficult to keep secrets from one another. Nevertheless, there was a girl in the class, a fellow M.O.T., whom I began to see outside of our little social circle. Sort of like dating, I guess, but not really. There was simply a mutual physical attraction for a short time, and that was that. Neither she nor I said a word about it to the larger group, however. Well, at least until, much later, long after the affair ended, I let someone else in the group, who by that time had become a very close friend whom I trusted, in on the secret. This someone else, who is not M.O.T., laughed heartily when I told him of the affair, saying – and I will never forget this – “Christ! I just thought it was a Jewish thing.”

The reason why I have never forgotten this is because the statement struck me to the core of my identity.  The fact that my classmate’s and my being chummy outside the circle was never a secret sort of surprised me: I thought we were discreet about it.  But that didn’t bug me so much, or at all.  It was the fact that it was generally recognized – nay, expected even – that Jewish people will naturally gravitate toward each other which really hit a nerve with me.

It hit a nerve because my friend was right.

After sharing some laughs and pitchers of Guinness with my good friend, I pondered his remark for a long time.  I thought about the friends I made in college, the ones with whom I still would go see at the drop of a hat.  I thought about the friends from high school with whom I kept in touch.  Pretty much M.O.T., across the board.  Not 100%, but very close to it.

Why was that?  It’s not as obvious as one thinks.  Yes, I did participate in some Jewish organizations [e.g., Hillel, USY], but they were mainly at the instigation of my mother and none of the friends as described above came from that participation.  Rather, I met many of these friends in the course of normal, non-Jewish activities [e.g., dorm parties,   class, etc.].  Yes, some of the most long-term friends were made in elementary school or earlier, when our parents [who obviously were Jewish] arranged what are now called playdates.  One or two others from Hebrew school.  But, really, nothing out of the ordinary which would exclude gentiles from the picture.

Further, although I was/am fairly stereotypically Jewish in many respects [big nose, academic inclinations, liberal, etc.], I was never particularly religious, especially in graduate school.  In fact, I would say that, at the time, my interest and knowledge of things Jewish was close to zero.  I had been to Israel [with 2 of the above-mentioned college friends], but even with family there, I have to say that I knew pretty much nothing about the place or its history.  I was a Jewish apathetic – I never really experienced anti-Semitism, being Jewish hardly ever got in my way, but I didn’t feel it did all that much for me.

So what was my thing with Jews, then?  Was it the shared experiences in our upbringing?  [Then why did I not make more friends from my hometown?]  Or is there something really tribal that just can’t be pinned down?  This is The Finkler Question which Howard Jacobson attempts, with great vigor, humor and honesty, to pick apart in his new, Man Booker Prize-winning novel.

In The Finkler Question, Jacobson tells the story of two middle-aged men who are mirror images of each other.  The first is Sam [formerly Samuel] Finkler, a Jewish man living in London who is a recent widower.  His late wife Tyler was Catholic but converted to Judaism for him.  Sam, however, is the leader of a group which calls themselves ASHamed Jews and are opposed to anything Israeli or Zionist.  He is also incredibly ignorant about his heritage.  That said, his social circle tends to include mostly like-minded Jews, save for his elderly friend Libor, who is also a recent widower.

The second man is Julian Treslove, a non-Jewish man who happens to be a philo-Semite.  [As Jeffrey Goldberg has observed, a philo-Semite is an anti-Semite that likes Jews.]  Treslove has never married and seems envious of his friends Finkler and Libor for becoming widowers. In fact, Treslove seems permanently envious of his friend Finkler, most notably for being a Jew.  It seems to Treslove, being a Jew is like being a member of a secret club and he’ll do almost anything to join and learn the secrets.

Julian’s adventure begins when he is mugged by a woman as he is out walking in London late one night.  While being mugged, he distinctly claims to have been told “You Jew!” by the mugger.  As Julian attempts to piece together who could have possibly carried out this attack [a spurned lover?], he comes to the conclusion that, somehow, some way, he must be Jewish.  And so he decides that he is a Jew.

Julian, a serial womanizer who typically goes with small, skinny blondes [whose names curiously all begin with “J”], falls for Libor’s great-great-niece who is named Hephzibah [“my delight is in she”].  Hephzibah is a strikingly large, non-skinny Jewish woman, twice-divorced from Jewish men, who is in charge of the building of a new Anglo-Jewish museum [right next to the zebra stripe of The Beatles’ Abbey Road].

The overriding theme of the novel is the question of identity, and what forms it in these middle-aged men.  Finkler, for example, is a philosopher.  In the old days, he would have been a scholar in his village, the highest-ranking sort of man to a tribe of Jews.  Finkler’s philosophy, however, is put to the service of delegitimizing Judaism and Zionism.  Treslove, until he meets Hephzibah, is a professional celebrity imitator [he used to work for the Beeb].  That is, his identity is a lack of one.

With Hephzibah, however, Treslove ditches the celebrity gig and decides to help Hephzibah with her museum.  Since there isn’t much for him to do, Treslove spends his time trying to learn Yiddish and Judaism.  He enjoys opportunities to put his new knowledge to use.

The problem is that nobody who matters, that is, nobody Jewish, Hephzibah included, is much interested.  Things begin to unravel when Finkler comes into the picture and behaves as if he and Hephzibah are old friends and speak a secret language.  With this, Treslove works to a slow boil as he wonders what on Earth being a Jew is all about.

Throughout the novel, Jacobson introduces Jews of all stripes: the Jews that hate Israel, the Jews that hate Jews, Jews who murder Arabs in the West Bank, Jews who feel attacked on all fronts by increasingly visible anti-Semitism, Jews who are tired of being linked with Gaza, and even a Jew who sleeps with a Holocaust Denier.  Yet somehow, there is something that draws them all [or at least most of them] together, despite, or frankly because of their conflicts with each other.  The novel manages to address some really serious issues in British society with respect to Jews [as to why British people, Jewish or non-, are so quick to link goings on in Gaza or the West Bank with Jews in London], and at the same time hit some really funny notes.  Jacobson is a comic author first and it shows.

I felt that, while exceedingly well-written, that things got off to a slow start.  You have to stick with the book for a while [likely most of Part One].  The payoff, however, is quite good and The Finkler Question manages the all-too-rare feat of getting better as you read.  I felt that there was not a wasted scene or character in the entire novel, and that the ending, after giving it some thought, was entirely satisfactory.

Would I recommend this, then, to non-Jews?  Yes.  First, the fact that this won the Man Booker prize [frankly, I am amazed at this for reason spelled out in the book of all places] points to general interest in the topic, at least in England.  That said, I think the Catholics I know face similar questions of identity.  The point is that many of us remain a tribal people despite the most canyonesque differences.  At least for me, Jacobson has reminded me of why I began my quest to understand my Jewish identity.