Notice that we have seen this sequence before when we discussed the sum of the natural numbers from 1 to n. The formula that gives this sum is
x(n) = .5(n)(n + 1),
and I wrote the proof of this for my third weekly assignment. This implies that x(n) is also the nth triangular number. This gives the first few triangular numbers,
0 1 3 6 10 15 21 28 36 ...
We talked about triangular numbers briefly in class, but I didn't realize the connection between them and x(n) until I read the wikipedia article on triangular numbers (edit: I now see it was also mentioned on the course page, so if I read that more thoroughly I'd have seen it there as well). Another interesting result I learned is that the sum of two consecutive triangular numbers is a square number. We see this in a few examples using the first triangular numbers:
0 + 1 = 1
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
15 + 21 = 36
21 + 28 = 49
28 + 36 = 64
In fact, the sum of the (n - 1)th and nth term is the square of n, which we see since
x(n - 1) + x(n) = .5(n - 1)(n) + .5(n)(n + 1)
= .5[(n - 1)(n) + n(n + 1)]
= .5(n2 - n + n2 + n)
which is also the square of the difference between the two triangular numbers.
In 1796, a 19-year-old Gauss proved that any given natural number can be written as the sum of three of these triangular numbers. Some sources state the theorem as at most three triangular numbers and exclude 0 from the list, but it amounts to the same thing. I don't particularly feel like doing yet another proof for my weekly assignment, so we'll just go down the list of natural numbers and see for a few examples that this works (and, after looking at an outline of the proof, I hope to never encounter it again).
0 = 0 + 0 + 0
1 = 0 + 0 + 1
2 = 0 + 1 + 1
3 = 1 + 1 + 1 = 0 + 0 + 3
4 = 0 + 1 + 3
5 = 1 + 1 + 3
6 = 0 + 0 + 6
7 = 0 + 1 + 6
8 = 1 + 1 + 6
9 = 0 + 3 + 6
10 = 1 + 3 + 6
35 = 1 + 6 + 28
This is a pretty interesting result, implying the entire set of natural numbers is spanned by the triangular numbers through addition. This is all the more remarkable since it's easy to see that the triangular numbers become relatively less frequent as we go into higher and higher numbers. Now, for the other half of the assignment, we'll look at a few notable minorities in mathematics. I noticed that other people have already written pretty extensively about women in mathematics, so I'll focus more on minority groups in general by giving a few brief biographies of mathematicians with minority backgrounds.
One of the many notable women in mathematics was Ada Lovelace, a 19th century mathematician most known for her contributions to Charles Babbage's work on his "Analytical Engine", a proposed (but never completed) computational device. Lovelace was the only legitimate child of Lord Byron who, despite being kind of terrible at being a family man, I'm fond of because of his role in the Romantic period.
Lovelace is credited with writing one of the first computer programs, which consisted of an algorithm for the Analytical Engine to compute Bernoulli numbers. Unfortunately, since the Babbage never completed his computer due to funding issues, Lovelace's algorithm was never tested. Some controversy surrounds Lovelace and her contributions, since many claim that Babbage prepared the algorithms several years earlier and had them published under Lovelace's name for some unknown reason. It's still not known how much of the work should be credited to Lovelace, but her role, no matter its true degree, deserves mention simply because of the extreme lack of recognition women received in mathematics. Although progress has certainly been made with regard to women in mathematics, there is still room for improvement and more inclusion. As noted in class, much of the skepticism seems to come from the public in general, and not from mathematicians themselves.
Another underrepresented minority in mathematics is African Americans (and blacks in general, regardless of nationality). The late 18th- and early 19th-century free African American, Benjamin Banneker, is one such mathematician. Although he received no formal education, Banneker is known as a scientist, surveyor, and almanac author.
When he was 22, Banneker designed a clock that rung on the hour that he modeled by scaling the pieces to a pocket watch he had. While working as a farmer for the Ellicott family, he was lent books on astronomy which he studied intently. In 1791, he was hired to help survey the boundaries of the newly-created District of Columbia using his knowledge of astronomy to aid in the location of coordinates. Using the same knowledge, Banneker accurately calculated phenomena such as sunrises and sunsets, lunar and solar eclipses, and other "remarkable days". He wrote almanacs that contained this and other information, and wrote journals that included many mathematical calculations and puzzles. Sadly, all but one of his journals were lost to a fire. Still, he is remembered for his contributions, and an obelisk was erected at his grave in 1977 and many facilities are named in his honor.
One last story worth noting is the unfortunate case of Alan Turing. Turing is most recognized for contributing to computer science, but he also published papers in mathematical biology, logic, and philosophy. He was hired by the British during World War II for his skills as a cryptanalyst, and afterward worked as a computer scientist for the National Physical Laboratory and continued his work in cryptography with the Government Communications Headquarters (GCHQ).
However, Turing entered into a homosexual relationship in 1952, which was illegal in the United Kingdom at the time. He was found guilty of indecency by the court, and was given the option of prison or probation with the condition that he receive hormonal treatment for his homosexuality. Turing opted for the latter, and as a result of his conviction lost his position with the GCHQ and was denied entry to the United States. Two years later, Turing was found dead at the age of 41 as a result of cyanide poising. Some mystery still surrounds his death because, although it was officially ruled a suicide, Turing displayed no signs of depression and appeared good-humored about his unfortunate circumstances. Turing had at least some interest in chemistry, and some claim that his death was the result of accidental cyanide inhalation while dissolving gold in potassium cyanide.
For being a field that requires the most careful use of logic and rationality, mathematics is certainly rife with prejudice. Although in most cases, it appears to the be entire culture, rather than the field itself, that is to be held responsible for these instances of injustice and exclusion. In my experience, those involved in mathematics are more open to differences and diversity than the general population. This holds true, I think, for higher education in general. If there is a lack of minority groups in the sciences and other fields, it is more likely due to the inability for these people to have the opportunities necessary to enter into them. For instance, our society views males as being better than females in the sciences and so each gender grows up feeling that they're either supposed to enter into the sciences or are somehow naturally inept at them and should go into something else. So it is our culture in general, not those in the sciences, that must grow more accepting. Whether our culture is simply the result of our biological nature (which would make changing much more difficult) or something we can grow out of, I don't know.
Likewise, many racial minorities grow up in lower-income areas and lack the same quality of education as more wealthy areas. Even within public education, there are vast differences in the quality of education and facilities among inner city schools and schools in wealthy suburbs. Excellent educators are less likely to teach in schools serving predominately low-income families since their skills are recognized and rewarded more by schools in better areas. Of course, I'll never be an educator and don't really have any suggestions how to make things better, so again I'm without answers. However, I think there has certainly been improvement in this area within the last century and as globalization continues and our societies evolve in response, there will be less distinction among races, at least in the developed world, and progress will continue.
Lastly, it seems that equality for homosexuals is well on its way in the West, and this demographic seems to suffer the least relative to the other two mentioned above (so long as the person in question is a white male, of course). In addition to growing acceptance, the relative ease of living a discrimination-free professional life being a homosexual likely comes from the fact that one can pretty easily "hide" such facts about themselves, unlike gender or racial ethnicity.
Again, those in the sciences seem to be generally more accepting of diversity than the average population, so our inclusion of minorities is limited by the overall social climate of our time. Only when (or if) our species is better able to accept and recognize the inherent need for differences among ourselves will minorities be treated as equals in their professions.