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Peter Dirichlet Essay

958 words - 4 pages

Peter G.L. Dirichlet was born in the time period of Napoleon’s great attempt at world domination, Thomas Jefferson’s inauguration, and the birth of Webster’s dictionary. Ultimately this period of time is known as the 1800s, and Peter G.L. Dirichlet was born in 1805. Dirichlet was not born into great wealth, (nor are many others, in this day and age either) his father was a postmaster in Germany where Dirichlet and his family lived. Though they didn’t have much spare change, it was recorded that Dirichlet would use any collected money to purchase math books. Dirichlet was said to be a model pupil, attentive in class and predominantly fascinated by history and mathematics. His parents ...view middle of the document...

The theorem had claimed that in order for “n > 2 there are no non-zero integers x, y, z such that xn + yn = zn.” (Connor) Mathematicians, Euler and Fermat had already proved n=3 and n=4, while Dirichlet worked to find n=5. To which he eventually did. However, this is not his most famous feat.
His most notable work is known as the Dirichlet Principle, also known as the Pigeonhole Principle. Dirichlet was studying the variance of possibilities in the amount of which a number could have. His example was of a Pigeonhole. There could be nine holes for each pigeon to go inside, however, if there were twenty pigeons, and only nine holes; then it must be assumed that there are multiple pigeons in some of the holes. Dirichlet used this principle to prove a result about approximating irrational numbers by rationals. His principle states:
“If one puts n+1 pigeons into n pigeonholes, then at least one pigeonhole will contain at least two pigeons.
More generally, if one puts kn+1 pigeons into n pigeonholes, then at least one pigeonhole will contain at least k+1 pigeons.”(Neale)
Dirichlet basically developed a sequence in which individuals could estimate similarities in situations. An example of one through the use of the Pigeonhole Principle would be: “For every 27 word sequence in the US constitution, at least two words will start will the same letter.” (Haslam. 16) Because there are only twenty six letters in the alphabet, and each of the twenty seven words must start with one of the twenty six letters, it can be assumed by the Pigeonhole Principle that at least two words must start with the same letter.
Dirichlet’s discovery of the Pigeonhole Principle aided today’s work in estimating the amount of expected errors, collisions, or malfunctions in machinery. This strategy is mainly noted in computer software, as well as large company facilities in which...

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