Contributions To Mathematics
Pascal continued to influence mathematics throughout his life. His Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle. The triangle can also be represented:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | |
| 2 | 1 | 3 | 6 | 10 | 15 | ||
| 3 | 1 | 4 | 10 | 20 | |||
| 4 | 1 | 5 | 15 | ||||
| 5 | 1 | 6 | |||||
| 6 | 1 |
He defines the numbers in the triangle by recursion: Call the number in the (m+1)st row and (n+1)st column tmn. Then tmn = tm-1,n + tm,n-1, for m = 0, 1, 2... and n = 0, 1, 2... The boundary conditions are tm, −1 = 0, t−1, n for m = 1, 2, 3... and n = 1, 2, 3... The generator t00 = 1. Pascal concludes with the proof,
In 1654, prompted by a friend interested in gambling problems, he corresponded with Fermat on the subject, and from that collaboration was born the mathematical theory of probabilities. The friend was the Chevalier de Méré, and the specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. From this discussion, the notion of expected value was introduced. Pascal later (in the Pensées) used a probabilistic argument, Pascal's Wager, to justify belief in God and a virtuous life. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz' formulation of the infinitesimal calculus.
After a religious experience in 1654, Pascal mostly gave up work in mathematics.
Read more about this topic: Blaise Pascal
Famous quotes containing the words contributions to and/or mathematics:
“The vast material displacements the machine has made in our physical environment are perhaps in the long run less important than its spiritual contributions to our culture.”
—Lewis Mumford (1895–1990)
“Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don’t happen to have all the data. In mathematics we have all the data ... and yet we don’t understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.”
—Simone Weil (1909–1943)