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TOK: Beauty, Elegance, and Intuition
Transcript of TOK: Beauty, Elegance, and Intuition
in Mathematics Many Proofs There may be different proofs of a theorem.
Mathematicians generally seek proofs that are clear, economical and elegant. Example 1 There are 1024 people in a knock-out tennis tournament. What is the total number of games that must be played before a champion can be declared? ANOTHER EXAMPLE!!! If you tie a string with a constant one inch distance from the circumference of a perfect tennis ball sized sphere, you will find that you use 2π inches more than you would if it were tightly wound.
If you repeated this with the Earth, how much extra string would you need to use? Example 2 What is the sum of the integers from 1-100? In other words...
Creative imagination and intuition play a key role in maths.
Clear and simple proofs are BEAUTIFUL. Complicated Solution: Another way to look at it would be to focus on the losers rather than the winners. In this sense, the former method is more beautiful (and elegant) since if you consider only the final winner, you only need to remember that every other player will play one match and lose. Thus, the total number of matches is 1024-1 (Winner)=1023 games
For the latter, you would need to add the number of losers each round, which is much more complicated 1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
... and so on until you get the answer Simple, Elegant Solution: 1 2 3 ... 48 49 50
100 99 98 ... 55 54 53
101 101 101 101 101 101 Since each pair of numbers sums up to 101, and we know that there are 50 pairs...
50 x 101 = 5,050 Generalized, the sum of n integers is 1/2 n(n+1) Simple, Elegant Solution Answer: 2π inches. Intuitions People who are not mathematically inclined may have been surprised.
However, a mathematician with an educated intuition might not be surprised since the formula for circumference is 2πr, so adding 1 to the radius ends up as 2π(r+1) and thus adding 1 increases the circumference by a factor of 2π As a great man once said... "Math is beautiful"
- Sabino Padilla Complicated Solution The matches as held in two's, so in the first round, there are 512 (which is 1024/2) games. Of these games, there will be 512 winners, the next 256 (512/2) and so on until there is one remaining winner.