stoutfellow (![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png)
stoutfellow) wrote2007-04-01 07:21 pm
stoutfellow) wrote2007-04-01 07:21 pm
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Ramble, Part 19: Going Down...
Like Blaise Pascal, Pierre de Fermat is mostly remembered for one thing and one thing only; and as with Pascal, this is less than fair. I'll have quite a bit to say about Fermat a bit later, but for the moment there's just one item I'd like to discuss.
In an earlier post, I gave the traditional proof that the square root of 2 is irrational. Here's another proof, using some rather different ideas. Suppose that a and b are two positive integers, and that a/b is the square root of 2. Then it's obvious that 0 < b < a < 2b. Set a1=2b-a, and b1=a-b. Then (taking into account that a2=2b2), we can compute that a12=(2b-a)2=4b2-4ab+a2=6b2-4ab, while b12=(a-b)2=a2-2ab+b2=3b2-2ab; that is, a12=2b12. Thus, a1/b1 is also the square root of 2. But a1 and b1 are smaller than a and b. Now we can do the same thing to a1 and b1, getting a2 and b2, which are smaller yet, and a2/b2 is still the square root of 2. We can repeat this ad infinitum, getting smaller and smaller positive integers whose ratio is the square root of 2. But this - this "infinite descent", as Fermat called it - is impossible; you cannot create an unending decreasing sequence of positive integers! Therefore, the initial assumption, that there was even one such pair of positive integers, must be false.
I'm not sure that Fermat used his "method of infinite descent" to prove that particular fact, but he did use it for a number of impossibility proofs, including, most likely, a proof that the equation a3+b3=c3 has no solution with a,b,c positive integers.
The reason this particular technique interests me is that it is still in use, in a slightly modified form, as the Well Ordering Principle: any nonempty collection of positive integers has a least member. Most budding mathematicians learn, in a discrete math course or a course on mathematical reasoning, that, though they are quite different in form, the Well-Ordering Principle is logically equivalent to the Principle of Mathematical Induction - which was the topic of the last Ramble, as having been first enunciated by Pascal. That the two principles were developed nearly simultaneously, in the same country, by a pair of mathematicians who had collaborated on other work is... rather amusing, I think.
I have more to say about old Pierre, but right now I think it's time to put Descartes before de Fermat...
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In an earlier post, I gave the traditional proof that the square root of 2 is irrational. Here's another proof, using some rather different ideas. Suppose that a and b are two positive integers, and that a/b is the square root of 2. Then it's obvious that 0 < b < a < 2b. Set a1=2b-a, and b1=a-b. Then (taking into account that a2=2b2), we can compute that a12=(2b-a)2=4b2-4ab+a2=6b2-4ab, while b12=(a-b)2=a2-2ab+b2=3b2-2ab; that is, a12=2b12. Thus, a1/b1 is also the square root of 2. But a1 and b1 are smaller than a and b. Now we can do the same thing to a1 and b1, getting a2 and b2, which are smaller yet, and a2/b2 is still the square root of 2. We can repeat this ad infinitum, getting smaller and smaller positive integers whose ratio is the square root of 2. But this - this "infinite descent", as Fermat called it - is impossible; you cannot create an unending decreasing sequence of positive integers! Therefore, the initial assumption, that there was even one such pair of positive integers, must be false.
I'm not sure that Fermat used his "method of infinite descent" to prove that particular fact, but he did use it for a number of impossibility proofs, including, most likely, a proof that the equation a3+b3=c3 has no solution with a,b,c positive integers.
The reason this particular technique interests me is that it is still in use, in a slightly modified form, as the Well Ordering Principle: any nonempty collection of positive integers has a least member. Most budding mathematicians learn, in a discrete math course or a course on mathematical reasoning, that, though they are quite different in form, the Well-Ordering Principle is logically equivalent to the Principle of Mathematical Induction - which was the topic of the last Ramble, as having been first enunciated by Pascal. That the two principles were developed nearly simultaneously, in the same country, by a pair of mathematicians who had collaborated on other work is... rather amusing, I think.
I have more to say about old Pierre, but right now I think it's time to put Descartes before de Fermat...
Previous Next
Ramble Contents
no subject
Planning on going back and reading through all your rambles at some point -- but for now, interesting!
And silly pun! ;-p