Pythogoreans vs Infinite descent

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The regular pentagram was the badge of the Pythogoreans.But the Pythagoreans first thought that all the rations were rational,i.e, $x = \frac{a}{b}$ . where a nd b are rational numbers.

Clearly this fraction must follow the equation for the golden ratio(It's a property of the pentagram),and hence we get $a^{2} = ab+b^{2}$

If $a$ and $b$ are of different parity clearly this leads to a contradiction.Also the assumption that $a$ and $b$ are odd also leads to a contradiction.

Hence both $a$ and $b$ must be even,that is $a = 2a_1$ and $b = 2b_1$ ,where $a_1 < a$ and $b_1 < b$ .

Substituting this into the equation and cancelling we get $a_1^{2} = a_1b_1 + b_1^{2}$ .

Now the same logic may be applied to this equation also which gives $a_1 = 2a_2$ and $b_1 = 2b_2$ where $a_2 <a_1$ and $b_2 < b_1$ .

So we obtain an infinitely decreasing sequence of positive integers,ie $a>a_1>a_2>a_3>.....$ and $b>b_1>b_2>b_3>...........$ .

Clearly no such infinitely decreasing sequence exists for positive integers.

For more details on this approach you may choose to read my note on Fermat's Method of Infinite Descent

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Nice note! But in my opinion, the note jumps out on you, does its job and then runs away.

Additional background, like what the golden ratio is, what a pentagram is and how these two are related would have been helpful.

This note proves that golden ratio can not be expressed in the form $\frac{a}{b}$ where $a$ and $b$ are positive integers. But it does not explicitly mentions this statement anywhere.

Despite all this, this is a nice demonstration of proofs that use the method infinite descent.

Mursalin Habib - 7 years ago

Thanks for the constructive review.Actually I didn't include the golden ration in this note because many people have posted about it before. Here is my note on Stochastic Programming.It will be extremely helpful for me if you write a small review of that note too.....I put a lot of effort into this one but it didn't get attention.Thanks in advance.

Eddie The Head - 7 years ago

can we say that infinite descent is a strong form of pentagram?

Niladri Dan - 7 years ago

The symbol is the pentagram and infinite descent is the proof technique....clck on the link if you're interested to learn more about it...

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$