Waste less time on Facebook — follow Brilliant.
×

Pythogoreans vs Infinite descent

img

img

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

Note by Eddie The Head
3 years, 6 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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

Sort by:

Top Newest

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 - 3 years, 6 months ago

Log in to reply

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 - 3 years, 6 months ago

Log in to reply

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

Niladri Dan - 3 years, 6 months ago

Log in to reply

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...

Eddie The Head - 3 years, 6 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...