I show that for the same reason that the golden ratio, $\phi=1.6180334..$, can be considered the most irrational number, that $1+\sqrt{2}$ can be considered the 2nd most irrational number, and indeed why $(9+\sqrt{221})/10$ can be considered the 3rd most irrational number.

This blog post was featured on the front page of Hacker News a while back. See here for the awesome discussion.

Let us imagine a game between two kids, Emily and Sam – both strong and determined in their own way who spend their entire lives trying to outwit each other, instead of doing their homework. (A real life Generative Adversial Network…)

Emily, proudly reminds us that she simultaneously bears the same first name as Emily Davison, the most famous of British suffragettes; Emily Balch, Nobel Peace Prize laureate; Emilie du Chatelet, who wrote the first French translation and commentary of Isaac Newton’s “*Principia*“; Emily Roebling, Chief Engineer of New York’s iconic Brooklyn Bridge; Emily Bronte author of *Wuthering Heights*; Emily Wilson, the first female editor of ‘*New Scientist*‘ publication; and also Emmy Noether, who revolutionized the field of theoretical physics.

On the other side we have Sam (and all his minion friends, who are aptly called Sam-002, Sam-003, Sam-004 ) who is part human / part robot and plays Minecraft or watches Youtube, 24/7.

They agree to play a game where Emily thinks of a number, and then Sam (with the possible help of his minions) has 60 seconds to find any fractions that are equal to Emily’s number.

And so the game begins…

Emily says “**8.5″**.

Sam & friends quickly reply with “**85/10″**,… “**34/4″**,… “**17/2″**,… “**425/50″**,…

They soon realize that all these answers are equally valid because they are all equivalent fractions. Being competitive they want to pick a single winner, so they all agree that the best answer is the one with the lowest denominator. And so, **17/2** is deemed the best answer.

This time, Emily tries to make it harder by picking ‘**0.123456**‘. After only a slight pause, Sam slyly says “**123456/1000000**“.

Emily’s annoyed with this answer. She knows that although the best answer would be the irreducible fraction **1929/15625,** Sam’s answer is still valid answer, and furthermore he will always be able to instantly answer like this if she picks any number with a terminating decimal expansion.

So this time, Emily picks “$\pmb{\pi}$”.

This time, it’s Sam turn to get annoyed and claims that this is cheating. They both know that there isn’t an answer because $\pi$ is an irrational number and so there isn’t any fraction that is exactly equal it.

Therefore they eventually agree, that the best answer will be whichever fraction is closest to her number.

With this new rule Sam’s minions suggest the following: ‘**3**‘,… ‘**31/10**‘,… ‘**22/7**‘,… ‘**16/5**‘,… ‘**3927/1250**‘.

Sam looks at all of them condescendingly and immediately calls out “**3141592/1000000**“

Again, Emily gets annoyed. She realizes that Sam has shown that for any irrational number, you can (easily) pick a fraction that is arbitrarily close to it. That is, the rationals are dense everywhere.

They both know that fractions with small denominators are more in the spirit of the game, but Sam says that because his answer was technically valid, it’s her responsibility (and not his) to think of a new rule to improve things.

Emily consider ways of giving each answer a score. Initially, she thought that for each fraction, the score could be the (absolute) difference between her number and the proposed fraction, and then multiplied by the denominator. (The lower the better).

However, after talking to some of her tech friends, she decided to make it even stricter to very severely penalize fractions with large denominators. Such as scoring system would definitely force Sam to have to work hard to find good fractions. And so finally she proposed that for each fraction, the score should be the absolute difference **multiplied by the denominator squared**. Lowest score wins.

So for the previous value of $\pmb{\pi}$, the Minions calculate their scores:

$$ 3 \rightarrow 0.141, \quad 31/10 \rightarrow 4.1, \quad 22/7 \rightarrow 0.06, \quad 16/5 \rightarrow 1.5, \quad 3927/1250 \rightarrow 36$$

Whilst, Sam comes last with a truly pathetic and embarrassing score of $3141592/1000000 \rightarrow 592$.

Sam now very annoyed, becomes very determined to find a simple method that will get a low score.

Later that week, whilst sitting at his computer, he had a flash of inspiration.

He first noted that $\pi = 3.14159…$.

So ignoring the fractional part of this implied that $\pi \simeq 3$.

For a better approximation he noted that the reciprocal of the previous fractional part equals $1/0.14159265 = 7.0625133$.

So, ignoring the fractional part of this last number, implies that $\pi \simeq 3 + \cfrac{1}{7} = \frac{22}{7}$.

For an even better approximation, he noted that the reciprocal of the previous fractional part is $1/ 0.0625133= 15.996594…$

Again, ignoring the fractional part of this implies that

$$ \pi \simeq 3 + \cfrac{1}{7+\cfrac{1}{15}}=\frac{333}{106} $$

For an even better approximation he noted that the reciprocal of the fractional part of this last number is $1/ 0.996594 = 1.00341723$.

Again, ignoring the fractional part of this implies that

$$ \pi \simeq 3 + \cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1}}}=\frac{355}{113} $$

Further, he realised that since the numerators of each of the fractions is 1, and there is always an addition between the various number the only important numbers are the ones going down the diagonal. That is, 3, 7,15 and 1. So he devises a short-hand way of writing these continued fractions.

$$ \pi \simeq [3; 7,15,1] =\frac{355}{113} $$

He calls each of these successive rational approximations, ‘convergents‘ because he knows that in the limit they converge to $\pi$. The score for each of these convergents is:

$$ \frac{3}{1} \rightarrow 0.141, \quad \frac{22}{7} \rightarrow 0.062, \quad \frac{333}{106} \rightarrow 0.935, \quad \frac{355}{113} \rightarrow 0.003, \quad \frac{103993}{33102} \rightarrow 0.633$$

Sam is super happy again, that he has found a systematic way of consistently low scores. He then writes a computer program to quickly calculate the continued fraction for any given number. For $\pi$ it was:

$$ \pi \simeq [3; 7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,1,4,2,6,6,99,1,,…] $$

Although he didn’t see any ongoing pattern, he was satisfied that (in principle) he could calculate this to as many digits as he desired.

His Minions can then calculate the convergents, each one based on this continued fraction to one level deeper than the one before. When they look at the scores for the first 100 convergents (see figure 1) they notice that all the scores are less than 1. This is great news, and it turns out be a consequence of

Amazing article. How easy is it to define the first 100 most irrational numbers?

Sorry about the delay (been on holidays!). The $n$-th most irrational number is equal to the $n$-th Lagrange number, $L_n$, where $L_n = \sqrt{9-\frac{4}{m_n^2}}$, where $m_n$ is the $n$-th Markov number. The first 1000 Markov numbers are listed in OEIS A002559.

The parent page of OEIS A002559 also gives Mathematica code for calculating arbitrary terms of the Markov-Lagrange spectrum.

Hope this helps!

I’m unable to produce your results on the first 3 numbers with that definition, can you show me what I’m doing wrong? I posted my struggles here:

https://math.stackexchange.com/questions/3498602/trying-to-calculate-the-top-n-most-irrational-numbers-as-continued-fractions

Aside from that, I’m interested in this question because I wonder… Has anyone done any work on understanding patterns in the continued fraction representations of the “most irrational” series? It seems like that’s potentially interesting. Or does it seem to be trivial?

[1; 1…]

[2; 2…]

[2; 2 1 1 2…]

At first glance, it almost looks like the start of a binary – is it that trivial once you know the pattern? Or is it more interesting? I guess I’m just wondering if there is a pattern in this notation alone that can determine the next most irrational number. From what I’ve gathered from your article there is not one known. Which makes me wonder if the third being [2; 2 1 1 2…] was not intuitive because no one has bothered to look at the pattern, or if there isn’t one?

Again, thanks for the article!

Hi Jordan,

Thank you for your keen interest in this topic, and you pose a very interesting question!

I generally think that many questions relating to this field have not been answered simply because not many people have studied it!

Regarding your specific technical question, I have to admit I glossed over some of the details for those trying to replicate and extend this pattern, as I was trying to balance teaching a math lesson with writing a smooth narrative!

In more detail:

1. The basis for calculating each new irrational class requires taking each of the markoff numbers below

$m_n = 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985,…$

2. You then substitute these into the Lagrange-Markov formula.

$$L_n = \sqrt{9-\frac{4}{m_n^2}}$$

3. You then need to find a quadratic equation with integer coefficients whose discriminant is $L_n$.

4. Then the solution to this quadratic equation give the irrational number

For your convenience, page 7 of the following paper https://arxiv.org/pdf/1702.05061.pdf does this for you and lists the first five most irrational classes:

$$ \frac{1}{2}(1+\sqrt{5}), \quad \sqrt{2}, \quad \frac{1}{10}(9+\sqrt{221}),\quad \frac{1}{26}(23+\sqrt{1517}, \quad \frac{1}{58}(5+\sqrt{7565}),…$$

Finally, the continued fractions for these are:

$$ [1, \overline{1}], \quad [1,\overline{2}], \quad [2,\overline{2,1,1,2}], \quad [2, \overline{2, 1, 1, 1, 1, 2}], \quad [1, \overline{1, 1, 2, 2, 2, 2},… $$

where the overline bar indicates that this subsequence is infinitely repeating.

Finally, the equivalence statement implies that for any of these irrational numbers, you can find an equivalently irrational number simply by changing any of the initial terms of the continued fraction!

This still might not be enough for you to test your theory, but hopefully it gives you enough of a boost to continue your explorations!

Martin

I don’t know what the point was of that opening other than to prostrate yourself before the altar of gender politics and declare loudly “women are smart!” in a way guaranteed to cast doubt on you actually believing it.

Hey, did you know women can be funny too? You guys, women are funny. They have a sense of humor. Isn’t that great? Why are you looking at me like that? I really think women can be funny. I swear, I love funny women. With their humor.

My daughter is called Emily, and my son is called Sam. I wrote it for her, and she really appreciated it.

lmaooooo

This is brilliant brilliant brilliant. Are you familiar with the role of irrational numbers and brainwaves? Part of the harmonic structure of the brain

https://www.ncbi.nlm.nih.gov/m/pubmed/20350536/

I didn’t read, but can someone tell me spoiler of what is beyond the golden ratio? I need to know! thanks.

beyond the golden ratio is the platinum ratio. No need to read the article. Who needs details and knowledge when a soundbite suffices right?

Why is the score “the absolute difference multiplied by the denominator squared”? I’ve noticed that the reciprocal of convergents’ absolute difference is around the value of the square of the denominator, but I don’t know why. Is it connected to the expected value for a random walk, which seems to have a similar formula?

These are the same!

If e is the absolute difference, and denominator is q,

Then score, S = eq^2, which means 1/e = q^2/S.

And for continued fraction convergence of quadratic irrationals, S is of order 1 (eg 1/sqrt[5], etc…)

Which means that 1/e ~ q^2.

My takeaway from the article is that ” the golden ratio is the most irrational number”, “the second most irrational number is 1 + √2”, and “the third most irrational number is (9 + √221)/10”. If you find the time, I do recommend reading the post as the journey to the conclusion is quite fun!

Also, I’m not terribly great at higher maths so if there was a deeper takeaway from the article, please feel free to correct me.

That is a perfect summary and it was the major conclusions that I wanted to convey!

The rest of the content was merely an attempt to explain the reasoning behind this idea, and provide possible pathways for those who wanted to pursue related branches of mathematics.

😉

Some genuinely interesting information, well written and broadly speaking user genial. How much have you written like this?