The story is apocryphal. It's likely that Hippasus was not in fact drowned for his discovery, and the Greek response was simply to exclude irrationals from the class of 'number', while admitting irrational magnitudes. The story does attest to the offence that irrationals caused to the Greek sensibility though, which had a preference for holism and perfect divisibility. Pappus of Alexandria, a late Greek mathematician, is on record:

"The school of Pythagoras was so affected by its reverence for these things [rationals] that a saying become current in it, namely, that he who first discovered the knowledge of incommensurables or irrationals and spread it among the common herd perished by drowning. This is most probably a parable, by which they sought to express their convictions: first, it is better to conceal every incommensurable, or irrational, or inconceivable in the world and second, the soul that, by by error of heedlesssness, discovers or reveals anything of this nature that is in it or in this world, wanders thereafter to and fro in the sea of non-identity, immersed in the stream of becoming and decay, where there is no standard of measurement".

rrational numbers are fascinating and there is a legendary story behind them. Back in time, over two thousand five hundred years ago, there was this incredibly brilliant mathematician named Pythagoras. — Michael Lee

Whatever you do, don't tell @Metaphysician Undercover. This information upsets him terribly.

It's just a legendary story and we really don't know what exactly happened because their school was so secretive in nature. I hope @Metaphysician Undercover will forgive me. :wink:

The story is apocryphal.

Of course it is @StreetlightX but it's entertaining.

Much of everything we know about mathematics was developed by that school. — Michael Lee

A bit of overstatement. But entertaining post. :cool:

Much of everything we know about mathematics was developed by that school. — Michael Lee

There is this widespread view of a long mathematical winter between Greek antiquity and the 12th century AD, i.e. a millennium-long standstill:

By the middle of the 1st Century BCE, the Roman had tightened their grip on the old Greek and Hellenistic empires, and the mathematical revolution of the Greeks ground to halt. Despite all their advances in other respects, no mathematical innovations occurred under the Roman Empire and Republic, and there were no mathematicians of note. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it (after Christianity became the official religion of the Roman empire) even less so. — storyofmathematics.com on Roman mathematics

Is that view a bit of an exaggeration or is it real?

There is this widespread view of a long mathematical winter between Greek antiquity and the 12th century AD, i.e. a millennium-long standstill

It is not just a view, but a fact, that the study of mathematics in the so called "western" society almost grounded to a halt. There were further advancement in the Arabic and Indian cultures - most famously the use of the decimal system.
The next major advancement came during the Age of Enlightenment with the development of the calculus by Newton and Leibniz - but there were notable mathematicians even during those intervening years: Fibonacci, Fermat, etc...
An anecdote that I really like to reflect upon - about the potential of all that was lost during those dark ages - is the finding of Archimedes Palimpsest. The finding of lost mathematical methods recycled into religious texts, is exemplary - in that regard that those centuries were not just a time in which mathematical development was repressed, but also a time when the raining culture erased most of the culture that came before it.

Whatever you do, don't tell Metaphysician Undercover. This information upsets him terribly. — fishfry

Jesus Christ! You would ruin my fucking day by bringing my attention to a number which is irrational, wouldn't ya? Don't try to lead me to the river, because you know I won't follow you.



PI is another very good example of an irrational ratio. What I think, is that the existence of these irrational ratios indicates that our method of representing spatial existence, with distinct dimensions, is fundamentally flawed.

Consider that a straight line, representing one dimension, is fundamentally incompatible with a curved line, representing two dimensions. So for example, take a line segment which exists between two supposed points. That line is straight, one dimensional. Now let's bring in a second dimension, and produce a curved line segment in relation to the straight line segment. We need to produce some sort of divergence from the straight line, and we might represent this as an angle of x degrees. The problem, is that no matter how hard we try, the angle produces two straight lines related to each other by an angle, it does not produce a curved line. There is no real relationship between a curved line and a straight line, they are incommensurable.

So the circle, arc, or curved line, is a two-dimensional representation which is fundamentally incompatible with the two-dimensional representation which is straight lines at angles to each other. Another example of this problem is the point of tangent. But the irrational nature of the square, which Pythagoras had so much trouble with, indicates that straight lines relative to points, with angles, is a less accurate representation of two-dimensional space than is the non-straight line.

However, we actually use points, angles, and vectors, so I believe we need to validate such principles, which are not validated by the empirical evidence of actual spatial existence. The root of the problem is the non-dimensional point. It does not represent any real space. If we want to allow the point to have compatibility with spatial representation, we can give it "space", make it an infinitesimal point or something like that. But this creates all sorts of problems. What is the infinitesimal point's size, it's shape, and how is it related to other infinitesimal points (straight line relations wouldn't make sense)? So this approach is not practical, and it really does not resolve the problem, which is how supposed points are related to each other. We still have to produce the method for relating these infinitesimal points to one another, which is the real problem.

Therefore we ought to return to the non-dimensional points, and find the empirical evidence required to validate them. The empirical evidence is that it works. Non-dimensional points are highly useful. The problem is that we do not understand the type of existence that they have, so we do not understand how they are related to each other. Making the point a dimensional infinitesimal creates problems, as does drawing straight lines at angles create problems. So it's back to the drawing board. Until we understand the nature of the non-dimensional point, we cannot develop the principles to model the medium which exists between these points.

The next major advancement came during the Age of Enlightenment with the development of the calculus by Newton and Leibniz - but there were notable mathematicians even during those intervening years: Fibonacci, Fermat, etc... — Marlon

You seem to have left out what is arguably the most important factor, the Copernican Revolution.

By the middle of the 1st Century BCE, the Roman had tightened their grip on the old Greek and Hellenistic empires, and the mathematical revolution of the Greeks ground to halt — storyofmathematics.com on Roman mathematics

That's an odd claim. I don't know much about the history of mathematics, but even I have heard of Alexandrian mathematicians who thrived well into the ADs, such as Diophantus (he of Diophantine equations) and Hypatia (who is pictured on our site's favicon).

Much of everything we know about mathematics was developed by that school. — Michael Lee

A bit of overstatement. But entertaining post — jgill

By overstatement I meant that of all the mathematical knowledge existing today, I speculate that over 99% was created or derived since 1700. It's true, the ancients got the ball rolling. If you can find figures that imply the Pythagoreans gave us a sizable percentage I'm interested.

The Romans had no use for pure mathematics, — storyofmathematics.com on Roman mathematics

I wonder if that was related to the use of Roman numbers as distinct from early decimal numerical systems derived from the East? (also mentioned above.)

Also there was resistance to the adoption of the concept of zero which I imagine would stymie any mathematical progress.

the soul that, by by error of heedlesssness, discovers or reveals anything of this nature that is in it or in this world, wanders thereafter to and fro in the sea of non-identity, immersed in the stream of becoming and decay, where there is no standard of measurement". — StreetlightX

Hey they’re talking about Sean Carroll :grin:

I corrected a logical error in the proof and posted it above.

@alcontali Do you have any idea why numbers like sqrt(2), pi, e, were given the name irrational? Is it because they can't be expressed as a ratio with integers as numerators and denominators?

Could it be that such numbers were considered as defying logic i.e. they are irrational in a logical sense?

Hi, rational is derived from the Latin word 'reckoning' or in accordance with reason. Because of the secretive nature of the Pythagorean School, we don't know what they called it. The sqrt(2) and many others are irrational in the sense they cannot be expressed as a ratio of integers. Since, according to the Pythagoreans, all numbers must be expressible that way they falsify the claim the Pythagoreans held dearly in their hearts. Similarly, Galileo's and Copernicus' work was declared irrational by the Catholic Church because it means Aristotle and the book of Joshua are wrong, and the Earth is not the centre of the Universe and humanity isn't as significant as we thought we were.

Thanks. :up: Do you have any idea how we ever calculate irrational numbers?

I mean, it's a given that they have a decimal expansion e.g
pi = 3.14159...I suspect this involves some kind of division e.g . pi = circumference/diameter and that's a fraction isn't it?

I mean, it's a given that they have a decimal expansion e.g
pi = 3.14159...I suspect this involves some kind of division e.g . pi = circumference/diameter and that's a fraction isn't it? — TheMadFool

$\pi$ is no longer defined geometrically. Here is the modern analytic definition.

First you establish the theory of convergent power series in one complex variable, analogously to how it's done in freshman calculus for a single real variable.

Then you define the function $\displaystyle \exp(z) = \sum_{n=0}^\infty \frac{z^n}{n!}$, which you can show converges for all complex numbers $z$.

You then define the functions

$\cos(z) = \frac{\exp(i z) + \exp(-i z)}{2}$

and

$\sin(z) = \frac{\exp(i z) - \exp(-i z)}{2 i}$

It's not hard to show that $exp$, $cos$, and $sin$ are the usual exponential, cosine, and sine functions of pre-calculus, extended to the complex domain. However no triangles or circles or geometric notions are involved at all. These definitions are purely analytic, meaning that they pertain only to the theory of complex analytic functions.

Then what is $\pi$? It's the smallest positive zero of the sine function. A glance at the unit circle shows that this property uniquely characterizes the number $\pi$ as we usually understand it.

This is how an analyst thinks about the number $\pi$. I mention this to give an example of how mathematicians go about "defining an irrational number" using the machinery of modern math.

https://en.wikipedia.org/wiki/Exponential_function#Complex_plane

In terms of calculating the digits of $\pi$, you could in fact use the power series for $exp$, the definition of $sin$, and a root-approximating algorithm to compute as many digits as you like. Of course I imagine calculator manufacturers use different algorithms but either way you can see that no triangles or circles are ever involved. It's all about convergent infinite power series.

A famous but inefficient algorithm for the digits of $\pi$ is based on the Leibniz formula

$\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \sum_{n = 0}^\infty (-1)^n \frac{1}{2 n + 1}$

It's fun to write a little program and watch it converge.

How did Archimedes calculate pi? I thought he used the method of exhaustion - increasing the number of sides of a polygon and doing the necessary division.

How did Archimedes calculate pi? — TheMadFool

I found this but I don't know anything about it beyond Googling around.

I found this but I don't know anything about it beyond Googling around. — fishfry

:up:

How did Archimedes calculate pi? I thought he used the method of exhaustion - increasing the number of sides of a polygon and doing the necessary division. — TheMadFool

Interestingly, that same method of inscribing or circumscribing polygons was used to prove some results in my integral calculus class. So there is a little truth to the hyperbole in the OP :)

Interestingly, that same method of inscribing or circumscribing polygons was used to prove some results in my integral calculus class. So there is a little truth to the hyperbole in the OP :) — SophistiCat

:up:

PI is indeed the ratio of the circumference divided by its diameter. But if you set the diameter equal to 1, and set the circumference to 3, you will underestimate its actual value of 3.14159... and if you set the circumference to 4, you will overestimate its actual value. To date, we have calculated it to well over 16 trillion digits and tried using many computer algorithms to try and find a pattern and nobody has been successful. I sometimes wonder why mathematicians bother in this silly endeavour.

PI is indeed the ratio of the circumference divided by its diameter. But if you set the diameter equal to 1, and set the circumference to 3, you will underestimate its actual value of 3.14159... and if you set the circumference to 4, you will overestimate its actual value. To date, we have calculated it to well over 16 trillion digits and tried using many computer algorithms to try and find a pattern and nobody has been successful. I sometimes wonder why mathematicians bother in this silly endeavour. — Michael Lee

:up:

You have a point but what about other numbers like 22 and 7. In high school, ages ago, we were taught to use 22/7 for pi. Archimedes too set a lower and an upper limit to pi using rational numbers.

You have a point but what about other numbers like 22 and 7. In high school, ages ago, we were taught to use 22/7 for pi. — TheMadFool

The Indiana legislature once attempted to define pi as 3.2

The Indiana legislature once attempted to define pi as 3.2 — jgill

Reminds me of pragmatism. Would it have worked? I mean if we did define pi = 3.2 and used it in, say, engineering calculations and other scientific fields would our results be accurate? For instance, just for the circle, using pi = 3.2, our calculations of circumference, diameter, radius and area would be off the mark. However, if the "error" is systematic i.e. fixed in value, I don't see how it could cause problems; after all Egyptian engineering didn't suffer from using pi = 3.1605.

I don't see how it could cause problems; after all the Egyptian engineering didn't suffer from using pi = 3.1605. — TheMadFool

I think GM has employed some of those guys from the pharaoh's think tank to design their 10 speed transmission. :sad: