If you have played with the square root function on an electronic calculator you may have noticed something odd. It looks like if you take the square root of ANY number that is not actually a perfect square, then you get an irrational number, with the decimal places spilling over the edge of your calculator’s display. Is this in fact true? Is the square root of any number that is not itself a perfect square an irrational? Let’s find out.
We start off in the conventional way by assuming that the square root of 2 IS rational and may be given as m/n where m & n are both integer.
1) √2 = m/n
2) m2 = 2 n2
That much we have seen before, but now comes the really clever bit. Factor m & n into their unique prime factors:
3) m1 m1 m2 m2 m3 m3 … mr mr = 2 n1 n1 n2 n2 n3 n3 … nr nr
Now here’s the problem. The prime number 2 will appear an even number of times on the left hand side of the equation (if it appears at all) and an odd number of times on the right. Since the decomposition into primes is unique the prime number 2 cannot appear an even number of times on one side of the equality and an odd number of times on the other. So the square root of 2 cannot be written in the form m/n with m & n integer. The square root of 2 is an irrational number.
A) Although the above was shown to be true for root 2, the same argument of course would hold true for any prime number. The square root of any prime number is an irrational.
B) Replace 2 with any integer that is not a perfect square. Now that integer may be decomposed into its prime factors, and, if the number is not a perfect square, then once again we will have an odd number of primes on one side of the equality and an even number of primes on the other. The square root of any integer that is not a perfect square is also an irrational.
C) Finally, replace 2 with any perfect square. With even numbers of primes on both sides the solution is trivial.
Admittedly Cantor was in his writings not very explicit about what he did take the set theoretic universe as a whole to be. One problem is that it is not in every instance clear whether he has a theological or a mathematical conception of absolute infinity in mind. Indeed, he argues that it is the task not of mathematics but of ‘speculative theology’ to investigate what can be humanly known about the absolutely infinite. The following passage, for example, leans heavily to the theological side: I have never assumed a “Genus Supremum” of the actual infinite. Quite on the contrary I have proved, that there can be no such “Genus Supremum” of the actual infinite. What lies beyond all that is finite and transfinite is not a “Genus”; it is the unique, completely individual unity, in which everything is, which comprises everything, the ‘Absolute’, for human intelligence unfathomable, also that not subject to mathematics, unmeasurable, the “ens simplicissimum”, the “Actus purissimus”. In this quotation, Cantor speaks of the necessity of ‘knowing’ the domain of variation through a ‘definition’. Surely Cantor is merely sloppy here, and we should discount the epistemological overtones. Another slip can be detected in Cantor’s use of the word ‘set’ in this quotation. Cantor means the argument to be applicable not just just to sets but also to absolute infinities. which is by many called “God”. All this is related to the fact that in an Augustinian vein, Cantor takes all the sets to exist as ideas in the mind of God.
And that last sentence rings a bell too 🙂 For Ramanujan said “An equation for me has no meaning, unless it represents a thought of God”.
We seem to be homing in on something here!
If God is the Absolute Infinite, the Ein Sof, then I think we are entering very interesting territory. Why? Because I don’t believe there is ANYTHING in the physical Universe that is infinite. I don’t believe there are an infinite number of photons, particles, quarks or neutrinos. Our Universe it appears is finite in size and contains a finite amount of stuff. So everything we can know (or ever know) in the real physical world appears to be made out of finite quantities – we won’t find God in the physical Universe.
Where do we find Infinities? The only place I know of where we find Infinity is in mathematics. Now that’s strange. We use mathematics to explain the real world to high accuracy, and we even carry out integrations over infinity to give answers that correspond to realities in the real world – and yet infinity does not seem to be part of the real world.
So am I saying God is Mathematics? No I am not. But can you see that Mathematics might give us a clue as to what God actually is? The Absolute Infinite was contemplated by Georg Cantor as an infinity that transcended the transfinite numbers. It should be noted that Cantor equated the Absolute Infinite with God! Cantor believed that the Absolute Infinite possessed mathematical properties including the reflection principle which states that every property of the Absolute Infinite is also held by some smaller object. It is sad to relate that Georg Cantor, along with several other famous mathematicians/physicists who dared to venture into the realm of the infinite encountered severe mental problems leading to death.
We are now coming to the end of this piece. How can a finite Mind contemplate and work with infinite quantities? I suppose the trite answer is that it cannot and it leads to madness, which in itself is extremely interesting, because as we all know “Whom the Gods would destroy, they first make mad”. But now consider the reason our finite minds can work with the Infinite is due to the reflection principle, so that every property of the Absolute Infinite is also held by some smaller object – ourselves!
Unfortunately it appears that it is physically impossible for there to be a quantum computational aspect to the brain.
For a while I thought that perhaps the “conscious” (i.e. classical computer) part of the brain could act as the “observer” and by an “entangled Quantum Zeno effect” it could make the decoherence time of a brain quantum computer sufficiently long to be physically possible. The Quantum Computer part of the brain would correspond to the “sub-conscious”. Sadly, a discussion with perhaps the greatest theoretical Quantum Physicist on the planet shows that this is not viable 🙁 It therefore appears that there is NOT a quantum computational aspect to the brain.
And just today the bloody obvious hit me like a hammer. If there WAS a quantum computational aspect to the brain – then why would we be stymied by the 2-slit experiment? And more importantly why would Feynman have been able to say that “nobody understands Quantum Mechanics”?
Looks like the brain is a massively parallel classical computer, still it gives very impressive results!!
Probably the most memorable line from the whole of the original series of Star Trek was the title of this post spoken by Captain Kirk. It still brings a shiver down my spine – it is SO perfect.
The poem is “Sea Fever” by John Masefield.
I must go down to the seas again, to the lonely sea and the sky,
And all I ask is a tall ship and a star to steer her by,
And the wheel’s kick and the wind’s song and the white sail’s shaking,
And a grey mist on the sea’s face, and a grey dawn breaking.
I must go down to the seas again, for the call of the running tide
Is a wild call and a clear call that may not be denied;
And all I ask is a windy day with the white clouds flying,
And the flung spray and the blown spume, and the sea-gulls crying.
I must go down to the seas again, to the vagrant gypsy life,
To the gull’s way and the whale’s way, where the wind’s like a whetted knife;
And all I ask is a merry yarn from a laughing fellow-rover,
And quiet sleep and a sweet dream when the long trick’s over.
In 1968 we were living at the top of the Sir John Soane’s museum in Lincoln’s Inn Fields, Holborn, London having just returned from 2 years in New Zealand. Believe me, the museum was the spookiest of places at the best of times.
However, I decided to take a hot bath one evening and settled in for a good soak. Something very strange happened that night. Something that I will never forget. Something that has stayed with me, clear as day for the past 46 years.
The water in the bath started to build up into waves, waves with a period of about a second. The amplitude of the waves increased very quickly over maybe 10 seconds or so, and they were so high at the walls of the bath that they actually collapsed over just like a wave in the sea. As you can imagine this was a bit unsettling for a 14 year old kid who just wanted a soak, and I sat up with a start. The waves stopped!! A bit shaken I laid down in the bath again, and no sooner had I sunken myself in my nice hot bath again – the waves started up! Big waves, breaking on the walls again – shit! Sat up again, and they stopped. This was weird. Even I could see this was weird. And at 14 with a highly active imagination and no Physics background – this was Supernatural and something spooky was going on in the Museum that night. I laid down in the bath again – same thing happened. This was all too much, got out and pulled the plug out – pretty shaken.
That evening stayed with me for quite a while. I think I told my parents who I believe told me I was talking rubbish, and that’s how it stood for maybe 20 years or so until I thought about it again – this time armed with a First Class Honours Degree in Physics, Maths and Astronomy from the University of Sussex. The Supernatural event became a natural event, but probably a very rare event. I must say that in the few baths I took since that day (preferring showers nowadays) I have never been able to repeat the observation – but this is what I think happened.
By pure luck (bad luck?) I had filled the bath to the point where it was in resonance with my heartbeat pumping the blood around my body. I had created a resonant cavity and the little “pushes” into the water caused by my heartbeat pumping the blood around were at resonance with the bath/water “cavity”. As for any resonant system, when pumping energy into the system at the resonant frequency there is a rapid build up in amplitude, whether it’s a voltage, or a wave amplitude. Very quickly the waves built up in the bath until they were splashing off the walls like a mini-storm at sea.
So that’s what I think happened with my increased Physics knowledge and my scientist hat on. But it could have been a Ghost of course 🙂 🙂
The Two Worlds post below was clearly alluding to C. P. Snow’s “The Two Cultures”. I have not read Snow’s book, but it is definitely next on the list. As an undergraduate at the University of Sussex I was required to take an “Arts/Science” course as part of my degree. True to Snow’s perception, this was taken as some sort of half-hearted attempt to raise the knuckles of scientists and engineers a little off the ground. It was however more than slightly revealing in that no “Science/Arts” course existed. I wonder why not? In fact, I didn’t wonder why not for a second, nor did any of my fellow science students. I just took a look at the WIKI entry for C. P. Snow to see if I could find anything regarding the two cultures that I could put into this post. Snow obviously preceded my own thoughts on the subject by a good few decades, and as his mastery of the language far surpasses my own, I’ll leave it to Snow to finish off this post:
- “A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is about the scientific equivalent of: ‘Have you read a work of Shakespeare‘s?’
- I now believe that if I had asked an even simpler question – such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, ‘Can you read?’ – not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their Neolithic ancestors would have had.”
People with strong Religious beliefs ignore the sciences at their peril because like it or not that’s the way the World operates. Scientists ignore the spiritual aspects of Homo Sapiens at their peril because like it or not that’s one of the main bases of the human condition. Until someone comes along that can effectively combine these two seemingly disparate disciplines we will always have a disconnect.
What’s really nice about the Curta is that by thinking about the maths problem you want to solve in a logical manner you can minimise the number of handle rotations to calculate the answer. So for example, finding the answer to 9×9, 99×99, 999×999, 9999×9999, 99999×99999, 999999×999999, or 9999999×9999999 all only take two handle rotations to get to the answer. But it gets even better than that. The Curta’s reign as calculator supremo of course came to an end with the advent of the electronic pocket calculator, but now look again at that last calculation in the set above. The answer the Curta (Type II) gave me to 9999999×9999999 was 99999980000001 which as you note is a 14-digit number. Now go and check on my (not inexpensive all-singing and dancing) electronic calculator and what happens? Can’t supply me with all the significant digits can it! Gives me 9.999998E+13 as an answer – not as accurate as the Curta! WOW! Actually have to go to the computer and run Mathematica to get the full run of significant digits – so there you go. O.K. so my electronic calculator can give me square roots, trig functions, and all the other fancy stuff at the press of a button – but my pocket difference engine can supply me with more significant digits in two handle rotations. Just how cool is that?