There I encountered Euler's formula, and was entranced. Now I know most of my readership won't be Pure Mathematicians, so I'll stick to the basics, and not provide a real complex explanation.
(I imagine that Mathematicians will see that pun and groan, but no matter.)
The Euler formula is eiπ + 1 = 0
It has been accurately described as The most remarkable formula in the world.
OK, now what the heck does that mean? The Picture comes from Geometry Step By Step:
Leonhard Euler (Swiss mathematician and physicist, 1707-1783) and his beautiful and extraordinary formula that links the 5 fundamental constants in Mathematics, namely, e, the base of the natural logarithms, i, the square root of -1, Pi, the ratio of the circumference of a circle to its diameter, 1 and 0, together!I think most people know about π. A few will know about e, the base of natural logarithms, but I'll assume most don't.
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...
Where 4! means 4 x 3 x 2 x 1 and 0! is defined as 1.
So e = 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ... and so on
Just as π is "3 and a bit", e is "about 2.7 plus a smidgin".
As for i, that's simple. It's the square root of -1. i x i = -1
Er... wait a sec... how can a negative number have a square root? Trust me on this, negative numbers all do, and these numbers are called imaginary numbers. When they first appeared in Mathematics, their Reality was doubted, they were thought of as something of a cheat, a "kludge" to make things work out. Then again, negative numbers were thought to be much the same, way back when. You can have 1 piece of chalk. You can even, at a stretch, have 0 pieces of chalk. But how can you have -1 pieces of chalk? Or even 1/2 a piece of chalk, as if you break a piece in two, you get two smaller pieces, not two "half" pieces. i is no more and no less respectable than those upstanding pillars of mathematical society, the fractions, and their slightly shadier cousins, the negatives. Those concepts are not my own, they're based in something Isaac Asimov wrote, something that I last read over 30 years ago and so doubt if I've plagiarised his words verbatim.
Imaginary numbers are all of the form X.i, where X can be a fraction, or a whole number. Typical imaginary numbers are 3i, 2.65i, -0.39i and so on.
When a number has a "real", genuine, non-imaginary bit, a respectable number like 3, or 37, or 19.45566, or 0, as well as the disreputable imaginary component, it's called a complex number. So a typical complex number might be 3.2+17i. Real + Imaginary = Complex.
When I first saw "eiπ + 1 = 0" I knew the Universe was trying to tell me something, if only I could figure out what!
And that brings me back to Reality. From Seed Magazine:
Quantum mechanics fundamentally concerns the way in which we observers connect to the universe we observe. The theory implies that when we measure particles and atoms, at least one of two long-held physical principles is untenable: Distant events do not affect one other, and properties we wish to observe exist before our measurements. One of these, locality or realism, must be fundamentally incorrect.Now cutting to the chase...
For more than 70 years, innumerable physicists have tried to disentangle the meaning of quantum mechanics through debate. Now Zeilinger and his collaborators have performed a series of experiments that, while neatly agreeing with the theory's predictions, are reinvigorating these historical dialogues. In Vienna experiments are testing whether quantum mechanics permits a fundamental physical reality. A new way of understanding an already powerful theory is beginning to take shape, one that could change the way we understand the world around us. Do we create what we observe through the act of our observations?
Most of us would agree that there exists a world outside our minds. At the classical level of our perceptions, this belief is almost certainly correct. ... The classical world is real, and not only in your head. Solipsism hasn't really been a viable philosophical doctrine for decades, if not centuries.
But none of us perceives the world as it exists fundamentally. We do not observe the tiniest bits of matter, nor the forces that move them, individually through our senses. We evolved to experience the world in bulk, our faculties registering the net effect of trillions upon trillions of particles or atoms moving in concert. We are crude measurers. So divorced are we from the activity beneath our experience that physicists became relatively assured of the existence of atoms only about a century ago.
Physicists attribute a fundamental reality to what they do not directly perceive. Particles and atoms have observable effects that are well described by theories like quantum mechanics. Single atoms have been "seen" in measurements and presumably exist whether or not we observe them individually. The properties that define particles—mass, spin, etc.—are also thought to exist before we measure them. In physics this is how reality is defined; particles and atoms have measurable properties that exist prior to measurement.
The data is tested against two theories: one that preserved realism but allowed strange effects from anywhere out there in the universe, and quantum mechanics.Yes, you read that right. The experiment doesn't require multi-billion dollar supercolliders, just some simple electronics. The apparatus can fit on a dining table.
During the 1980s and 1990s, the foundations of quantum mechanics slowly returned to vogue. The theory had been shown, with high certainty, to be true, though loopholes in experiments still left some small hope for disbelievers. However, even to believers, nagging questions remained: Was the problem with quantum mechanics locality, realism, or both? Could the two be tested?
As he spoke, Zeilinger reclined in a black chair, and I leaned forward on a red couch. "Quantum mechanics is very fundamental, probably even more fundamental than we appreciate," he said, "But to give up on realism altogether is certainly wrong. Going back to Einstein, to give up realism about the moon, that's ridiculous. But on the quantum level we do have to give up realism."
With eerie precision, the results of Gröblacher's weekend experiments had followed the curve predicted by quantum mechanics. The data defied the predictions of Leggett's model by three orders of magnitude. Though they could never observe it, the polarizations truly did not exist before being measured. For so fundamental a result, Zeilinger and his group needed to test quantum mechanics again. In a room atop the IQOQI building, another PhD student, Alessandro Fedrizzi, recreated the experiment using a laser found in a Blu-ray disk player.
In mid-2007 Fedrizzi found that the new realism model was violated by 80 orders of magnitude; the group was even more assured that quantum mechanics was correct.The Universe is trying very hard to tell us something, isn't it? What though?
Leggett agrees with Zeilinger that realism is wrong in quantum mechanics, but when I asked him whether he now believes in the theory, he answered only "no" before demurring, "I'm in a small minority with that point of view and I wouldn't stake my life on it." For Leggett there are still enough loopholes to disbelieve. I asked him what could finally change his mind about quantum mechanics. Without hesitation, he said sending humans into space as detectors to test the theory. In space there is enough distance to exclude communication between the detectors (humans), and the lack of other particles should allow most entangled photons to reach the detectors unimpeded. Plus, each person can decide independently which photon polarizations to measure. If Leggett's model were contradicted in space, he might believe. When I mentioned this to Prof. Zeilinger he said, "That will happen someday. There is no doubt in my mind. It is just a question of technology." Alessandro Fedrizzi had already shown me a prototype of a realism experiment he is hoping to send up in a satellite. It's a heavy, metallic slab the size of a dinner plate.
Brukner and Kofler had a simple idea. They wanted to find out what would happen if they assumed that a reality similar to the one we experience is true—every large object has only one value for each measurable property that does not change. In other words, you know your couch is blue, and you don't expect to be able to alter it just by looking. This form of realism, "macrorealism," was first posited by Leggett in the 1980s.
Late last year Brukner and Kofler showed that it does not matter how many particles are around, or how large an object is, quantum mechanics always holds true. The reason we see our world as we do is because of what we use to observe it. The human body is a just barely adequate measuring device. Quantum mechanics does not always wash itself out, but to observe its effects for larger and larger objects we would need more and more accurate measurement devices. We just do not have the sensitivity to observe the quantum effects around us. In essence we do create the classical world we perceive, and as Brukner said, "There could be other classical worlds completely different from ours."
I'll close this post with another quote from Geometry Step by Step :
Benjamin Peirce (1809-1880, American mathematician, professor at Harvard) gave a lecture proving "Euler's equation", and concluded:Isn't it wonderful to be living in such a fascinating Universe, and one that has room in it for Love too? Now you know why I describe myself as "an agnostic with a tendency to commit Buddhism". It is because of the Real, the Complex, and the Imaginary."Gentlemen, that is surely true,
it is absolutely paradoxical;
we cannot understand it,
and we don't know what it means.
But we have proved it,
and therefore we know it must be the truth."