This has no meaning in the physical world, even though we know it’s true (I’ve demonstrated this in another post). For a start, i is not really a number (even though it’s defined by i = √-1) because you can’t have an i number of things. I prefer to think of it as a dimension that’s perpendicular to all other dimensions, because that’s how it’s represented graphically. In fact, mathematically, it’s not Real by definition. You have Real numbers and imaginary numbers and they are described in complex algebra as z = a + ib, where z has a Real component and an imaginary component. Notice that they don’t get mixed up, yet they do in Euler’s identity. Euler’s identity is so weird, for want of a better word, that it has a special status.
My point is that Euler’s identity only has meaning in an abstract realm or transcendental realm, which is apt, considering that π and e are called transcendental numbers, which means they can never be calculated in full. They can only exist in a transcendental realm – the Universe can’t contain them. Even God doesn’t know the last digit of π (or e, for that matter).
On my right arm I have Schrodinger’s equally famous equation, which I’ve also expounded upon in depth in another post. It’s been called the most important equation in physics: i
This is a poor representation but it’s close enough for my purposes. The tattoo on my arm is a much better rendition. Notice that it also includes the number i because complex algebra is essential to quantum mechanics and this is a seminal equation in QM. It is the complement or opposite of the equation on my left arm, in as much as it only has meaning in the physical world (the same as E = mc2, for example). Outside the Universe it has no meaning at all; whereas Euler’s identity would still be true even if the Universe didn’t exist and there was no one to derive it.
Schrodinger derived this equation from a hunch; it’s not derived from anything we know (as Richard Feynman once pointed out). It describes the wave function of a particle that’s not yet 'observed', which makes it truly remarkable, and therefore it can only give us probabilities of finding it. Nevertheless, it’s been found to be very accurate in those probabilities. Schrodinger’s wave function is now incorporated into QED (quantum electrodynamics) which effectively describes everything we can see and touch and is arguably the most successful mathematical theory in physics, comparable only to Einstein’s general theory of relativity. In principle, you could have a Schrodinger equation for the entire universe, but you’d probably need a computer the size of the Universe to calculate it.
So on my left arm I have a mathematical connection to a transcendental (or Platonic) realm, and on my right arm I have a mathematical connection to the physical Universe.
But there is more, because Euler’s identity is the solution of an equation called Euler’s equation: eiθ = cosθ + isinθ; which becomes Euler’s identity when θ = π. The point is that this equation provides the key ingredient to Schrodinger’s wave function, ψ(psi), so these equations are linked. The transcendental world is linked to the physical world, arguably without the need of human consciousness to make that link.
Footnote: A friend of mine wrote a poem about my tattoos.
Addendum: I came across this description by Clifford A Pickover in his opus, The Mαth βook:
Schrodinger's wave equation - which describes reality and events in terms of wave functions and probabilities - may be thought of as the evanescent substrate on which we all exist.