Genady

I don't. But if it is not, then we take r=sqrt(2)=s, and rs is rational. Which answer the OP. One of the two has to be rational, and this answers the question.

The following answer doesn't require number theory, Euler, or complex powers, just algebra: r=sqrt(2)sqrt(2) , s=sqrt(2) Either r is rational or rs is rational. So, the answer to the OP is, Yes.

Yes, it answers it. If we know that logπ2 is irrational. I got a simpler proof, without that knowledge: r=sqrt(2)sqrt(2) , s=sqrt(2).

I am interested to know if there are two real irrational numbers r and s such that rs is rational. The comment you refer to is a reply to an attempted proof above that:

OK. I just don't see yet that (-1)-i is a rational number as per definition "a rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q."

I thought about it, but I don't know if πi is rational or irrational. To make it well-defined, let's stay in the real numbers.

Can the thing in the title be rational?

This is exactly what math is -- investigation of such features which do not depend on implementation of a system. This point seems to connect back to another recent thread. Perhaps, Navier-Stokes equations are not derivable from the molecular level, but a system behavior that these equations describe is. As if we had a gigantic computer which could simulate evolution of a system of billions molecules of water, a macroscopic behavior of water would appear in the output.

A state of two quantum-entangled particles is irreducible? unpredictable? novel? emergent? The question marks are to indicate that I'm not certain in meaning of these terms. However, I'm certain in the meaning of terms "state of two quantum-entangled particles".

I wonder, is it in fact useful? When? Where?

This is correct. Approaching a neutron star, there would be a similar but less intense curvature. Moreover, approaching any star (or anything else for that matter) there is a similar, but much less intense curvature. The gravity in empty space around any radially symmetric mass has the same shape and only differs in intensity.

Yes, it would be the same gravitationally. Same with the black hole, too.

(1) THE SHINING (1980) - "All work and no play makes Jack a dull boy" [HD] - Bing video jk However, Computer Engineering? It is better than accounting, but still... Did you consider that may be this subject is not your forte? or your bliss?

In fact, it is overexplained. There are many different explanations, sometimes overlapping, sometimes inconsistent. My OP generated a small survey of what some members here pick as their favorite explanation. It turned out to be a subset of explanations existing "on the market."

First of all, thank you for the correction. I like it. And, yes, I also think that it (something being emergent) refers to phenomenon rather than to our description of it, including all the circumstances that the said phenomenon needs in order to occur. Then, the criteria of a phenomenon being or not being emergent should not depend on how we describe it and what we know or need to know to describe it. In other words, the criteria should be about the phenomenon and not about us.

Thus, there are things which are emergent as per less specific criteria but not emergent as per the more specific one.

Now I see where all (my) confusion in the earlier discussion came from -- I never seriously related to the word "unreasonable" and saw it just as metaphoric. In the article Wigner talks only about unexplained effectiveness.

So, this criterion is different from:

Does a phenomenon become 'emergent' after we find a way to describe how it emerges? If a condition for a phenomenon to be emergent is to "be described without knowing or needing from what they exactly emerge", and the evolution of a wave function is a phenomenon that is "described without knowing or needing from what they exactly emerge", how come that this phenomenon does not qualify as being emergent?

Do 'regularity' and 'existence and uniqueness' mean the same?

Does a qualification of being 'emergent' apply to a phenomenon or to an equation describing it?

Schrödinger equation describes the phenomenon of a wave function evolution without knowing or needing from what it exactly emerges. Does it make this phenomenon an emergent one?

Which regularity makes differential geometry effective in describing GR?

Aren't there two different concepts: one is 'probability', the other - 'probability density'?

Twenty seven years ago, our Marine Park started a long-term "experiment" in coral development, albeit inadvertently. They installed concrete mooring blocks, 1m x 1m x 1m, along the town shoreline for yachts and boats to moor. Coral colonies began inhabiting the block faces. Years later, I’ve noticed that this intriguing growth is not randomly uniform but instead follows patterns, some quite puzzling. Here is one of them. There are two lines of the moorings between South and North ends of the downtown boulevard. One line consists of a dozen moorings constructed from three blocks each and is located about 50m from shore, just near the reef drop-off. Another line consists of a dozen moorings constructed from two blocks each and is located on the sand-flat, half way between the reef drop-off and the shore. I've compared exposed vertical faces of the blocks near the drop-off and those of the blocks near the shore. Where would one expect to find more corals? Wouldn’t corals prefer to grow near the reef and away from the town shoreline with all its polluted water and sewage runoff? The corals provided surprises. More coral colonies grew on the faces of the blocks close to the shore than on those close to the reef. Only 12% of the faces of the off-shore blocks are covered with live corals, like here: Twice as much, 25%, of the close-to-shore block faces are covered with live corals, like here: What is going on? Are shore effects good for the corals? Are some off-shore effects bad for them? What factors are responsible? Any ideas? Questions? More info?