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I have no doubt that if we encountered an intelligent alien species, we could determine it was "alive" even if it had a completely different cellular structure.However, there are some things that would be difficult to judge. Could a planet be "alive"? A few of the Star Trek episodes dealt with planet sized or nebula sized life forms. Likewise, we have a metabolic rate giving us a lot of movement and a lifespan of about 100 earth-years. It would be hard to judge intelligence in something that moved at the rate of an Oak Tree, or conversely with a lifespan of a few seconds.We also have great difficulty in judging the intelligence of non tool building species such as the blue whale, right hear on Earth.
Yes, Star Trek is fiction, but it does bring up, at least a few ideas that would be worth considering. And, while most of their "aliens" look very much like humans, some don't. Not that everything is believable.
As far as the number 137, or 1/137. It is only being used as an estimate of an irrational number.
Likewise, one might choose to send the number 22/7.
Which, of course, is also an estimate for a better known irrational number.
However, one wouldn't wish to convey to the "aliens" that we did not understand the fundamental difference between rational and irrational numbers.
If we were able to in fact make contact via radio, that in itself would be a feat which would demonstrate an amount of intelligence. One wouldn't have to send math puzzles to convince them of intelligence which would be obvious from the ability to make contact.
What will be the cycle time between sending a message and getting a response? 50 years? Plenty of time to send an unabridged dictionary, and every page on Wikipedia while waiting for the next response.
The Arecibo message was broadcast into space a single time via frequency modulated radio waves at a ceremony to mark the remodeling of the Arecibo radio telescope on 16 November 1974. It was aimed at the globular star cluster M13 some 25,000 light years away because M13 was a large and close collection of stars that was available in the sky at the time and place of the ceremony.
How long was Noah's Ark btw?300 cubits = 450 feet = 137.16 meters
no matter where you are in the universe, and no matter what units you use to measure all of the above ... the answer will always be '1/137'.