Using mathematical constants is a good way just to let the aliens know our current level of advancement. But of course you'd need to establish context before you can really communicate meaning.
Message 1: 11 111 11111 1111111 11111111111
The first five primes, to give the aliens a hint for the next message.
Message 2: 10 11 101 111 1011
The same five primes in binary since they have the hint from the first message we're talking about primes, now they know we are using binary numbers to communicate.
Message 3:
11 001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000000110111000001110011010001
Pi in binary. They already understand the hint that we use a binary counting system, so they calculate what those numbers might mean, and recognize it as Pi.
Message 4:
(Picture of animal) 0001. (Picture of human) 0001 11. (Picture of building) 110110. (And so on)
They already understand we're using binary numbers to express things. Now they have a hint we're using them as language and sharing them the basic vocabulary.
And so on. A process of first establishing context by showing them something simple they already recognize, then building on what you taught them.
Good idea, but how do you know that you didn't just insult the Slug Queen's Mother on Sluggonia XVIII?
https://www.slashgear.com/strange-radio-transmissions-are-emanating-from-proxima-centauri-19651887/
Strange radio transmissions are emanating from Proxima Centauri
Something strange is going on at the closest star to the sun. A mysterious radio signal is coming from Proxima Centauri, which is too dim to be seen from Earth with the naked eye. Despite being too dim for us to see from Earth, it’s a cosmic stone throw away at only 4.2 light-years away. A mysterious signal is emanating from the star, which was discovered in archived data gathered last year.
[The signal](
https://www.scientificamerican.com/...over-mysterious-signal-from-proxima-centauri/) was discovered this fall and appears to come from the direction of Proxima Centauri. As of now, researchers have been unable to dismiss the signal as interference from Earth. While detecting a strange signal coming from a nearby star certainly has tantalizing possibilities, scientists on the project caution, there is still lots of work to be done.
Researcher Andrew Siemion from the University of California, Berkeley has said that the signal has properties that have caused it to pass many checks so far, and it can’t be explained. One of the most curious aspects of the signal is that it occupies an extremely narrow radio spectrum of 982 megahertz. That is a spectral range that is typically empty of any transmissions from human-made satellites and spacecraft.
One of the most curious aspects of the signal is that it occupies an extremely narrow radio spectrum of 982 megahertz. That is a spectral range that is typically empty of any transmissions from human-made satellites and spacecraft.
Space Weather Phenomena
https://www.swpc.noaa.gov/phenomena
In the article there are numerous topics relating to space weather phenomena, but none are below the 1,000 megahertz range.
The anomaly definitely isn't a solar flare from Proxima Centauri's sun.
Solar flares occur in a power-law spectrum of magnitudes; an energy release of typically 1020 joules of energy suffices to produce a clearly observable event, while a major event can emit up to 1025 joules.
The researcher says that there is no known natural way to compress electromagnetic energy into a single band frequency. Siemion does say that there could be some unknown exotic work of plasma physics that could account for the signal. However, he admits that for now, the only source that is known for such a signal is technological.
If the signal however does prove to be a natural effect of plasma physics compressing EM energy into a single band frequency, then let the good times roll.
