What E.164 is
E.164 is the international format for phone numbers: a plus sign, the country calling code and the national number, with no spaces or punctuation and at most 15 digits. +442079460123 is a UK number written in E.164.
Sign-up forms, contact databases and services that send SMS store numbers this way, because in E.164 one number has exactly one spelling.
What the check can and cannot tell you
It checks the format: that the country code exists and that the number has a length that country uses, against the numbering plans published in Google's libphonenumber.
It does not dial the number, send it anywhere or ask any network. So it cannot tell who owns a number, whether it is in service, or whether it can receive SMS.
Checking a list
Paste up to 1,000 numbers, one per line or separated by commas or semicolons, or choose a CSV file: the first column that holds phone numbers is read. Numbers written without + are read in the country you pick. Download the result as a CSV with the E.164 form beside each input.
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A private number from SMS Activate App receives SMS and calls in your dashboard, and it is yours alone while you keep it.