print "input three numbers separated by spaces\n";
my $user_input = <>;
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This is good as far as it goes: you're getting something | a string from  STDIN that is terminated by a newline.

chomp $user_input;
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This is good too, except you do it twice more and this is where you start to go off the rails. chomp removes a newline (if present) from the end of a string (well, it's a little more involved than that, but let it stand for now), $user_input in this case, so doing it repeatedly will not usually get you anything. For learning/development/debugging, I would advise placing a print statement after this step so you can be sure exactly what you are dealing with:
    print "A: '$user_unput' \n";
(In general, develop your code step-by-step and use print statements at each successive step to see just what is happening. If weird stuff starts to happen, understand the problem before proceeding.)

my ($nr1) = $user_input, "$a";
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This has no meaning. You are assigning  $user_input to  $nr1 (why?) and interpolating  $a (which has no value at this point) into a string, which you then throw away: what do you think is happening here? (Update: Two other, similar statements have the same problem.)

Please consider these and other points raised so far, write some more code and post again.

Update: Do continue to  use strict; and  use warnings; in your code (use Modern::Perl; includes both of these). They will make learning Perl much easier.