How important are binary number to a programmer?
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Moderately important. It depends exactly what you're counting. If you mean the ability to be able to do arithmetic in binary, it's not important (but it is easy: it works just like base-10 math, except there are fewer addition/multiplication facts to know. A lot fewer).
Being able to get specific bits from fields, translate data from one format to another, pack a bunch of booleans together, be able to figure out what the maximum value that can be stored in a particular type of data, or send messages with specific bits on to hardware can all come in handy and are all under the broad umbrella of "binary."
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