What absolute minimum should a programmer know about binary numbers and arithmetic?

Question

Although I know the basic concepts of binary representation, I have never written code that uses binary arithmetic and operations.

I want to know

• What are the basic concepts a programmer should know about binary numbers and arithmetic ?, and

• In what "practical" ways can binary be used in programming. I saw some cool options for changing operators and XOR, etc., but there are some typical problems when using binary operations - the obvious choice.

Please indicate pointers to good reference material.

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