Number Systems and Digital Logic

A Number System is a mathematical method for representing quantities and digits using a specific set of symbols or base values. In computer science, the fundamental number systems include Decimal (base 10), Binary (base 2), Octal (base 8), and Hexadecimal (base 16). Computers exclusively understand binary due to the electronic states of transistors (on/off, 1/0), while hexadecimal and octal are used as shorthand notations for compacting long binary strings. A major focus of board exams in this section is mastering base conversions—such as converting decimal fractions and integers to binary, octal, and hex, as well as performing binary arithmetic (addition, subtraction using 1's and 2's complement).

At the physical level, computer hardware relies on Boolean Algebra and Logic Gates to process binary data. Logic gates are electronic circuits that operate on one or more input signals to produce a single standard output based on logical relationships. The three primary (basic) logic gates are AND, OR, and NOT, while universal gates like NAND and NOR can construct any Boolean function. For examinations, students must be thoroughly prepared to draw truth tables, write Boolean expressions, and simplify circuits using Boolean postulates and De Morgan's Laws.

The chapter concludes with combinational logic design principles, emphasizing how complex operations are built from basic gates. Key concepts frequently tested include the construction of half-adders and full-adders for binary addition, as well as decoders, encoders, and multiplexers. Examiners regularly test both theoretical knowledge (e.g., "State De Morgan's theorems") and practical problem-solving (e.g., "Simplify the given Boolean expression and draw its logic circuit diagram"), making precision in drawing diagrams and step-by-step mathematical proofs essential for full marks.