Number Systems and Boolean Algebra
What this domain tests
This domain covers positional number systems, base conversion, signed number representation, binary arithmetic, logic gates and Boolean simplification. Questions are usually short and exact: convert, subtract, complete a truth table, prove an identity, or reduce an expression.
Core rules
- In base b the value is the sum of digit x b^position; digits run 0 to b-1. Binary, octal and hexadecimal convert in groups of 3 and 4 bits.
- Negative numbers in two's complement: invert all bits, then add 1; the MSB carries weight -2^(n-1), so an n-bit range is -2^(n-1) to 2^(n-1)-1.
- BCD uses 4 bits per decimal digit; ASCII is 7-bit; Unicode covers large character sets.
- Gates: AND (.), OR (+), NOT ('), NAND, NOR, XOR. NAND and NOR are universal.
- De Morgan: (A.B)' = A' + B' and (A + B)' = A'.B'. Use these plus distributive and absorption laws for simplification.
- SOP is built by ORing minterms where output is 1; Karnaugh maps group 1s in powers of two.
Worked example
Subtract 25 from 19 in 8-bit two's complement. 19 = 00010011, 25 = 00011001. Two's complement of 25: invert to 11100110, add 1, giving 11100111. Add: 00010011 + 11100111 = 1 11111010. Drop the carry out, leaving 11111010. The MSB is 1, so it is negative; its magnitude is the two's complement of 11111010, which is 00000110 = 6. The answer is -6.
Traps that cost marks
- Forgetting the +1 step after inverting in two's complement.
- Reading the sign bit as a magnitude bit, which gives 250 instead of -6.
- Writing hex digits beyond 9 as lowercase text or as two decimal digits.
- Assuming XOR gives 1 when both inputs are 1; it gives 1 only for an odd count of 1s.
- Leaving a simplification answer in non-canonical or partially reduced form.