Number Systems
What the exam tests
Number Systems tests positional notation, conversion between binary, octal, decimal and hexadecimal, binary arithmetic, complements and codes such as BCD and ASCII. Expect direct conversion and range questions.
Core rules
- In base r, each digit has weight r^position; positions increase left from the radix point and decrease right.
- Decimal to binary: repeated division by 2, collecting remainders bottom-up. Binary to decimal: sum weights of 1 bits.
- Group binary bits from the radix point: 3 bits = 1 octal digit; 4 bits = 1 hexadecimal digit. Pad with zeros at the outer ends, not across the point.
- n-bit unsigned range: 0 to 2^n − 1. n-bit two's complement range: −2^(n−1) to 2^(n−1) − 1.
- To negate in two's complement: invert all bits and add 1.
- BCD encodes each decimal digit in 4 bits; it is not the same as binary. ASCII is a character code; Unicode covers larger scripts.
Worked example
Convert 45 decimal to binary, octal and hexadecimal. Repeated division by 2 gives remainders 1,0,1,1,0,1; reading bottom-up: 101101₂. Group from the right in threes: 101 101 = 55₈. Group in fours: 0010 1101 = 2D₁₆. Check: 32 + 8 + 4 + 1 = 45.
Traps
Do not group from the left across the radix point. A leading zero does not change value. 101101₂ is not 45 in BCD; BCD would be 0100 0101. In two's complement, the negative range has one more value than the positive range. Do not use 2^n − 1 as the two's complement maximum.