Number Systems and Conversions Quiz

Study Notes

Digital circuits rely on binary numbers (0 and 1).
Decimal number system uses base 10.
- Value of the nth digit from the right = nth digit × (base)^n-1.
Binary number system uses base 2 (0 and 1).
Octal number system uses base 8 (0-7).
Hexadecimal number system uses base 16 (0-9, A-F).

Decimal to binary: Successive division by 2.
- Remainders form the binary number.
Binary to decimal: Multiply each bit by 2^n-1.
Decimal to octal: Successive division by 8.
Octal to decimal: Multiply each digit by 8^n-1.
Decimal to hexadecimal: Successive division by 16.
Hexadecimal to decimal: Multiply each digit by 16^n-1.
Binary to octal: Group binary digits in sets of 3, starting from the right, and replace with corresponding octal digit.
Octal to binary: Replace each octal digit with its 3-bit binary equivalent.
Binary to hexadecimal: Group binary digits in sets of 4, starting from the right, and replace with corresponding hexadecimal digit.
Hexadecimal to binary: Replace each hexadecimal digit with its 4-bit binary equivalent.

r's complement: r^m - N
(r-1)'s complement: (r^m - 1) - N (where m is the number of digits)
- Useful for subtraction by adding the complement.
- In binary, 1's complement flips 0s to 1s and 1s to 0s; 2's complement adds 1 to the 1's complement.
End-around carry: if the carryout to the most significant digit is needed, add it to the least significant digit.

Sign-magnitude: Leftmost bit is the sign, remaining bits are the magnitude.
1's complement: Representing a negative number by flipping the bits of its positive representation
2's complement: Represent a negative number by taking the 1's complement and adding 1. Common in computers.