Number Systems - Foundations of Computer Science

Flashcards covering concepts of number systems, including positional and non-positional systems, decimal, binary, hexadecimal, and octal systems, and conversion methods between these bases.

Number System

Defines how a number can be represented using distinct symbols.

Positional Number System

A system where the position a symbol occupies in the number determines the value it represents.

Non-positional Number System

A system where the position a symbol occupies in the number normally bears no relation to its value, and the value of each symbol is fixed. The total value is typically found by adding the values of all symbols.

Base (Radix)

The number of distinct symbols or digits used in a positional number system.

Decimal System

A positional number system with base b = 10, using ten symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

Binary System

A positional number system with base b = 2, using two symbols: {0, 1}. Its symbols are often referred to as 'bits'.

Hexadecimal System

A positional number system with base b = 16, using sixteen symbols: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}, where A-F represent values 10-15 respectively.

Octal System

A positional number system with base b = 8, using eight symbols: {0, 1, 2, 3, 4, 5, 6, 7}.

Bits (Binary Digits)

The symbols {0, 1} used in the binary system.

Conversion (Any Base to Decimal)

To convert a number from any base to decimal, multiply each digit by its corresponding place value (power of the base) and sum the results.

Conversion (Decimal Integer to Any Base)

To convert a decimal integer to another base, continuously divide the decimal number by the destination base, recording the remainders. The result is the sequence of remainders from last to first.

Conversion (Decimal Fractional Part to Any Base)

To convert a decimal fraction to another base, continuously multiply the fractional part by the destination base, recording the integral part of each result. The result is the sequence of integral parts from first to last.

Binary-Hexadecimal Conversion

Binary numbers can be converted to hexadecimal by grouping bits into patterns of four, starting from the decimal point, and then converting each 4-bit pattern to its hexadecimal equivalent.

Binary-Octal Conversion

Binary numbers can be converted to octal by grouping bits into patterns of three, starting from the decimal point, and then converting each 3-bit pattern to its octal equivalent.

Roman Numerals

A non-positional number system example where symbols (I, V, X, L, C, D, M) have fixed values, but with specific rules for interpretation (e.g., IV = 4, VI = 6) that introduce exceptions to simple addition.

Minimum Number of Digits (K)

The minimum number of digits 'K' required to represent a decimal number 'N' in a specific base 'b' is given by K = ⌈logb N⌉.

Minimum Digits (Cross-Base Conversion)

The minimum number of digits 'x' needed in a destination base 'b2' to represent numbers up to 'k' digits in a source base 'b1' is given by x = ⌈k × (logb1 / logb2)⌉.