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Numeric Base Mathematics: Positional Representation and Base Conversions
Every number system uses positional notation where the value of a digit depends on its position relative to the radix (base). Binary is a base-2 system using digits \(\{0, 1\}\), decimal is base-10 using \(\{0..9\}\), and hexadecimal is base-16 using \(\{0..9, A..F\}\).
To convert a decimal integer to binary or hexadecimal, we use the division-remainder method. The decimal number is repeatedly divided by the target base, and the remainders are recorded in reverse order. For example, to convert decimal 65 to binary: $$65 / 2 = 32\text{ R } 1$$, $$32 / 2 = 16\text{ R } 0$$, ..., reading the remainders upwards gives the binary representation `01000001`.