Binary Study Guide

Hexadecimal (base-16) notation is widely used in computing as a human-friendly representation of binary (base-2) data. It's often impractical and error-prone for humans to read or write long strings of binary digits. Each hexadecimal digit corresponds directly to four binary digits (bits), making conversion between the two systems straightforward. For example, 1111 in binary is F in hexadecimal. This compact representation saves space when displaying data, such as memory addresses, MAC addresses, colour codes, or error messages, while still directly reflecting the underlying binary structure. To convert from binary to hexadecimal, you group binary digits into sets of four, starting from the right. If the leftmost group has fewer than four digits, you pad it with leading zeros. Each group of four binary digits is then converted to its equivalent single hexadecimal digit. Conversely, to convert from hexadecimal to binary, each hexadecimal digit is converted to its four-bit binary equivalent.

Computers process numbers in binary form. For whole non-negative numbers, called unsigned integers, each binary digit (bit) contributes to the value based on its position, similar to how decimal numbers work, but with powers of two. For example, an 8-bit unsigned integer can represent values from 0 (all zeros) to 255 (all ones). When a bit pattern represents a signed integer, one bit is typically reserved to indicate the sign. Two’s complement is the most common method for representing signed integers in computing. In two’s complement, the most significant bit (leftmost bit) indicates the sign: 0 for positive and 1 for negative. However, for positive numbers, the representation is the same as unsigned. For negative numbers, the two’s complement value is found by inverting all the bits of the corresponding positive number and then adding one. This system allows for efficient arithmetic operations because addition and subtraction can be performed using the same circuitry, regardless of the sign of the numbers involved, and it avoids the issue of having two representations for zero (which happens in signed magnitude and one's complement).