How Computers Represent Data

A bit is one 0 or 1. Eight bits make a byte, which is the smallest chunk most systems can address individually. You already understand how binary numbers work, because you already understand place value: in decimal, 305 means three hundreds, no tens and five ones, and each column is worth ten times the one to its right because there are ten digits. Binary has two digits, so each column is worth twice the one to its right. The place values from the right are 1, 2, 4, 8, 16, 32, 64, 128. Nothing else about the arithmetic changes.

To go from binary to decimal, write the place values above the bits and add up only the columns holding a 1. To go from decimal to binary, the division method is the one that is hardest to get wrong: divide by 2 repeatedly, writing down each remainder, then read the remainders from the bottom up.

Binary is correct but unreadable — nobody dictates 11010110 over the phone without making a mistake. Hexadecimal is base 16, and because 16 is $2^4$, exactly four bits make exactly one hex digit with no arithmetic carrying across the boundary. A byte becomes two characters, a colour becomes six, a memory address becomes eight. That is why every error code, MAC address and web colour you will ever see is written in hex. The digits above 9 are letters: A is 10, B is 11, C is 12, D is 13, E is 14, F is 15.

Above the byte the prefixes climb, but there are two competing sets of them and that difference causes a question that appears in every course like this one. Storage manufacturers and network engineers use decimal prefixes: a kilobyte is 1000 bytes. Memory sizing and most operating systems use binary prefixes: 1024 bytes, because memory is addressed in powers of two. The proper names for the binary ones are kibibyte, mebibyte and gibibyte, but many programs still label them KB, MB and GB.

A bit pattern means nothing on its own. The byte 01000001 is the number 65, the letter A, a very dark red, or a fragment of a machine instruction, and which one it is depends entirely on what the program reading it expects. ASCII assigns a number from 0 to 127 to each English character, fitting in 7 bits stored inside one byte. The layout is deliberate: A is 65 and a is 97, exactly 32 apart, so flipping one bit changes case. The digit characters start at 48, so subtracting 48 from a digit's code gives the number it stands for.

ASCII covers English and nothing else, which is why Unicode exists. Unicode assigns a code point to every character in every writing system, plus emoji. UTF-8 encodes those code points using one to four bytes each, and its first 128 values are deliberately identical to ASCII, so every ASCII file is already valid UTF-8. Images work by the same idea in two dimensions: a grid of pixels, each pixel storing a colour as a number, with colour depth deciding how many bits each pixel gets. Sound is the same idea in one dimension: an analog waveform sampled thousands of times a second, each sample stored as a number.

This is not background trivia — it is the reason Java has more than one kind of whole number. Every primitive type is a promise about exactly how many bits the variable gets, and a fixed number of bits can only hold a fixed number of distinct values. Java's integer types are signed, which means one bit's worth of the range is given over to negatives, so an n-bit signed type runs from $-2^{n-1}$ to $2^{n-1}-1$.

int big = 2147483647;        // this is Integer.MAX_VALUE
System.out.println(big + 1); // prints -2147483648, NOT 2147483648

long safe = 2147483647L;     // the L makes it a long literal
System.out.println(safe + 1);// prints 2147483648

char letter = 'A';
int code = letter;           // 65 - a char really is a number
System.out.println((char)(letter + 32)); // prints a