Integers and their representation

Positional systems

The lesson Variables and memory presented the variable as a cell with a name, a type and a value. Now, memory does not store decimal digits: it stores bits, each of which is 0 or 1, grouped into bytes of eight bits (the standard requires at least eight; all current systems use exactly eight). To know which values an integer variable can take, one must therefore know how a number is written with only two digits, and that is what a positional system of base 2 does.

In a positional system of base $b$, each digit counts according to the position it occupies: the last one multiplies $b^0 = 1$, the one before it $b^1$, the one before that $b^2$, and so on. In base 10, $2026$ means $2 \cdot 10^3 + 0 \cdot 10^2 + 2 \cdot 10^1 + 6 \cdot 10^0$. In base 2 the digits are 0 and 1, and the weights are the powers of 2: $1, 2, 4, 8, 16, 32, 64, 128, \ldots$ From this follows the rule for going from binary to decimal: add the weights of the positions that hold a 1.

The reverse path, from decimal to binary, is traveled by repeated division: divide the number by 2, write down the remainder (0 or 1) and repeat with the quotient until reaching 0; the remainders, read from the last to the first, are the binary digits. For 45: $45 = 2 \cdot 22 + 1$, $22 = 2 \cdot 11 + 0$, $11 = 2 \cdot 5 + 1$, $5 = 2 \cdot 2 + 1$, $2 = 2 \cdot 1 + 0$, $1 = 2 \cdot 0 + 1$; read backward, the remainders give 101101, which with two zeros on the left is the pattern of the figure. The reason is that each remainder is the units digit of the number being divided, and each quotient is that number shifted one position to the right. The same method works for any base: for base 16 one divides by 16, and the remainders from 10 to 15 are written with the letters a to f.

Base 16, or hexadecimal, exists for convenience: since $16 = 2^4$, each hexadecimal digit corresponds exactly to four bits, and a byte is written with two digits. A 32-bit number takes thirty-two digits in binary, which are hard to read, and only eight in hexadecimal, which convert to binary without calculation, digit by digit. That is why programmers read memory in hexadecimal. The mathematical foundation of positional systems is studied in the course Algebra and Geometry I, in the lesson “Positional numeral systems”.

C++ accepts integers written directly in these bases: a literal that begins with 0b is in binary, and one that begins with 0x, in hexadecimal. Inside any literal an apostrophe can be placed as a digit separator, which the compiler ignores and the reader appreciates: 0b0010'1101, 1'000'000. In the output, the format specification {:b} writes an integer in binary, {:x} in hexadecimal with lowercase letters and {:X} with uppercase letters; the sign # prepends the prefix, and a width preceded by a zero pads with zeros on the left.

#include <print>​int main(){    const int decimal{45};    const int binary{0b0010'1101};    const int hexadecimal{0x2d};    std::println("{} {} {}", decimal, binary, hexadecimal);    std::println("binary: {:b}", decimal);    std::println("eight bits: {:08b}", decimal);    std::println("hexadecimal: {:x} {:X} {:#x}", decimal, decimal, decimal);}

45 45 45binary: 101101eight bits: 00101101hexadecimal: 2d 2D 0x2d