Chapter 12: Numeral Systems, Scientific Notation, and Escape Sequences

Numeral Systems

Binary, Octal, Decimal, and Hexadecimal

In C++, you can represent numbers in different numeral systems.

Binary

#include <bitset>

std::bitset<8> bin1{ 0b1100'0101 }; // binary literal for binary 1100 0101

Octal

int x{ 012 }; // 0 before the number means this is octal

Decimal

Decimal is the default numeral system.

Hexadecimal

int x{ 0xF }; // 0x before the number means this is hexadecimal

Conversion Table

Decimal	10	11	12	13	14	15	16	17
Hexadecimal	A	C	D	E	F	10	11

Examples

int bin = 0b11; // binary 0000 0000 0000 0011
bin = 0b1; // same as binary 1

int hex = 0xF770; // binary 1111 0111 0111 0000
int bin2 = 0b11110000; // binary 0000 0000 1111 0000

Digit Separator

long value { 2'132'673'462 }; // much easier to read than 2132673462

Formatting Binary Output

#include <format> // C++20
#include <print> // C++23

std::cout << std::format("{:b}\n", 0b1010);  // C++20, {:b} formats the argument as binary digits
std::cout << std::format("{:#b}\n", 0b1010); // C++20, {:#b} formats the argument as 0b-prefixed binary digits

std::println("{:b} {:#b}", 0b1010, 0b1010);  // C++23, format/print two arguments (same as above) and a newline

Scientific Notation

Scientific notation is a way to represent numbers that are too large or too small to be conveniently written in decimal form.

Examples

Example 1

600.410 → 6.00410e2

Example 2

0.0078900 → 7.8900e-3

Detailed Example

Start with: 42030 (assuming the trailing zero is significant)
Slide decimal left 4 spaces: 4.2030e4
No leading zeros to trim: 4.2030e4
Keep trailing zeros: 4.2030e4 (5 significant digits)

Start with: 0.0078900
Slide decimal right 3 spaces: 0007.8900e-3
Trim leading zeros: 7.8900e-3
Don't trim trailing zeros: 7.8900e-3 (5 significant digits)

Notation Rules

Keep trailing zeros after a decimal point.
Drop trailing zeros if there is no decimal point.

Escape Sequences

Escape sequences are used to represent special characters in a string.

Table of Escape Sequences

Name	Escape Sequence	Description
Alert	\a	Makes an alert (beep)
Backspace	\b	Moves cursor back one space
Formfeed	\f	Moves cursor to next logical page
Newline	\n	Moves cursor to next line
Carriage return	\r	Moves cursor to beginning of line
Horizontal tab	\t	Prints a horizontal tab
Vertical tab	\v	Prints a vertical tab
Single quote	\'	Prints a single quote
Double quote	\"	Prints a double quote
Backslash	\\	Prints a backslash
Question mark	\?	Prints a question mark
Octal number	\number	Translates into char represented by octal
Hex number	\xnumber	Translates into char represented by hex number

Example Usage

std::cout << "Hello, World!\n"; // Prints "Hello, World!" followed by a newline
std::cout << "This is a tab:\tSee?\n"; // Prints a tab between "This is a tab:" and "See?"