\((\text {Number Systems Table})\) Write a program that prints a table of the binary, octal and hexadecimal equivalents of the decimal numbers in the range \(1-256 .\) If you are not familiar with these number systems, read Appendix D, first. [Hint: You can use the stream manipulators dec, oct and hex to display integers in decimal, octal and hexadecimal formats, respectively.]

Understanding how to convert between binary, octal, hexadecimal, and decimal number systems is a significant aspect of computer science education. Decimal is the standard base-10 number system that uses symbols 0 through 9. Binary is a base-2 system, using only 0 and 1 to represent numbers, and is foundational in computing since computers operate with binary data.

Octal is less commonly used in practice, but is a base-8 number system that utilizes digits 0 through 7. Hexadecimal, a base-16 system, extends beyond the decimal digits to include letters A through F, representing values 10 to 15, making it a compact form of binary representation often used in programming and system design.

Conversion between these systems is important, especially when interfacing with low-level computer processes. To convert from one system to another, methods such as division-remainder for binary and octal, and grouping binary digits for hexadecimal are commonly used.

Handling the format of output in C++ is facilitated by the iomanip library, a component of the standard library which provides a plethora of input/output stream manipulators. These manipulators include functions like setw(), which sets the width of the next input/output field, and manipulators such as setprecision() for floating-point values control, which aid in creating readable and well-formatted textual data.

The iomanip library becomes particularly useful when displaying data in specific layouts, such as tables, where consistent alignment is key to readability. In the context of number system conversions, iomanip simplifies formatting the outputs in a table without the need for complex calculations or manual spacing.

Stream manipulators in C++ enable programmers to control the format, width, and precision of the data streamed to input/output streams. For instance, std::dec, std::oct, and std::hex are manipulators used to designate the numeral system for stream output. When dealing with the output of numeric data, std::left, std::right, and std::internal can dictate the alignment of data within the given field width, established via setw().

The ease of spatting out numbers in various systems stems from utilizing these stream manipulators. For example, by injecting std::hex into the stream, subsequent numbers are displayed in hexadecimal format until the manipulator is changed or the stream is reset. This allows for seamless transitions between displaying numbers in different number systems.