Question

Write a program that prints a table of the binary, octal and hexadecimal equivalents of the decimal numbers in the range 1 through \(256 .\) If you are not familiar with these number systems, read Appendix D, Number Systems, first.

Short answer

Loop from 1 to 256, convert to binary, octal, hexadecimal, and print.

Step-by-step solution

Understand the Number Systems

Binary, octal, and hexadecimal are alternative number systems to decimal. Binary is base 2 using digits 0-1, octal is base 8 using digits 0-7, and hexadecimal is base 16 using digits 0-9 and A-F. We'll convert numbers between 1 and 256 into these systems.

Plan the Program Structure

Design the program to loop through numbers 1 to 256. For each number, convert it to binary, octal, and hexadecimal formats and print the results in a tabulated form.

Implement the Loop

Use a `for` loop in your programming language of choice to iterate from 1 to 256. This will allow you to process each integer in the range.

Convert Numbers and Print

Within the loop, for each number, use built-in functions to convert the decimal number to binary, octal, and hexadecimal. In many programming languages, these functions are `bin()`, `oct()`, and `hex()`. Format and print these conversions.

Execute and Test the Program

Run the program to ensure that each conversion is correct for all numbers from 1 to 256. Verify selections by cross-checking with a calculator if necessary.

Key concepts

Binary Numbers

Binary numbers are a crucial concept in computer science because computers inherently operate using this system. Binary is a base 2 number system that uses only two digits: 0 and 1.
These digits are the building blocks for all data and instructions processed by digital systems.

To convert a decimal number to binary, repeatedly divide the number by 2 and record the remainder. This remainder represents the binary digit (bit).
Continue dividing until you reach a quotient of 0.

• For example, to convert the decimal number 5 to binary, divide 5 by 2 to get a quotient of 2 and a remainder of 1. Next, divide 2 by 2 to get a quotient of 1 and a remainder of 0. Lastly, divide 1 by 2 to get a quotient of 0 and a remainder of 1. Reading the remainders from bottom to top gives you 101 in binary.

Binary numbers are often used in programming loops, as computers efficiently compute binary operations.

Octal Numbers

The octal number system is based on 8, using digits from 0 to 7. Although not as prevalent in modern computing as binary or hexadecimal, octals are used in some applications, especially those involving Unix file permissions.
Each octal digit represents three binary digits (or bits).

To convert a decimal number to octal, repeatedly divide the number by 8 and write down the remainder.

• For example, to convert decimal 65 to octal: divide 65 by 8 to get a quotient of 8 and a remainder of 1. Then, divide 8 by 8 to get a quotient of 1 and a remainder of 0.
• Read the remainders from last to first, resulting in octal 101.

The close ties between binary and octal make converting between the two straightforward by grouping bits into threes. Understanding octal numbers can simplify operations in digital electronics where binary might be cumbersome.

Hexadecimal Numbers

Hexadecimal numbers are integral to understanding how computers work because they provide a more human-friendly representation of binary-coded values.
Hexadecimal (base 16) uses digits 0-9 and letters A-F, where A represents the decimal 10, B represents 11, up to F for 15.

This higher base system allows for a compact representation of long binary numbers, making it popular in programming and debugging.
To convert a decimal number to hexadecimal, divide the number by 16 and write down the remainder, continuing this process until the quotient is zero.

• For instance, converting decimal 255 to hexadecimal involves dividing 255 by 16 to get a quotient of 15 (with a remainder 15, which is F) and then dividing 15 by 16 to get 0 (with a remainder again of F).
• Thus, 255 converts to FF in hex.

Hexadecimals neatly package bytes, as one hex digit represents four binary digits. This is efficient for programming tasks and memory address referencing.

Programming Loops

Programming loops are a fundamental concept in software development for automating repetitive tasks.
They allow programmers to execute a block of code multiple times, facilitating operations like iterating through number ranges or collections of data.

In the case of printing number conversions, a loop enables you to handle numbers systematically from 1 to 256. For loops in many programming languages control the iteration with a counter variable that starts from a specified point and moves with each iteration until a condition is met. Here’s how you can apply a for loop:

• Start with a `for` loop that initializes a variable at 1 and increments this variable up to 256.
• Within this loop, you would typically call functions to convert the current decimal number to binary, octal, and hexadecimal.
• The conversions are then formatted and printed, achieving the required task concisely.

Understanding loops will enhance your ability to programmatic control flow, offer optimization, and handily manage operations that involve multiple iterations.