Question

Write an application that displays 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 E first.

Short answer

Convert each decimal from 1 to 256 to binary, octal, and hexadecimal using programming functions, then display in a table.

Step-by-step solution

Understanding Number Systems

To solve this exercise, it is important to first understand the different number systems: binary, octal, and hexadecimal. Binary is base-2, using digits 0 and 1. Octal is base-8, using digits 0 to 7. Hexadecimal is base-16, using digits 0 to 9 and letters A to F. We will convert decimal numbers (base-10) from 1 to 256 into these three number systems.

Setting Up the Program Environment

Start by setting up a programming environment where you can write and execute your code. Languages like Python, Java, or C++ are suitable for this task. Each language has built-in functions or can easily handle conversions between number systems.

Loop through the Decimal Numbers

Using a programming loop (e.g., a 'for' loop), iterate through decimal numbers from 1 to 256. For each number in this range, perform the conversion to binary, octal, and hexadecimal formats.

Convert Decimal to Binary

For each decimal number, convert it to binary. For instance, in Python, you can use the 'bin()' function. Other languages like C++ might use bitwise operations or libraries. Simply remove the prefix usually like '0b' in Python, if needed, for clean display.

Convert Decimal to Octal

Convert each decimal number to its octal equivalent. In Python, use the 'oct()' function or similar capabilities in other languages. Again, strip the '0o' prefix for a clear output if necessary.

Convert Decimal to Hexadecimal

Each decimal number is then converted to hexadecimal. Use the 'hex()' function in Python or equivalent in other languages. Typically, you would convert letters to uppercase (A-F) if not already.

Display the Table

Once conversions are done, format and display the output in a tabular form. Ensure that each row of the table corresponds to a single decimal number and its binary, octal, and hexadecimal equivalents. Use justified widths for clean alignment and presentation.

Key concepts

Binary Numbers

Binary numbers are essential in computing and digital electronics. They use a base-2 number system, which means they consist only of two digits: 0 and 1. Think of binary as a simple on/off system, where 0 indicates "off" and 1 indicates "on." These digits represent powers of 2, with the least significant bit (rightmost) representing \(2^0\), the next \(2^1\), and so on, increasing by power as you move to the left.
For example, to convert the decimal number 5 to binary, you write it as \(101_2\). This means \(1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\), which equals 4 + 0 + 1 = 5 in decimal.
Binary is crucial because computers operate using binary logic, which includes operations like AND, OR, and NOT.

Octal Numbers

Octal numbers use a base-8 system, involving digits from 0 to 7. Each position in an octal number represents a power of 8. It's less commonly used than binary or hexadecimal, but it can be useful for simplifying binary numbers.
To convert a decimal number to octal, repeatedly divide the decimal number by 8, recording the remainders. Write these remainders from bottom to top. For example, the decimal number 65 converts to octal as \(101_8\), which means \(1 \times 8^2 + 0 \times 8^1 + 1 \times 8^0\), totaling 64 + 0 + 1 = 65 in decimal.
Octal is handy in computing when grouping binary digits, as each octal digit corresponds to exactly three binary digits.

Hexadecimal Numbers

Hexadecimal is a base-16 number system, which uses digits 0-9 and the letters A-F (where A=10, B=11, ... F=15). This system is prevalent in computing because it is a compact way of representing binary-coded values.
To convert a decimal number into hexadecimal, you can use a similar method to octal conversion: divide the number by 16 and record the remainders. Write the remainders in reverse order. For example, decimal 255 converts to hexadecimal as \(FF_{16}\), indicating \(15 \times 16^1 + 15 \times 16^0\), equivalent to 240 + 15 = 255 in decimal.
Each hexadecimal digit corresponds to four binary digits, making hex a valuable tool for debugging and coding in areas like color representation in web design.

Decimal Conversion

Decimal conversion is the process of converting a number from the decimal (base-10) system to another system, such as binary, octal, or hexadecimal, and vice versa. Our daily numerical expressions are primarily in the decimal system, using digits 0-9.
To convert a number from decimal to another system, you apply division and calculate remainders using the target base until you can't divide anymore. To illustrate, converting decimal 10 to binary involves dividing 10 by 2 to get remainders, which produces \(1010_2\).
While converting back, each digit is multiplied by its base power, and the results are summed. Decimal conversion helps in bridging human-friendly numbers with machine-level operations.

Programming Loops

Programming loops are a fundamental concept in computer science, utilized for executing a block of code repeatedly until a specific condition is false. Loops are vital when working with number system conversions, such as iterating over a range of numbers to its various forms.
There are different types of loops like 'for', 'while', and 'do-while'. For example, using a 'for' loop in Python to iterate through numbers 1 to 256 for conversion statements looks like this: `for num in range(1, 257):`.
Loops efficiently automate tasks like converting a list of numbers from one base system to another. They allow programmers to handle large data sets without writing repetitive code, thus saving time and reducing errors.