Question

Write an application that inputs an integer containing only 0 s and \(1 s\) (i.e., a binary integer) and prints its decimal equivalent. [Hint: Use the remainder and division operators to pick off the binary number's digits one at a time, from right to left. In the decimal number system, the rightmost digit has a positional value of 1 and the next digit to the left a positional value of \(10,\) then \(100,\) then \(1000,\) and so on. The decimal number 234 can be interpreted as \(4^{*} 1+3^{*} 10+2^{*} 100 .\) In the binary number system, the rightmost digit has a positional value of \(1,\) the next digit to the left a positional value of \(2,\) then \(4,\) then \(8,\) and so on. The decimal equivalent of binary \(\left.1101 \text { is } 1^{*} 1+0^{*} 2+1^{*} 4+1^{*} 8, \text { or } 1+0+4+8 \text { or, } 13 .\right]\)

Short answer

To convert a binary number to its decimal equivalent, initialize a base value to 1 and a decimal value to 0. Extract each digit from right to left, continuously multiply the extracted digit by the base value, add the result to the decimal value, divide the binary number by 10, and multiply the base value by 2. Repeat until the binary number is fully consumed.

Step-by-step solution

Initialize variables

Define variables to store the decimal equivalent of the binary number, the current digit, and the base position. Initialize 'decimalValue' to 0, 'currentDigit' to 0, and 'base' to 1, which represents the positional value of the least significant digit in a binary number.

Input the binary number

Prompt the user to input a binary number ensuring the input consists only of 0's and 1's. Store this value in a variable 'binaryNumber'.

Extract digits and convert to decimal

Use a loop to extract each digit from the binary number starting from the rightmost digit. For each digit: a) Find the remainder of 'binaryNumber' divided by 10 to get the current digit. b) Multiply the current digit by 'base' and add this to 'decimalValue'. c) Divide 'binaryNumber' by 10 to remove the rightmost digit. d) Multiply 'base' by 2 to get the next positional value in binary (since binary is base 2). Continue this process until 'binaryNumber' is reduced to 0.

Output the decimal equivalent

Print out 'decimalValue', which now contains the decimal equivalent of the binary number.

Key concepts

Java Programming

Java is a versatile and powerful programming language that allows developers to create complex applications with relative ease. For problems like binary to decimal conversion, Java offers a robust set of tools and constructs that make the process straightforward.

In the context of our exercise, Java's primitive data types and operations are particularly useful. For instance, an int can be used to represent our binary number, and operations like division (/) and modulus (%) prove essential for the step-by-step extraction of each binary digit.

Additionally, Java provides a while or for loop construct that efficiently handles the repetitive task of digit extraction and accumulation. The loop continues to extract and process each digit of the binary number until the full conversion is achieved.

To facilitate user interaction, Java's Scanner class is convenient for obtaining input from the user. Our program would typically start with a prompt for the user to enter a binary number, after which the input is processed as outlined in the exercise solution.

Integer Manipulation

Handling integer values effectively is central to many programming tasks, including our exercise on converting binary to decimal. Integer manipulation in programming usually refers to performing operations such as addition, subtraction, multiplication, division, and modulus to achieve a particular outcome.

In our binary to decimal conversion, the modulus operation is key. It allows us to isolate the rightmost digit of the binary number (currentDigit = binaryNumber % 10;). After obtaining this digit, we multiply it by the current base value, which aligns with the binary number's positional properties, and add the result to the accumulating decimal value (decimalValue += currentDigit * base;).

The loop structure then proceeds to remove the processed digit by dividing the binary number by 10 (binaryNumber /= 10;) and prepares for the next iteration by doubling the base (base *= 2;). This doubled base reflects the binary system's base-2 progression, crucial for the conversion process.

Positional Number Systems

Understanding positional number systems is essential for grasping how binary to decimal conversion works. In these systems, the position of a digit in a number affects its value. The two most common positional systems are decimal (base 10) and binary (base 2).

In the decimal system, each position corresponds to a power of 10. Starting from the right, the first position represents 10 to the power of 0 (which is 1), the second position represents 10 to the power of 1 (which is 10), and so on.

The binary system works similarly but uses a base of 2. The rightmost position represents 2 to the power of 0 (which is 1), the next represents 2 to the power of 1 (which is 2), followed by 4, 8, and continuing in powers of 2.

This concept is used in our conversion exercise when we assign a base initially set to 1, and then double it after processing each binary digit. It mirrors the increasing positional values in the binary number system and ensures that every binary digit is attributed the correct decimal value during the conversion process.