(Printing the Decimal Equivalent of a Binary Number) 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 \(\left.1101 \text { is } 1^{*} 1+0^{*} 2+1^{*} 4+1^{*} 8, \text { or } 1+0+4+8 \text { or, } 13 .\right]\)

Step by step solution

01

- Understand the Binary System

The binary system is base-2, using only digits 0 and 1. Each digit is a power of 2, based on its position from the right, starting at 0. For example, the third position from the right is the value of 2 raised to the power of 2 which is 4.

02

- Process the Input Binary Number

Start from the rightmost digit, keep track of the position starting from 0. Initialize the decimal number to 0. For each digit in the binary number, multiply the digit by 2 raised to the power of its position and add it to the decimal number.

03

- Calculate Each Digit's Contribution

For each binary digit, calculate the contribution to the decimal equivalent by multiplying the binary digit by 2 raised to the power of its position index. If the binary digit is 1, add the result to the decimal equivalent; if it is 0, add nothing.

04

- Complete the Calculation for the Entire Binary Number

Repeat Step 3 for each digit in the binary number. Keep summing the result until all binary digits have been processed.

05

- Output the Result

After processing all the binary digits, the computed sum is the decimal equivalent of the binary number.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Number System

The binary number system is a cornerstone of computing, relying on just two numerals: 0 and 1. Each digit in this base-2 system is called a bit and represents a power of two, with the rightmost bit having the positional value of 1, and each bit to the left doubling in value—2, 4, 8, and so on. For instance, the binary number '1010' can be broken down as
\(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\), which simplifies to
\(8 + 0 + 2 + 0\). When these values are added together, they yield a decimal result of 10. The efficient binary-to-decimal conversion is not just academic but critical in programming, circuit design, and various digital systems where binary data needs to be understood or displayed in a more human-friendly format.

Decimal Number System

Conversely, the decimal number system, also known as base-10, is the standard system for denoting integer and non-integer numbers. It's the most familiar numbering system for humans, due to our ten fingers, and uses digits from 0 to 9. In this system, each digit's value depends on its position, or place value, which is a power of 10. The number '234', for example, is calculated as
\(4 \times 10^0 + 3 \times 10^1 + 2 \times 10^2\), resulting in
\(4 + 30 + 200\), or 234 in decimal. Understanding the decimal system is essential, as it's the basis for most calculations in daily life, as well as in scientific and financial computations.

Integer Data Type

In programming, the integer data type refers to a data type that holds whole numbers without fractional components. Integers can be both positive and negative, defined by a specific range determined by the number of bits allocated for that data type in memory. For example, a 32-bit signed integer can represent values from
\(-2^{31}\) to
\(2^{31}-1\). Understanding integers in programming is vital since they are used extensively to control loops, index arrays, and represent discrete values in algorithms. Unlike real numbers, there is no danger of rounding errors with integers, making them more reliable for count-based calculations.

Arithmetic Operations in Programming

Arithmetic operations such as addition, subtraction, multiplication, and division are fundamental in programming. These operations work similarly to those in mathematics but have some intricacies due to data types and operator precedence. When performing calculations, it's crucial that data types match to prevent errors or unintended type conversions, known as typecasting. For example, dividing two integers in some programming languages may result in an integer, truncating the decimal part and potentially leading to a loss of information. Understanding the quirks of arithmetic operations in specific programming contexts is critical for writing accurate and efficient code.