Question

Input an integer containing only 0 s and 1 s (i.e., a "binary" integer) and print its decimal equivalent. Use the modulus and division operators to pick off the "binary" number's digits one at a time from right to left. Much as in the decimal number system, where the rightmost digit has a positional value of 1, the next digit left has a positional value of \(10,\) then \(100,\) then \(1000,\) and so on, in the binary number system the rightmost digit has a positional value of \(1,\) the next digit left has a positional value of \(2,\) then \(4,\) then \(8,\) and so on. Thus the decimal number 234 can be interpreted as \(2 * 100+3 * 10\) \(+4 * 1 .\) The decimal equivalent of binary 1101 is \(1 * 1+0 * 2+1 * 4+1 *\) 8 or \(1+0+4+8,\) or \(13 .\) [Note: The reader not familiar with binary numbers might wish to refer to Appendix D.]

Short answer

The decimal equivalent of binary 1101 is 13.

Step-by-step solution

Extract the Rightmost Digit

Start by extracting the rightmost digit of the binary number. You can achieve this by using the modulus operator. For example, if the binary number is 1101, performing 1101 % 10 will give you 1. This is the first digit we'll convert.

Determine Positional Value

Recognize that each binary digit has a positional value, starting from 1 for the rightmost digit. We will iterate over the digits, and this value will be a power of 2. For position 0, the positional value is \(2^0 = 1\).

Convert Rightmost Digit to Decimal

Multiply the extracted digit from Step 1 by its corresponding positional value to convert it to a decimal equivalent. For example, \(1 \times 1 = 1\).

Update Total Decimal Value

Add the result from Step 3 to a running total which represents the decimal equivalent. Initialized at 0, after the first conversion it becomes 1.

Remove the Rightmost Digit

Remove the rightmost digit by dividing the binary number by 10. This simplifies the number, making the next digit the new rightmost digit. Using the example, 1101 // 10 gives 110.

Continue for Next Digits

Repeat Steps 1 to 5 for the remaining digits, updating the positional value to the next power of 2. For the binary number 110 (the current number after removing the last digit), extract the digit (0), determine the position (2^1 = 2), and update the total decimal value accordingly.

Complete the Process

Continue this process until there are no more digits left, ensuring each is converted and contributes to the total decimal value. For 1101, continue this for binary digits 0, 1, and 1, resulting in the decimal number.

Key concepts

Modulus and Division Operators

When converting a binary number to a decimal, the process of extracting digits is crucial. We achieve this by utilizing the modulus and division operators. The **modulus operator** ( %) is used to obtain the remainder when one number is divided by another. In the context of binary conversion, using the modulus operator helps us extract the rightmost digit of the binary number. For instance, when you divide a binary number (e.g., 1101) by 10, 1101 % 10 gives you the remainder, which is the rightmost digit (1, in this example).

**Division operator** (//) complements this process by removing the rightmost digit. By performing an integer division of the binary number by 10 (e.g., 1101 // 10), we eliminate the extracted digit, leaving the rest of the number (e.g., 110). This updated number is then considered for the extraction of the next rightmost digit. By repeating these steps, you can isolate each digit of the binary number from right to left.

Binary Number System

The binary number system is a base-2 numeral system, utilizing only two symbols: 0 and 1. Each binary digit represents a power of two, which is key to converting a binary number into its decimal form. The structure of a binary number is such that the **rightmost digit** is assigned the power of 0, or 2^0, equating to 1 in positional value. The **next digit to the left** is 2^1, equating to 2, then 2^2 equating to 4, and so forth, doubling with each step to the left.

Understanding the binary number system is crucial for conversion tasks, as it allows us to assign each digit its specific positional value. This sequence of powers of two forms the foundation of calculating the decimal equivalent, where the binary number is essentially a sum of these powers multiplied by the binary digits. This system contrasts with the decimal system, where each positional value is a power of 10.

• The rightmost digit has positional value 1.
• The next digit's positional value is 2.
• This sequence continues: 4, 8, 16, and so on.

Decimal Equivalent Calculation

Calculating the decimal equivalent of a binary number involves combining each digit's value in the binary system with its corresponding positional value. Here's how you perform this calculation:
1. **Extract the Digit**: As described, begin with the rightmost digit using the modulus operator.
2. **Determine the Positional Value**: Using powers of two, assign the correct value to each digit based on its position. For instance, in binary number 1101, the first digit has a positional value of 2^0, the second 2^1, and so on.
3. **Multiply for Individual Values**: Multiply each binary digit by its respective positional value. This calculates the decimal contribution of each digit separately.

• For example, the binary "1101":
  • 1 (from rightmost) × 1 (positional value) = 1
  • 0 × 2 = 0
  • 1 × 4 = 4
  • 1 × 8 = 8

4. **Combine Results**: Sum these products to find the total decimal value. The sum of all the values calculated is the decimal equivalent of the binary number. Using the example, 1 + 0 + 4 + 8 results in the decimal number 13.

By following these steps consistently, you ensure each binary number translates accurately to its decimal form.