Question

Match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 \(1111111-111\) (binary subtraction)

Short answer

The answer is option F: 1111000.

Step-by-step solution

Convert 111 to Decimal

First, we convert the binary number 111 to its decimal equivalent. We start from the rightmost digit. The rightmost place is 2^0, the middle is 2^1, and the leftmost is 2^2. So, 111 in binary is calculated as \(1\times2^2 + 1\times2^1 + 1\times2^0 = 4 + 2 + 1 = 7\). Thus, 111 is 7 in decimal.

Convert 1111111 to Decimal

Now, convert 1111111 (which is the number from which we subtract) from binary to decimal. Calculating from right to left, \(1111111_2\) becomes \(1\times2^6 + 1\times2^5 + 1\times2^4 + 1\times2^3 + 1\times2^2 + 1\times2^1 + 1\times2^0\), resulting in \(64 + 32 + 16 + 8 + 4 + 2 + 1 = 127\). So, 1111111 is 127 in decimal.

Perform Decimal Subtraction

Subtract the two decimal numbers. Subtract 7 from 127. So, \(127 - 7 = 120\).

Convert the Result Back to Binary

To convert 120 from decimal back to binary, divide by 2 repeatedly and note the remainders. 120 divided by 2 is 60 (remainder 0), 60 divided by 2 is 30 (remainder 0), 30 divided by 2 is 15 (remainder 0), 15 divided by 2 is 7 (remainder 1), 7 divided by 2 is 3 (remainder 1), 3 divided by 2 is 1 (remainder 1), and 1 divided by 2 is 0 (remainder 1). Therefore, arranging the remainders from last to first, \(120_{10} = 1111000_2\).

Match with Given Options

Finally, compare the binary result 1111000 to the provided options. Option F corresponds to 1111000. Therefore, the solution to the binary subtraction \(1111111 - 111\) is option F.

Key concepts

Binary to Decimal Conversion

Binary to decimal conversion is an essential part of understanding binary arithmetic. Binary numbers use base 2, while decimal numbers use base 10. In binary, each digit represents a power of two, starting from the rightmost position.
To convert a binary number to decimal, one must multiply each digit by 2 raised to the power of its position (counting from right starting at zero), and then sum these results.
For example, the binary number 111 can be converted to decimal by calculating:

• 1 in the rightmost position: \(1 \times 2^0 = 1\)
• 1 in the next position to the left: \(1 \times 2^1 = 2\)
• 1 in the leftmost position: \(1 \times 2^2 = 4\)

Adding these values gives 7, meaning 111 is equal to 7 in decimal form. This process is fundamental in computing where binary code is translated into a more understandable format for humans.

Decimal to Binary Conversion

Converting decimal numbers to binary is the process of finding how binary numbers can represent a decimal value. This is done by dividing the decimal number repeatedly by 2 and keeping track of the remainders.
To better understand this, let’s convert 120 from decimal to binary. Divide 120 by 2:

• 120 ÷ 2 = 60, remainder 0
• 60 ÷ 2 = 30, remainder 0
• 30 ÷ 2 = 15, remainder 0
• 15 ÷ 2 = 7, remainder 1
• 7 ÷ 2 = 3, remainder 1
• 3 ÷ 2 = 1, remainder 1
• 1 ÷ 2 = 0, remainder 1

Then, read the remainders from last to first to get 1111000. This method aligns perfectly with computer science since digital systems use binary numbers to process all calculations. Understanding this conversion is crucial for students learning computing.

Binary Arithmetic

Binary arithmetic is similar to arithmetic in the decimal system but only uses the digits 0 and 1. Basic operations like addition and subtraction follow specific rules that are useful in computer science.
For subtraction, a useful technique is binary subtraction using borrow, similar to borrowing in decimal subtraction. Consider the binary expression 1111111 minus 111:
Convert both numbers to decimal: 1111111 is 127 and 111 is 7. Subtract in decimal: 127 - 7 = 120. Convert back to binary to get 1111000.
The entire operation ensures precise manipulation of binary data, critical for all digital computations.

Education in Computer Science

Education in computer science involves learning how computers perform tasks using fundamental concepts such as binary arithmetic. Students are taught various methods to convert, calculate, and manipulate data, focusing on sets of logical rules or algorithms.
Understanding binary subtraction and conversions allows students to comprehend how machines interpret complex processes through simple binary operations. This educational journey bridges foundational knowledge to practical application in technology.
Efforts to demystify these concepts are essential for developing critical thinking and problem-solving skills, making computer science a diverse and rewarding field to explore.