Question

Match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 \(1100111-111\) (binary subtraction)

Short answer

The binary subtraction is 1010000 (error noted). Only option close is D, but actual solution doesn't match given options accurately.

Step-by-step solution

Understanding Binary Subtraction

Binary subtraction works similarly to decimal subtraction but uses only the digits 0 and 1. When subtracting binary numbers, you may need to perform borrowing like in decimal subtraction.

Lining Up the Numbers

Align the two binary numbers for subtraction:\[\begin{array}{c}1100111 \-111 \\hline\end{array}\]

Perform the Subtraction

Start subtracting from the right:1. Subtract the rightmost column: 1 - 1 = 0.2. Move left. Since 1 - 1 = 0, write down 0.3. Since there are no more digits left in the number to subtract from, bring down the rest of the number:\[\begin{array}{c}1100111 \-111 \\hline1010000\end{array}\]

Confirming Result

Check the result 1010000 against the options provided: A. 10001100, B. 10011110, C. 1101010, D. 1100000, E. 1010001, F. 1111000. The closest we get to the determined result is option D, 1100000. But the correct number obtained from subtraction actually appears to be a mistake or misalignment in problem interpretation.

Key concepts

Binary Numbers

Binary numbers form the foundation of computer data processing. In binary, we only use two digits: 0 and 1. This might seem simple, but it is incredibly powerful. Just like how we use the decimal system every day, which goes from 0 to 9, binary counts start from 0, go to 1, and then move to the next digit as 10, which actually means 'two' in decimal terms.
Binary numbers are crucial because they align with the on/off state of computer systems. Each binary digit, or bit, represents two possible states, making it a perfect fit for digital circuits. Computers use this system to perform complex calculations and processes.
In computers, a bit is the smallest unit of data. Multiple bits together form larger units, such as bytes (8 bits). Larger groups, like kilobytes, megabytes, etc., are just made up of more and more bits, allowing for massive amounts of data to be represented in binary form.

Borrow in Subtraction

Borrowing in binary subtraction is similar to borrowing in decimal subtraction, but there's a twist. When you subtract one binary number from another and the number above is smaller, you must "borrow" from the next higher bit position.
For example, if you need to subtract 1 from 0, which can't be done under normal circumstances, you borrow from the next column. The borrowed bit from the next column adds 2 to the current column since we're in base 2. This means 0 becomes 10.

• This borrowed 1 from the next higher bit is equivalent to adding 2 to our computation for that column.
• Performing the operation in binary requires us to move sequentially from right to left, managing all borrows as we go.

The borrowing process in binary can initially be tricky, but with practice, it becomes as routine as its decimal counterpart.

Binary Math Education

Learning binary math is essential for students venturing into computer science and technology fields. Binary math provides the basis for understanding how computers perform operations. It involves concepts such as addition, subtraction, multiplication, and division, all executed in binary form.
To effectively teach binary math, students should engage with real-world examples. Begin with simple binary addition before advancing to subtraction where borrowing becomes necessary.
Employing visual aids can greatly help understanding. Tools like showing operations on a grid or using software for digital simulation can offer clarity. These visual exercises reinforce retention and comprehension. Practicing these operations not only enhances computational skills but also builds a solid foundation for advanced computer science topics.

Number Systems in Computer Science

Computer science heavily relies on different number systems beyond just binary. These systems, such as octal (base 8), decimal (base 10), and hexadecimal (base 16), play vital roles in various computing processes.
While binary is used for direct instruction-processing and logical output, hexadecimal simplifies binary representation. It's particularly useful due to its compact form, which is easier for humans to read and interpret than long binary strings.

• Each hexadecimal digit represents four binary digits (bits), making it a streamlined representation that helps in efficient programming and debugging.
• Octal is sometimes used, especially in older systems, due to its compatibility through simple conversion from binary, by grouping three binary digits together.

Understanding these systems and how to effectively switch between them is crucial for tackling complex computing challenges. They form the backbone of understanding how data is managed, processed, and manipulated within various software and hardware environments.