Question

Match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 \(1010101+10101\) (binary addition)

Short answer

The solution is C. 1101010.

Step-by-step solution

Align the Numbers

First, align the two binary numbers for addition. Write the numbers vertically, with the numbers aligned to the right. \[\begin{array}{c}1010101 \+ 0010101 \\hline\end{array}\]

Add from the Right

Start adding the numbers from the rightmost digit (least significant bit). Remember that in binary addition:- 0 + 0 = 0- 1 + 0 = 1- 0 + 1 = 1- 1 + 1 = 0 (and carry over 1 to the next column)Adding from the right:- Add 1 + 1 = 0 (write 0, carry 1)- Next digit: 0 + 0 + 1 (carry) = 1 (write 1)- Next digit: 1 + 1 = 0 (write 0, carry 1)- Next digit: 0 + 0 + 1 (carry) = 1 (write 1)- Next digit: 1 + 1 = 0 (write 0, carry 1)- Next digit: 0 + 0 + 1 (carry) = 1 (write 1)- Last digit: 1 + 0 = 1 (write 1)\[\begin{array}{c}1010101 \+ 0010101 \\hline1101010 \\end{array}\]

Verify the Answer

Check that the resulting sum from our addition matches one of the given options. We found that the sum is 1101010. Compare this with the provided options: C. 1101010 Since it matches, this is the correct solution.

Key concepts

Binary Numbers

Binary numbers are the fundamental building blocks of binary arithmetic. They are a numerical system that uses only two digits, 0 and 1. This is different from the decimal system, which uses ten digits (0 to 9). Each digit in a binary number is referred to as a "bit," which is short for binary digit. In computers, binary numbers are used because they align with the digital nature of their operation, which is based on two states: on (1) and off (0).

• Binary numbers are read from right to left.
• The rightmost bit is known as the least significant bit (LSB).
• The leftmost bit is called the most significant bit (MSB).

Every bit position in a binary number represents a power of 2. Thus, understanding binary numbers is essential for engaging with binary arithmetic.

Binary Arithmetic

Binary arithmetic is the method of performing mathematical operations using binary numbers. This includes addition, subtraction, multiplication, and division. In the context of this problem, we focus on binary addition, which is similar to decimal addition but follows specific rules due to binary's limited digit set.
Binary arithmetic operations are foundational to computer processing and memory operations. By mastering binary arithmetic, one can delve deeper into how calculations are made at the lowest level within digital systems.

• Addition follows these rules: 0 + 0 = 0, 1 + 0 = 1, 0 + 1 = 1, and 1 + 1 = 0 (carry 1).
• Binary subtraction, multiplication, and division follow a similar set of rules, adjusted for binary logic.
• Understanding binary arithmetic allows for efficient problem-solving in computer science and engineering.

Problem-Solving Steps

Problem-solving in binary arithmetic often follows a structured approach to ensure accuracy and understanding. Let’s break down the exercise:
1. **Align the Numbers**: Start by writing the two binary numbers vertically, aligning them to the right. This ensures that the least significant bits are correctly aligned for accurate addition.
2. **Add from the Right**: Just like in decimal addition, begin adding from the rightmost digits. Follow the binary addition rules and carry over where necessary.
3. **Verify the Answer**: After summing the numbers, compare the result to the given options or problem requirements to confirm its correctness.
Using a step-by-step approach enables clear visualization and verification each time.

• Each step is essential for methodical progression in problem-solving.
• Rechecking steps can prevent mistakes and solidify understanding.
• Using structured steps is crucial for mastering binary operations and applying them in computing tasks.

Carry in Binary Addition

Carry in binary addition is an essential concept that occurs when two 1s are added together. In binary, when 1 + 1 equals 0, a carryover of 1 occurs to the next significant bit. Understanding how to handle this carry is crucial for accurate binary addition.
Let's look at how carry works in the given exercise: When adding the binary numbers, notice how each column where a 1 + 1 occurs, results in a carry. This carry influences the sum of the following column, just like in decimal addition but simplified to two base numbers.

• A carry results because the total "2" in base 2 translates to "10," meaning "0" in the current column and "1" carried over.
• Carries continue cascading to the left as long as two 1s are encountered.
• Practicing binary addition with carry reinforces a deeper understanding of how binary systems operate.

Mastering the carry in binary addition helps solve more complex arithmetic problems and deal with computer calculations more effectively.