What is the 8-bit two's complement for the following integer?
−53

To find the 8-bit two's complement for \(-53\), follow these steps:

### Step 1: Find the binary representation of \(53\) (positive integer)
First, convert \(53\) into binary:

\[
53 \div 2 = 26 \, \text{R}1 \\
26 \div 2 = 13 \, \text{R}0 \\
13 \div 2 = 6 \, \text{R}1 \\
6 \div 2 = 3 \, \text{R}0 \\
3 \div 2 = 1 \, \text{R}1 \\
1 \div 2 = 0 \, \text{R}1 
\]

So, \(53\) in binary is \(110101\). To represent it in 8 bits, pad with zeros:

\[
53 = 00110101_2
\]

### Step 2: Find the one's complement
To find the one's complement, invert all the bits (change \(0\)s to \(1\)s and \(1\)s to \(0\)s):

\[
\text{One's complement of } 00110101_2 = 11001010_2
\]

### Step 3: Add 1 to get the two's complement
Now, add \(1\) to the one's complement:

\[
11001010_2 + 1_2 = 11001011_2
\]

### Conclusion:
The 8-bit two's complement representation of \(-53\) is:

\[
11001011_2
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How do you convert a positive decimal to binary?
2. What is the process for calculating one's complement?
3. How do you check if an 8-bit binary number is negative in two's complement?
4. What is the two's complement representation for positive numbers?
5. How does two's complement handle overflow in 8-bit arithmetic?

**Tip:** Always ensure your final two's complement result is in the correct bit length (e.g., 8-bit, 16-bit) by padding with leading zeros if necessary!

What is the 8-bit two's complement for the following integer? −53

Learn how to find the 8-bit two's complement representation for −53 with a step-by-step solution. This problem covers binary number conversion, one's complement, and two's complement techniques. Perfect for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation