Add (-10)10 + (-12)10 using 5-bit signed 2’s complement format. Check for overflow. If the over follow happens, modify to obtain the correct result and show this result in the decimal format.

To solve the problem of adding (-10)₁₀ + (-12)₁₀ using 5-bit signed 2's complement format, we'll follow these steps:

### 1. Convert the decimal numbers to 5-bit signed 2's complement:

#### (-10)₁₀ in 5-bit 2's complement:
1. The positive binary of 10 in 5 bits: 
   - \(10_{10} = 01010_2\)
2. To find the 2's complement, invert the bits and add 1:
   - Inverted: \(10101_2\)
   - Add 1: \(10101_2 + 1 = 10110_2\)

So, (-10)₁₀ in 5-bit 2's complement is:
   \[
   (-10)_{10} = 10110_2
   \]

#### (-12)₁₀ in 5-bit 2's complement:
1. The positive binary of 12 in 5 bits:
   - \(12_{10} = 01100_2\)
2. To find the 2's complement, invert the bits and add 1:
   - Inverted: \(10011_2\)
   - Add 1: \(10011_2 + 1 = 10100_2\)

So, (-12)₁₀ in 5-bit 2's complement is:
   \[
   (-12)_{10} = 10100_2
   \]

### 2. Add the two 5-bit signed numbers:
\[
10110_2 + 10100_2
\]

Performing the binary addition:

\[
\begin{array}{c@{}c@{}c@{}c@{}c@{}c}
 & 1 & 0 & 1 & 1 & 0 \\
+ & 1 & 0 & 1 & 0 & 0 \\
\hline
 1 & 0 & 1 & 0 & 1 & 0 \\
\end{array}
\]

Result:
\[
1 01010_2
\]

### 3. Check for overflow:
Since this is a 5-bit system, the leftmost 1 (the 6th bit) is an overflow, meaning the sum cannot be represented correctly in 5 bits.

### 4. Correct the result:
Ignoring the overflow bit, the result is:
\[
01010_2
\]

Convert this back to decimal:
\[
01010_2 = 10_{10}
\]

This is the **wrong result** because of the overflow.

### 5. Correct result (adjusting the sum):
Since overflow occurred, we need to interpret the overflowed result differently. The sum of (-10)₁₀ + (-12)₁₀ is supposed to be:
\[
(-10) + (-12) = -22_{10}
\]

Now, we need to represent -22 in 5-bit signed 2's complement format.

However, -22 cannot be represented in 5 bits since the range of 5-bit signed 2's complement is from -16 to +15. Therefore, **it is impossible to represent the correct result (-22) in 5-bit format**.

### Conclusion:
There is an overflow because the sum (-22) is out of the range for 5-bit signed numbers. The correct sum of (-10) + (-12) is:
\[
(-22)_{10}
\]

This cannot be represented in a 5-bit signed 2's complement system.

---

Would you like more details or have any further questions?

### Expanding Questions:
1. What is the range of numbers that can be represented in 5-bit signed 2's complement?
2. How do you detect overflow in binary addition of 2's complement numbers?
3. Can overflow occur when adding two positive or two negative numbers?
4. How do you extend 5-bit representation to a larger number of bits (e.g., 8-bit or 16-bit)?
5. What other ways can binary arithmetic errors manifest besides overflow?

#### Tip:
When detecting overflow in 2's complement addition, always check if two positive numbers give a negative result or two negative numbers give a positive result.

Add (-10)₁₀ + (-12)₁₀ using 5-bit signed 2’s complement format. Check for overflow. If the overflow happens, modify to obtain the correct result and show this result in the decimal format.

Learn how to add (-10)₁₀ + (-12)₁₀ using 5-bit signed 2’s complement format and detect overflow. Follow a step-by-step process to convert decimal numbers to binary, perform binary addition, and check for overflow in 2’s complement representation. This is suitable for students in Grades 10-12.

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