6.	Do the following arithmetic as if these were five-bit signed representations and indicate if overflow occurs and, if so, why.  Note:  Remember that you want to add.  So, for signed subtraction, convert the subtrahend (the number being subtracted) to its 2's complement and add it.  
 
     a.  10110 + 01101
     b.  11001 + 00101
     c.  10110 - 01101
     d.  11111 - 01011


When performing these arithmetic operations with five-bit signed representations, it's important to remember that:

- In five-bit signed numbers, the first bit is the sign bit:
  - 0 = positive
  - 1 = negative
- The range of values for five-bit signed numbers is \(-16\) to \(+15\).
  - Positive numbers range from \(00000\) (0) to \(01111\) (+15)
  - Negative numbers range from \(11111\) (-1) to \(10000\) (-16)

We'll convert each binary number to its decimal equivalent, perform the operations, and check for overflow. Overflow happens when the result is outside the range \([-16, 15]\).

### a) \(10110 + 01101\)

1. Convert to decimal:
   - \(10110 = -10\) (two's complement)
   - \(01101 = +13\)
   
2. Perform the addition:
   \(-10 + 13 = 3\)

3. Check for overflow:
   - The result, \(3\), is within the valid range \([-16, 15]\), so **no overflow**.

**Result: \(3\), no overflow.**

### b) \(11001 + 00101\)

1. Convert to decimal:
   - \(11001 = -7\) (two's complement)
   - \(00101 = +5\)
   
2. Perform the addition:
   \(-7 + 5 = -2\)

3. Check for overflow:
   - The result, \(-2\), is within the valid range \([-16, 15]\), so **no overflow**.

**Result: \(-2\), no overflow.**

### c) \(10110 - 01101\)

For subtraction, convert the subtrahend (\(01101\)) to its two's complement form and add it.

1. Convert to decimal:
   - \(10110 = -10\) (two's complement)
   - \(01101 = +13\)

2. Find two's complement of \(01101\):
   - Flip the bits: \(10010\)
   - Add 1: \(10011 = -13\) (two's complement)

3. Perform the addition:
   \(-10 + (-13) = -23\)

4. Check for overflow:
   - The result, \(-23\), is **outside** the valid range \([-16, 15]\), so **overflow occurs**.

**Result: Overflow occurs because \(-23\) is out of range.**

### d) \(11111 - 01011\)

For subtraction, convert the subtrahend (\(01011\)) to its two's complement form and add it.

1. Convert to decimal:
   - \(11111 = -1\) (two's complement)
   - \(01011 = +11\)

2. Find two's complement of \(01011\):
   - Flip the bits: \(10100\)
   - Add 1: \(10101 = -11\) (two's complement)

3. Perform the addition:
   \(-1 + (-11) = -12\)

4. Check for overflow:
   - The result, \(-12\), is within the valid range \([-16, 15]\), so **no overflow**.

**Result: \(-12\), no overflow.**

---

Would you like any details or further clarifications? Here are some related questions for further exploration:

1. What are the benefits of using two's complement for signed number representation?
2. How do you detect overflow in signed binary addition?
3. What would happen if we used 6-bit signed numbers instead of 5-bit?
4. Can you explain the difference between one's complement and two's complement representation?
5. How is two's complement subtraction different from addition?

**Tip:** In two's complement, to negate a binary number, you simply flip the bits and add 1, making arithmetic operations like subtraction simpler!

Do the following arithmetic as if these were five-bit signed representations and indicate if overflow occurs and, if so, why.  Note:  Remember that you want to add.  So, for signed subtraction, convert the subtrahend (the number being subtracted) to its 2's complement and add it. 

 a.  10110 + 01101
 b.  11001 + 00101
 c.  10110 - 01101
 d.  11111 - 01011

Explore how to perform arithmetic using 5-bit signed binary numbers and detect overflow. Includes detailed solutions for binary addition and subtraction using two's complement representation. Perfect for high school students in Grades 10-12.

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