Two's complement representation in ARM Architecture - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding Two's Complement Range

What is the range of values that can be represented by an 8-bit two's complement number?

A 0 to 255

B -256 to 255

C -127 to 128

D -128 to 127

💡 Hint

Remember that two's complement uses one bit for sign and the rest for magnitude.

An 8-bit two's complement number uses 1 bit for sign and 7 bits for magnitude, allowing values from -128 to 127.

📋 Factual

intermediate

Two's Complement Negative Conversion

How do you find the two's complement (negative) of a binary number in ARM architecture?

A Invert all bits only

B Invert all bits and add 1

C Add 1 only

D Subtract 1 and invert all bits

💡 Hint

Think about how to represent negative numbers in binary.

To get the negative of a number in two's complement, invert all bits and then add 1.

🔍 Analysis

advanced

Interpreting Binary Values

Given the 8-bit binary number 11110110, what decimal value does it represent in two's complement?

A -10

B 246

C -6

D 250

💡 Hint

Check the sign bit and convert accordingly.

The sign bit is 1, so it's negative. Invert bits and add 1: 00001001 + 1 = 00001010 (10 decimal). So the value is -10.

❓ Comparison

advanced

Difference Between Sign-Magnitude and Two's Complement

Which of the following statements correctly describes a key difference between sign-magnitude and two's complement representations?

A Two's complement has only one representation for zero, sign-magnitude has two

B Sign-magnitude uses less bits than two's complement for the same range

C Two's complement cannot represent negative numbers

D Sign-magnitude uses inversion and addition to find negatives

💡 Hint

Think about how zero is represented in both systems.

Sign-magnitude represents zero as both positive and negative zero, while two's complement has a single zero representation.

❓ Reasoning

expert

Overflow Detection in Two's Complement Addition

In ARM architecture, when adding two 8-bit two's complement numbers, which condition indicates an overflow has occurred?

A When the carry out of the least significant bit is 1

B When the sum is zero

C When the carry into the sign bit differs from the carry out of the sign bit

D When both operands are negative

💡 Hint

Overflow happens when the sign bit result is incorrect due to carry mismatch.

Overflow in two's complement addition occurs if the carry into the sign bit is not equal to the carry out of the sign bit.

Practice

1. What is the main purpose of using two's complement in ARM architecture?

easy

A. To represent both positive and negative integers in binary

B. To store only positive numbers efficiently

C. To convert decimal numbers to hexadecimal

D. To encrypt data for security

Solution

1. Step 1: Understand number representation in computers

   Computers use binary to represent numbers, but need a way to show both positive and negative values.

2. Step 2: Role of two's complement

   Two's complement allows representing negative numbers by inverting bits and adding one, enabling simple arithmetic.

3. Final Answer:

   To represent both positive and negative integers in binary -> Option A

4. Quick Check:

   Two's complement = Signed integer representation [OK]

Hint: Two's complement = signed numbers in binary [OK]

Common Mistakes:

• Thinking it only stores positive numbers
• Confusing with encryption or encoding
• Assuming it converts decimal to hex

2. Which of the following is the correct 4-bit two's complement representation of -3?

easy

A. 1101

B. 0011

C. 1011

D. 1110

Solution

1. Step 1: Find binary of positive 3 in 4 bits

   3 in binary is 0011.

2. Step 2: Calculate two's complement for -3

   Invert bits: 1100, then add 1: 1101.

3. Final Answer:

   1101 -> Option A

4. Quick Check:

   -3 in 4-bit two's complement = 1101 [OK]

Hint: Invert bits and add 1 for negative numbers [OK]

Common Mistakes:

• Using positive binary without complement
• Adding instead of inverting bits first
• Confusing bit length or dropping bits

3. Given an 8-bit two's complement number 11111010, what decimal value does it represent?

medium

A. 22

B. -22

C. 250

D. -6

Solution

1. Step 1: Identify if number is negative

   Most significant bit is 1, so number is negative.

2. Step 2: Find magnitude by two's complement

   Invert bits: 00000101, add 1: 00000110 (6 decimal).

3. Final Answer:

   -22 -> Option B

4. Quick Check:

   11111010 two's complement = -22 [OK]

Hint: If MSB=1, invert bits +1 to find negative value [OK]

Common Mistakes:

• Reading as unsigned binary (250)
• Ignoring two's complement rules
• Mistaking bit inversion step

4. A programmer wrote this 4-bit two's complement number: 1001. They claim it equals +9. What is the error?

medium

A. 1001 equals +9 correctly

B. 1001 is invalid in two's complement

C. 1001 represents -7, not +9

D. Two's complement cannot represent 4-bit numbers

Solution

1. Step 1: Check sign from MSB

   MSB is 1, so number is negative in two's complement.

2. Step 2: Calculate decimal value

   Invert bits: 0110, add 1: 0111 (7 decimal), so value is -7.

3. Final Answer:

   1001 represents -7, not +9 -> Option C

4. Quick Check:

   1001 two's complement = -7, not +9 [OK]

Hint: MSB 1 means negative, check two's complement value [OK]

Common Mistakes:

• Assuming binary is always positive
• Ignoring sign bit meaning
• Confusing two's complement with unsigned

5. In ARM 8-bit two's complement, what is the result of adding 00010110 (+22) and 11101010 (-22)?

hard

A. 00011100 (28)

B. 11111111 (-1)

C. Overflow error

D. 00000000 (0)

Solution

1. Step 1: Understand the numbers

   00010110 is +22, 11101010 is -22 in two's complement.

2. Step 2: Add the binary numbers

   00010110 + 11101010 = 00000000 (ignoring carry out), which is 0.

3. Final Answer:

   00000000 (0) -> Option D

4. Quick Check:

   +22 + (-22) = 0 [OK]

Hint: Adding number and its negative gives zero [OK]

Common Mistakes:

• Expecting overflow when adding opposites
• Misreading negative number value
• Ignoring carry out in two's complement addition