Two's complement representation in ARM Architecture - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is two's complement representation?

Two's complement is a way to represent positive and negative whole numbers in binary. It allows easy addition and subtraction by encoding negative numbers as the binary of their positive value, flipped and plus one.

beginner

How do you find the two's complement of a binary number?

To find the two's complement, first invert all bits (change 0 to 1 and 1 to 0), then add 1 to the result.

intermediate

Why is two's complement preferred over sign-magnitude representation in computers?

Two's complement simplifies arithmetic operations because addition and subtraction work the same for positive and negative numbers, avoiding separate rules for signs.

beginner

What is the range of values for an 8-bit two's complement number?

An 8-bit two's complement number can represent values from -128 to 127.

beginner

How does two's complement handle zero?

In two's complement, zero is represented by all bits set to 0. There is only one zero, unlike sign-magnitude which has positive and negative zero.

What is the first step to find the two's complement of a binary number?

A Invert all bits

B Add 1

C Subtract 1

D Flip the sign bit only

The first step is to invert all bits (change 0 to 1 and 1 to 0). Then you add 1.

What decimal number does the 8-bit two's complement binary 11111111 represent?

A -1

B -128

C 127

D 255

In two's complement, 11111111 represents -1.

Why is two's complement useful in computer arithmetic?

A It uses fewer bits

B It represents only positive numbers

C It allows the same addition rules for positive and negative numbers

D It avoids using binary

Two's complement allows addition and subtraction to work the same way for positive and negative numbers.

What is the range of a 4-bit two's complement number?

A 0 to 15

B -7 to 8

C -15 to 0

D -8 to 7

A 4-bit two's complement number ranges from -8 to 7.

How many zeros exist in two's complement representation?

A Two

B One

C None

D Depends on the bit length

Two's complement has only one zero, represented by all bits zero.

Explain how to convert a positive binary number to its negative equivalent using two's complement.

Think about flipping bits and adding one to get the negative.

Describe why two's complement is preferred for representing signed integers in ARM architecture.

Focus on how it makes math easier for the processor.

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