Two's complement representation in ARM Architecture

Introduction

Two's complement is a way to represent positive and negative whole numbers in computers. It helps the computer do math with both positive and negative numbers easily.

When you want to store negative and positive numbers in a computer's memory.

When performing arithmetic operations like addition or subtraction on signed numbers.

When designing or understanding how processors like ARM handle signed integers.

When debugging or reading low-level machine code that uses signed numbers.

When converting between binary and decimal numbers including negatives.

Core Concept

ARM Architecture

For an n-bit number:
- Positive numbers: same as normal binary.
- Negative numbers: invert all bits of the positive number, then add 1.

The leftmost bit is the sign bit: 0 means positive, 1 means negative.

Two's complement allows one unique representation of zero.

Key Points

Positive 5 is just normal binary. Negative 5 is found by flipping bits of 5 (00000101 -> 11111010) and adding 1 -> 11111011.

ARM Architecture

8-bit two's complement of +5: 00000101
8-bit two's complement of -5: 11111011

In 4 bits, the smallest number is -8 (1000) and largest is +7 (0111). Negative 3 is 1101.

ARM Architecture

4-bit two's complement range: -8 to +7
Example: -3 in 4-bit: 1101

Detailed Explanation

This function converts a number to its two's complement binary string for a given bit size. It shows how positive numbers stay the same, and negative numbers are converted by adding 2^bits.

ARM Architecture

def twos_complement(value: int, bits: int) -> str:
    if value >= 0:
        return format(value, f'0{bits}b')
    else:
        return format((1 << bits) + value, f'0{bits}b')

# Examples
print(twos_complement(5, 8))   # Positive 5
print(twos_complement(-5, 8))  # Negative 5
print(twos_complement(-1, 4))  # Negative 1 in 4 bits
print(twos_complement(-8, 4))  # Smallest 4-bit number

Important Notes

Two's complement makes addition and subtraction simpler because the same hardware can handle both positive and negative numbers.

Overflow can happen if a number is too big or too small for the chosen bit size.

ARM processors use two's complement for signed integer arithmetic.

Summary

Two's complement is a method to represent signed integers in binary.

Negative numbers are formed by inverting bits and adding one.

This system simplifies arithmetic and is widely used in computer processors like ARM.

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