Two's complement representation in ARM Architecture - Step-by-Step Execution

Concept Flow - Two's complement representation

Start with binary number

↓

Check if number is positive or negative

↓

Use as is

↓

Add 1 to inverted bits

↓

Result is two's complement

↓

Use in calculations

This flow shows how to convert a number to two's complement: positive numbers stay the same, negative numbers are inverted and incremented by one.

Execution Sample

ARM Architecture

Original binary: 00000101
Is negative? No
Use as is: 00000101

Original binary: 00000101
Is negative? Yes
Invert bits: 11111010
Add 1: 11111011

Shows step-by-step how 5 (positive) and -5 (negative) are represented in 8-bit two's complement.

Analysis Table

Step	Input Binary	Is Negative?	Action	Result Binary
1	00000101	No	Use as is	00000101
2	00000101	Yes	Invert bits	11111010
3	11111010	Yes	Add 1	11111011
4	-	-	-	Stop

💡 Step 4: Conversion complete, two's complement representation obtained.

State Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	Final
Input Binary	00000101	00000101	00000101	11111010	11111011
Is Negative?	N/A	No	No	Yes	Yes
Result Binary	N/A	00000101	N/A	11111011	11111011

Key Insights - 2 Insights

Why do we invert bits and add 1 for negative numbers?

Inverting bits flips all bits, and adding 1 completes the two's complement process, allowing negative numbers to be represented and used in binary arithmetic correctly, as shown in steps 2 and 3 of the execution_table.

Why is the positive number used as is without changes?

Positive numbers have a leading 0 bit and do not need conversion; they are represented directly, as shown in step 1 of the execution_table.

Visual Quiz - 3 Questions

Look at the execution_table, what is the result binary after adding 1 to the inverted bits?

A 00000101

B 11111011

C 11111010

D 00000001

💡 Hint

Check step 3 in the execution_table under Result Binary.

At which step does the process stop according to the execution_table?

A Step 2

B Step 3

C Step 4

D Step 1

💡 Hint

Look for the row with 'Stop' in the Action column.

If the input binary is positive, what action is taken according to the execution_table?

A Use as is

B Invert bits

C Add 1

D Stop

💡 Hint

Refer to step 1 where Is Negative? is No.

Concept Snapshot

Two's complement represents negative numbers by inverting all bits and adding 1.
Positive numbers remain unchanged.
This allows easy binary addition and subtraction.
The leftmost bit indicates sign: 0 for positive, 1 for negative.
Used widely in ARM and other architectures.

Full Transcript

Two's complement is a way to represent negative numbers in binary. If the number is positive, we keep its binary form as is. If the number is negative, we invert all bits and then add 1 to get its two's complement form. This method allows computers to perform subtraction and addition using the same binary addition process. The leftmost bit shows the sign: 0 means positive, 1 means negative. This is commonly used in ARM architecture and many other systems.

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