Addition of Binary Numbers

Addition of Binary Numbers (Adding Two Positive 8-bit Numbers)

Binary Addition Rules

Bit 1	Bit 2	Carry Happens? (Carry Out = 1)	Sum
0	0	❌ No	0
0	1	❌ No	1
1	0	❌ No	1
1	1	✅ Yes (Carry Out = 1)	0

What is a Carry?

When adding bits, if the total is 2 (which is '10' in binary), write 0 and carry 1 to the next left bit.

If total is 3 ('11' in binary), write 1 and carry 1.

Binary Addition examples:

a) Add 45 + 100 (8-bit)

Step 1: Convert to 8-bit binary

45 = 00101101

100 = 01100100

Step 2: Add bit by bit

Bit Position	7	6	5	4	3	2	1
Number 1	0	1	0	1	1	0	1
Number 2	1	1	0	0	1	0	0
Carry In	0	1	0	1	1	0	0
Sum	1	0	1	1	0	1	1

Result:
Sum (8-bit): 01011011 (91 in decimal)

b) Add 12 + 34 (8-bit)

Step 1: Convert to 8-bit binary

12 = 00001100

34 = 00100010

Step 2: Add bit by bit

Bit Position	6	4	3	2
Number 1	0	1	1	0
Number 2	1	0	0	1
Carry In	0	1	1	0
Sum	1	1	1	1

Result:
Sum (8-bit): 00101110 (46 in decimal)

c) Add 78 + 57 (8-bit)

Step 1: Convert to 8-bit binary

78 = 01001110

57 = 00111001

Step 2: Add bit by bit

Bit Position	8	7	6	5	4	3	2	1
Number 1	0	1	0	0	1	1	1	0
Number 2	0	0	1	1	1	0	0	1
Carry In	0	0	1	0	1	1	1	0
Sum	1	0	0	0	0	1	1	1

Result:
Sum (8-bit): 10000111 (135 in decimal)

d) Add 150 + 50 (8-bit)

Step 1: Convert to 8-bit binary

150 = 10010110

50 = 00110010

Step 2: Add bit by bit

Bit Position	8	7	6	5	4	3	2
Number 1	1	0	0	1	0	1	1
Number 2	0	0	1	1	0	0	1
Carry In	0	1	0	1	0	1	1
Sum	1	1	0	0	1	0	0

Result:
Sum (8-bit): 11001000 (200 in decimal)

Some examples with Overflow

If the sum requires more bits than available (e.g., result is 9 bits but only 8 bits allowed), the extra bit is called overflow.

Overflow means the result can’t fit in the assigned number of bits.

Working Example:

a) Add 01101110(110) + 11011110(222)

Bit Position	8	7	6	5	4	3	2
Number 1	0	1	1	0	1	1	1
Number 2	1	1	0	1	1	1	1
Carry In	1	1	1	1	1	1	0
Sum	0	0	1	0	0	1	0

Result:

Binary Sum: 00101100

Decimal:

01101110 = 110

11011110 = 222

Total = 332 (overflow)

8-bit Truncated: 00101100 = 44 (due to overflow)

Key Point: The 9th carry bit was discarded, causing overflow (332 > 255). The 8-bit result is incorrect.

b) Add 89 + 175 (8-bit)

Step 1: Convert to 8-bit binary

89 = 01011001

175 = 10101111

Step 2: Add bit by bit

Bit Position	8	7	6	5	4	3	2	1
Number 1	0	1	0	1	1	0	0	1
Number 2	1	0	1	0	1	1	1	1
Carry In	1	1	1	1	0	1	1	0
Sum	0	0	0	0	1	0	0	0

Result:
Short Answer:
Sum (8-bit): 00001000 (8 in decimal)
Overflow: Yes (carry out from MSB) → True sum (264) exceeds 8-bit range (0-255).

Explanation:

89 + 175 = 264, but 8-bit max is 255.

Binary addition yields 00001000 (8) with a carry out, indicating overflow.

Correct result requires 9 bits: 100001000 (264).

c) Add 168 + 99 (8-bit)

Here's the improved version with the explanation integrated:

Step 1: Convert to 8-bit binary

89 = 01011001

175 = 10101111

Step 2: Add bit by bit

Bit Position	8	7	6	5	4	3	2	1
Number 1	0	1	0	1	1	0	0	1
Number 2	1	0	1	0	1	1	1	1
Carry In	1	1	0	1	1	0	0	0
Sum	1	0	0	0	0	1	0	0

Result:
8-bit Result: 00001000 = 8 (decimal) ❌
(This is obtained by taking only the rightmost 8 bits of the sum and discarding the 9th overflow bit)

Actual sum in decimal: 89 + 175 = 264 → exceeds 255 → unsigned overflow occurs.
(The correct 9-bit result is 100001000, but 8-bit systems can only store 00001000)

d) Add 88 + 215 (8-bit)

Step 1: Convert to 8-bit binary

88 = 01011000

215 = 11010111

Step 2: Add bit by bit with carries

Bit Position	8	7	6	5	4	3	2	1
Number 1	0	1	0	1	1	0	0	0
Number 2	1	1	0	1	0	1	1	1
Carry	1	1	1	1	0	0	0	0
Sum	0	0	1	1	0	1	1	1

Result:
8-bit Result: 00110111 = 55 (decimal) ❌
(Only the rightmost 8 bits are kept, discarding the overflow carry)

Actual sum in decimal: 88 + 215 = 303 → exceeds 255 → unsigned overflow occurs.
(Full 9-bit result would be 100110111, but 8-bit truncation gives incorrect 00110111)

Key Observation:
The 9th carry bit (1) indicates overflow, making the 8-bit result invalid.

16-bit Addition Practice Solutions

a) Add

0111 1111 1111 0001 + 0101 1111 0011 1001

Bit Position	16	15	14	13	12	11	10	9	8	7	6	5	4	2	1
Number 1	0	1	1	1	1	1	1	1	1	1	1	1	0	0	1
Number 2	0	1	0	1	1	1	1	1	0	0	1	1	1	0	1
Carry In	0	1	1	1	1	1	1	1	0	0	1	1	0	0	0
Sum	1	1	0	1	1	1	1	1	0	0	1	0	1	1	0

Result:
16-bit Result: 1101 1111 0010 1010 (0xDF2A) = 57,130 ✅
Actual sum: 32,753 + 24,377 = 57,130 → No overflow (≤65,535)

Note: All carries propagate correctly without final overflow (no 17th bit generated). The result matches manual calculation:

0x7FF1 (32,753) + 0x5F39 (24,377) = 0xDF2A (57,130)

b) Add

1110 1110 0000 1011 + 1111 1101 1101 1001

Bit Position	16	15	14	13	12	11	10	9	8	7	5	4	2	1
Number 1	1	1	1	0	1	1	1	0	0	0	0	1	1	1
Number 2	1	1	1	1	1	1	0	1	1	1	1	1	0	1
Carry In	1	1	1	1	1	1	1	0	0	0	0	1	1	1
Sum	1	1	1	0	1	1	0	0	0	0	0	0	0	0

Result:
16-bit Result: 1110 1100 0000 0000 (0xEC00)
Actual sum in decimal: 60,939 + 64,985 = 125,924
Overflow: ❌ (Result requires 17 bits, but only 16 are available)

Verification:

Full 17-bit result would be 1 1110 1100 0000 0000 (121,344)

16-bit truncation gives 1110 1100 0000 0000 (60,416)

Correct sum (125,924) > 65,535 → Overflow occurs

Key Observation:
The discarded 17th carry bit (1) indicates overflow, making the 16-bit result invalid. The truncated value (60,416) is incorrect due to overflow.