Binary, BCD, Octal, and Hexadecimal Arithmetic

Why Do Computers Need Special Arithmetic?

Inside the CPU, the Arithmetic Logic Unit (ALU) performs millions of calculations every second.

Computer chips cannot understand signs like + or -.
They only understand high and low electrical voltages (1s and 0s).
Designing separate electronic circuits for addition and subtraction is costly and wastes chip area.

Instead, computer engineers use a clever trick called 2’s complement. With 2’s complement, the same adder circuit can perform both addition and subtraction!

💡 Real-Life Indian Student Analogy:
Think of an old mechanical bike odometer.
If it reads 00 km and you roll the wheel backward by 1 km, it shows 99 km!
Complement arithmetic works the exact same way. It turns subtraction into addition by rolling numbers around a circle.

1. Binary Arithmetic Basics

A. Binary Addition Rules

There are only 4 simple rules for adding binary numbers:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (with a Carry of 1, because 2 in binary is 10₂)
1 + 1 + 1 (when there is a previous carry) = 1 (with a Carry of 1, because 3 in binary is 11₂)

Example: Add (1011)₂ and (1101)₂:

   Carry:  1 1 1
           1 0 1 1  (= 11 in decimal)
         + 1 1 0 1  (= 13 in decimal)
         ---------
         1 1 0 0 0  (= 24 in decimal)

B. How Computers Store Negative Numbers

How does computer memory know if a number is positive (+5) or negative (-5)?

Sign-Magnitude Form:
- The leftmost bit is the Sign Bit (0 = Positive, 1 = Negative).
- The remaining bits show the number value.
- Example (8-bit): +5 = 00000101, -5 = 10000101.
- Problem: It gives two different codes for zero (+0 and -0), which confuses the CPU.
1’s Complement:
- Simply flip every bit: turn all 0s to 1s, and 1s to 0s.
- +5 = 00000101 → 1’s complement is 11111010 (represents -5).
2’s Complement (Used in All Modern Computers):
- Formula: 2’s Complement = 1’s Complement + 1
- Let’s find -5:
  - +5 = 00000101
  - 1’s complement = 11111010
  - Add 1: 11111010 + 1 = 11111011 (This is -5).
- Why is it best? It has only one zero (00000000) and standard addition hardware works directly.

2. Subtraction using 2’s Complement (A - B)

To calculate A - B, the computer actually calculates:
A + (2’s complement of B)

The 3 Easy Steps:

Find the 2’s complement of the number you are subtracting (B).
Add this value to the first number (A).
Check the extra carry at the left:
- If Carry = 1: Discard it! The answer is positive and ready.
- If Carry = 0: The answer is negative. Take 2’s complement of the result and put a minus sign (-) in front.

Solved Example: Calculate 28 - 14 using 8-bit 2’s Complement

A = 28 in binary = 00011100
B = 14 in binary = 00001110
Find 2’s complement of 14:
- 1’s complement: 11110001
- Add 1: 11110001 + 1 = 11110010
Now add A and (2’s complement of B):

    00011100  (28)
  + 11110010  (-14)
  ----------
  1 00001110
  ↑
  End Carry = 1 (Discard it!)

Final answer: 00001110₂ = +14 in decimal (Correct!)

3. BCD (Binary Coded Decimal) Arithmetic

In BCD (8421 code), each decimal digit from 0 to 9 is written separately as a 4-bit group.

Example: Decimal 25 is written as 0010 0101.
A 4-bit group can count up to 15 (1111₂).
But decimal digits stop at 9 (1001₂).
Therefore, patterns from 10 to 15 (1010 to 1111) are INVALID BCD.

The Golden Rule of BCD Addition (+6 Rule):

When adding two BCD numbers:

Add them using standard binary addition.
If any 4-bit group is greater than 9 OR produces a carry out:
- Add 6 (0110₂) to that group!
- Why 6? Because 16 - 10 = 6. Adding 6 skips over the 6 invalid codes and fixes the decimal carry.
- Pass any resulting carry to the next 4-bit group on the left.

Solved Example: Add 35 + 48 in BCD

35 = 0011 0101
48 = 0100 1000

      0011    0101   (35)
    + 0100    1000   (48)
    ----------------
      0111    1101   (Right group is 1101 = 13, which is > 9: INVALID!)
    + 0000    0110   (Add +6 correction to fix it)
    ----------------
      1000    0011
        8       3    = 83 in decimal! (Correct!)

4. Octal and Hexadecimal Arithmetic

Octal Arithmetic (Base 8)

Uses digits 0 to 7.
Addition: If the column sum is 8 or more, subtract 8 and carry 1 to the left.
Subtraction: If you borrow from the left column, you borrow an amount of 8 (not 10).

Example: Add (57)₈ + (34)₈:

Right Column: 7 + 4 = 11. Since 11 ≥ 8: 11 - 8 = 3 with carry 1.
Left Column: 1 (carry) + 5 + 3 = 9. Since 9 ≥ 8: 9 - 8 = 1 with carry 1.
Result = (113)₈

Hexadecimal Arithmetic (Base 16)

Uses digits 0 to 9 and letters A to F.
Addition: If the column sum is 16 or more, subtract 16 and carry 1 to the left.
Subtraction: If you borrow from the next column, you borrow 16.

Example: Add (4B)₁₆ + (7A)₁₆:

Right Column: B + A = 11 + 10 = 21.
- Since 21 ≥ 16: 21 - 16 = 5 with carry 1.
Left Column: 1 (carry) + 4 + 7 = 12 = C.
Result = (C5)₁₆

💡 DAVV Exam Preparation Tips

5-Mark Question: “Perform the subtraction 35 - 50 using 2’s complement 8-bit arithmetic.”

Convert both numbers to 8-bit binary:

35 = 00100011

50 = 00110010

Take 2’s complement of 50: 11001101 + 1 = 11001110.

Add: 00100011 + 11001110 = 11110001.

State clearly: End carry = 0, so the result is negative.

Take 2’s complement of answer: 00001110 + 1 = 00001111 = 15.

Final result is -15.

10-Mark Question: “Explain BCD addition rules. Why is the correction factor of 6 (0110) added? Illustrate with two examples.”

Define BCD (8421 code) where each digit 0-9 takes 4 bits.

Explain why +6 is needed: A 4-bit adder counts to 16, but decimal base is 10. The difference is 16 - 10 = 6 invalid codes.

Give two clear examples:

Example 1: Sum ≤ 9 (no correction needed, e.g., 23 + 14 = 37).

Example 2: Sum > 9 (correction with +0110, e.g., 35 + 48 = 83).

Binary, BCD, Octal & Hexadecimal Arithmetic