Boolean algebra is a mathematical structure used to perform operations on binary variables (0s and 1s). It is fundamental in digital logic design and computer science.

Basic Boolean Operations

Boolean algebra consists of three primary operations:

1. AND Operation (⋅)
   • Symbol: A ⋅ B or A AND B

   • Truth Table:

     A	B	A ⋅ B
     0	0	0
     0	1	0
     1	0	0
     1	1	1

2. OR Operation (+)
   • Symbol: A + B or A OR B

   • Truth Table:

     A	B	A + B
     0	0	0
     0	1	1
     1	0	1
     1	1	1

3. NOT Operation (¯)
   • Symbol: Ā or NOT A

   • Truth Table:

     A	Ā
     0	1
     1	0

Boolean Identities

Boolean algebra follows several important identities:

• Identity Laws:
  A + 0 = A, A ⋅ 1 = A
• Null Laws:
  A + 1 = 1, A ⋅ 0 = 0
• Idempotent Laws:
  A + A = A, A ⋅ A = A
• Complement Laws:
  A + Ā = 1, A ⋅ Ā = 0
• Distributive Laws:
  A ⋅ (B + C) = A ⋅ B + A ⋅ C
• Absorption Laws:
  A + (A ⋅ B) = A, A ⋅ (A + B) = A

---

Conversion of Decimal Numbers into Binary Numbers

Converting a decimal number to binary involves dividing the number by 2 repeatedly and recording the remainders.

Steps to Convert Decimal to Binary:

1. Divide the decimal number by 2.
2. Record the remainder (0 or 1).
3. Repeat the process with the quotient.
4. Reverse the sequence of remainders to obtain the binary number.

Example 1: Convert 13 to Binary

1. 13 ÷ 2 = 6, remainder 1
2. 6 ÷ 2 = 3, remainder 0
3. 3 ÷ 2 = 1, remainder 1
4. 1 ÷ 2 = 0, remainder 1
5. Read from bottom to top: 1101

Example 2: Convert 25 to Binary

1. 25 ÷ 2 = 12, remainder 1
2. 12 ÷ 2 = 6, remainder 0
3. 6 ÷ 2 = 3, remainder 0
4. 3 ÷ 2 = 1, remainder 1
5. 1 ÷ 2 = 0, remainder 1
6. Read from bottom to top: 11001

Shortcut: Binary of Small Decimal Numbers

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010

These concepts are essential for understanding digital logic design and computer arithmetic.