A binary number is a number written in base-2, which uses only the digits 0 and 1. Each position represents a power of 2 (1, 2, 4, 8, 16…), just like decimal uses powers of 10.

Example:
1011₂ = (1×8) + (0×4) + (1×2) + (1×1) = 11₁₀

Place Value in Binary

Place	2⁴	2³	2²	2¹	2⁰
Value	16	8	4	2	1

Converting Decimal to Binary

Steps:

1. Divide the decimal number by 2.


2. Record the remainder (0 or 1).


3. Continue dividing the quotient by 2 until it is 0.


4. Write the remainders from bottom to top.

Examples:

Convert 19₁₀ to binary
19 ÷ 2 = 9 remainder 1

9 ÷ 2 = 4 remainder 1

4 ÷ 2 = 2 remainder 0

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Binary = 10011₂

Convert 6₁₀ to binary

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Binary = 110₂

Convert 13₁₀ to binary

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1
v Binary = 1101₂

Convert 25₁₀ to binary

25 ÷ 2 = 12 remainder 1

12 ÷ 2 = 6 remainder 0

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Binary = 11001₂

Convert 9₁₀ to binary

9 ÷ 2 = 4 remainder 1

4 ÷ 2 = 2 remainder 0

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Binary = 1001₂

Converting Binary to Decimal

Multiply each binary digit by the power of 2 for its position and sum them.

Examples:

10110₂ to decimal: (1×16) + (0×8) + (1×4) + (1×2) + (0×1) = 16 + 0 + 4 + 2 + 0 = 22₁₀

111₂ to decimal: (1×4) + (1×2) + (1×1) = 4 + 2 + 1 = 7₁₀

1001₂ to decimal: (1×8) + (0×4) + (0×2) + (1×1) = 8 + 0 + 0 + 1 = 9₁₀

1101₂ to decimal: (1×8) + (1×4) + (0×2) + (1×1) = 8 + 4 + 0 + 1 = 13₁₀

1010₂ to decimal: (1×8) + (0×4) + (1×2) + (0×1) = 8 + 0 + 2 + 0 = 10₁₀

A. Addition of Two or Three-Digit Binary Numbers

Rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0 (carry 1 to next column)

Examples:

11₂ + 10₂
   11
+  10
------
   01 (carry 1)
   11 + carry → 101₂
Decimal check: 3 + 2 = 5

101₂ + 011₂
   101
+  011
------
   000 (carry 1)
   1000₂
Decimal check: 5 + 3 = 8 

110₂ + 101₂
   110
+ 101
------
   011 (carry 1)
  1011₂
Decimal: 6 + 5 = 11 

111₂ + 001₂
   111
+ 001
------
   000 (carry 1)
  1000₂
Decimal: 7 + 1 = 8 

101₂ + 110₂
   101
+ 110
------
   011 (carry 1)
  1011₂
Decimal: 5 + 6 = 11 

B. Subtraction of Two or Three-Digit Binary Numbers

Rules: 0 − 0 = 0, 1 − 0 = 1, 1 − 1 = 0, 0 − 1 = 1 (borrow from next column)

Examples:

10₂ − 01₂
  10
- 01
----
  01₂
Decimal: 2 − 1 = 1 

101₂ − 011₂
  101
- 011
-----
  010₂
Decimal: 5 − 3 = 2 

111₂ − 101₂
  111
- 101
------
  010₂
Decimal: 7 − 5 = 2 

110₂ − 010₂
  110
- 010
------
  100₂
Decimal: 6 − 2 = 4 

101₂ − 001₂
  101
- 001
------
  100₂
Decimal: 5 − 1 = 4 

C. Multiplication of Two-Digit Binary Numbers

Rules: 0 × 0 = 0, 0 × 1 = 0, 1 × 0 = 0, 1 × 1 = 1

Examples:

11₂ × 10₂
   11
 × 10
 ----
   00  (11 × 0)
  11   (11 × 1, shift left)
 ----
  110₂
Decimal: 3 × 2 = 6 

101₂ × 11₂
   101
 ×  11
 -----
   101  (101 × 1)
  101   (101 × 1, shift left)
 -----
  1111₂
Decimal: 5 × 3 = 15 

10₂ × 11₂
   10
 × 11
 ----
   10   (10 × 1)
  10    (10 × 1, shift left)
 -----
  110₂
Decimal: 2 × 3 = 6 

11₂ × 11₂
   11
 × 11
 ----
   11
  11
 ----
 1001₂
Decimal: 3 × 3 = 9 

101₂ × 10₂
   101
 × 10
 ----
   00
  101
 ----
 1010₂
Decimal: 5 × 2 = 10 

• Computers and digital devices store all data in binary.


• Foundation of computer science, coding, and electronics.


• Helps students understand number systems beyond decimal.