Binary Number System - Comprehensive Study Notes

1. Binary to Decimal Conversion Basic Method:

Write powers of 2 under each digit (right to left)

Multiply each binary digit by its position value

Sum all results

Calculation Binary: 1 1 0 0 0 0 0 1 Position: 128 64 32 16 8 4 2 1 (2^7)(2^6)(2^5)(2^4)(2^3)(2^2)(2^1)(2^0) Result: 128 + 64 + 0 + 0 + 0 + 0 + 0 + 1 = 193

Essential Powers of 2: 2^0 = 1 2^1 = 2 2^2 = 4 2^3 = 8 2^4 = 16 2^5 = 32 2^6 = 64 2^7 = 128 2^8 = 256

2. Decimal to Binary Conversion

Division Method:

Divide by 2 continuously

Record remainders

Read remainders bottom to top

Example: 125 to binary 125 ÷ 2 = 62 r 1 62 ÷ 2 = 31 r 0 31 ÷ 2 = 15 r 1 15 ÷ 2 = 7 r 1 7 ÷ 2 = 3 r 1 3 ÷ 2 = 1 r 1 1 ÷ 2 = 0 r 1

Result: 1111101

3. Important Subnet Mask Values Descending Pattern: Decimal Binary Gap Down CIDR 255 11111111 (-1) /32 254 11111110 (-2) /31 252 11111100 (-4) /30 248 11111000 (-8) /29 240 11110000 (-16) /28 224 11100000 (-32) /27 192 11000000 (-64) /26 128 10000000 /25

Ascending Pattern: 128 (+64) = 192 192 (+32) = 224 224 (+16) = 240 240 (+8) = 248 248 (+4) = 252 252 (+2) = 254 254 (+1) = 255

4. Quick Reference Patterns Common Binary Values: text 128 = 10000000 192 = 11000000 224 = 11100000 240 = 11110000 248 = 11111000 252 = 11111100 254 = 11111110 255 = 11111111

5. Memory Tips and Tricks For Binary to Decimal:

Remember powers of 2: 1,2,4,8,16,32,64,128

Only add where there's a 1

Practice common patterns

For Decimal to Binary:

Keep dividing by 2

Write remainders bottom to top

Stop at 0

For Subnet Masks:

Each gap doubles going down

Each step removes one rightmost 1

The gap equals the removed power of 2

6. Practice Examples Binary to Decimal: 10101010 = 128 + 32 + 8 + 2 = 170 11001100 = 128 + 64 + 8 + 4 = 204

Decimal to Binary: 150 = 10010110 200 = 11001000

7. Common Applications

Subnet mask calculations

IP addressing

Network/host calculations

CIDR notation

Network design

Binary operations in programming

8. Key Points to Remember

Powers of 2 are fundamental

Practice pattern recognition

Understand the relationship between gaps

Memorize common values

Use the doubling/halving pattern

These notes cover the essential concepts for binary calculations without a calculator, perfect for networking exams and practical applications.

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