17. Base Conversion (Advanced) – Base-2/8/10/16 + Base-n

Base conversion advanced: base-n to decimal, decimal to base-n, binary↔octal↔hex shortcuts, base arithmetic basics, SSC CGL level examples & practice with answers.

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1) General conversion rules (must know)

A) Base-b → Decimal (place value)

(a_k...a₁a₀)_b = Σ a_i×bⁱ

Example: (231)₄ = 2×4² + 3×4 + 1 = 32 + 12 + 1 = 45

B) Decimal → Base-b (repeated division by b)

Number ko b se repeatedly divide karo, remainders reverse order me likho.

Example: 45 to base-4

• 45 ÷ 4 = 11 remainder 1
• 11 ÷ 4 = 2 remainder 3
• 2 ÷ 4 = 0 remainder 2
• So 45 = (231)₄

2) Octal (base-8) & Hex (base-16) basics

• Octal digits: 0–7
• Hex digits: 0–9, A(10), B(11), C(12), D(13), E(14), F(15)

3) Binary ↔ Octal (fast shortcut)

Rule: Binary ko right se 3-3 bits me group karo. Har group (000 to 111) ko 0 to 7 me convert kar do.

Example: 1101011₂ to octal

• Group: 1 101 011 ⇒ pad left: 001 101 011
• 001=1, 101=5, 011=3 ⇒ (153)₈

Example: 745₈ to binary

• 7=111, 4=100, 5=101 ⇒ (111100101)₂

4) Binary ↔ Hexadecimal (fast shortcut)

Rule: Binary ko right se 4-4 bits me group karo. Har 4-bit group ko 0–15 me convert karke hex digit likho.

Example: 11101110₂ to hex

• Group: 1110 1110 ⇒ 1110=E, 1110=E ⇒ (EE)₁₆

Example: 3A₁₆ to binary

• 3=0011, A=1010 ⇒ (00111010)₂ ⇒ (111010)₂ (leading zeros drop)

5) Octal ↔ Hex conversion (via binary)

Direct octal↔hex fast method: octal → binary → hex (or reverse).

Example: 57₈ to hex

• 57₈ → binary: 5=101, 7=111 ⇒ 101111₂
• Binary group 4: 0010 1111 ⇒ 2F₁₆

6) Validity check (exam trick)

Base-b me allowed digits 0 to b−1 hote hain. Agar kisi number me digit ≥ base, number invalid.

• (128)₈ invalid (because digit 8 not allowed in base-8)
• (1021)₂ invalid (because digit 2 not allowed in base-2)

7) Base arithmetic (very basic SSC)

Base-b me carry b par banta hai. Example base-5: 4+3 = 7 (decimal) = 12₅ (carry 1, digit 2).

Example: (243)₅ + (34)₅

• 243 + 034
• Units: 3+4=7 ⇒ 12₅ ⇒ write 2 carry 1
• Next: 4+3+1=8 ⇒ 13₅ ⇒ write 3 carry 1
• Next: 2+0+1=3
• Answer: (332)₅

8) Common traps

• Grouping bits always from right side (LSB), left side padding with zeros allowed.
• Hex digits A–F ko decimal values (10–15) yaad rakho.
• Base-b me digit “b” allowed nahi (0..b−1 only).

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9) Practice (SSC CGL) + Answers

1. Convert (231)₄ to decimal.
2. Convert 45₁₀ to base-4.
3. Convert 1101011₂ to octal.
4. Convert 745₈ to binary.
5. Convert 3A₁₆ to decimal.
6. Is (129)₈ valid? (Yes/No)

Show Answers

1. 45
2. (231)₄
3. (153)₈
4. (111100101)₂
5. 3A₁₆ = 3×16 + 10 = 58
6. No (digit 9 not allowed in base-8)