16. Base System (Binary/Decimal) Basics

Base system ka basic concept: base-10 (decimal) vs base-2 (binary), place value, conversion (binary↔decimal), quick tricks, common errors, SSC CGL level examples & practice with answers.

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1) Base / Radix kya hota hai?

Base (radix) = kitne digits allowed hain.

• Decimal (base-10): digits 0–9
• Binary (base-2): digits 0–1
• General base-b me digits: 0 to (b−1)

2) Place value (general formula)

Agar number (a_ka_k-1...a₁a₀)_b ho, to value: Σ a_i × bⁱ (i=0 to k)

Example: (345)₁₀

3×10² + 4×10¹ + 5×10⁰ = 300 + 40 + 5 = 345

Example: (1011)₂

1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11

3) Binary → Decimal (direct method)

Binary digits ko powers of 2 ke weights se multiply karke add kar do.

4) Decimal → Binary (division by 2 method)

Number ko 2 se repeatedly divide karo. Jo remainders aayen, unko reverse order me likho.

Example: 13 to binary

Remainders bottom-to-top: 1101 ⇒ (1101)₂

5) Quick binary facts (SSC handy)

• Binary number with n digits has max value = 2ⁿ − 1 (all ones)
• (1000...0)₂ (1 followed by k zeros) = 2^k
• Even binary ends with 0; odd binary ends with 1
• Binary addition:
  • 0+0=0
  • 0+1=1
  • 1+0=1
  • 1+1=10 (carry 1)

6) Binary addition (mini example)

(1011)₂ + (110)₂ = ?

1011 +0110 =10001 ⇒ (10001)₂ = 17

7) Common traps

• Base-2 me digit 2,3,... allowed nahi.
• Conversion me place value powers right se start: 2⁰, 2¹, ...
• Leading zeros value change nahi karte: (00101)₂ = (101)₂

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8) Practice (SSC CGL) + Answers

1. Convert (10101)₂ to decimal.
2. Convert 25 to binary.
3. Convert (11100)₂ to decimal.
4. Is (1001)₂ odd or even?
5. Find decimal value of (1 00000)₂ (i.e., 1 followed by 5 zeros).
6. Add: (111)₂ + (1)₂.

Show Answers

1. (10101)₂ = 16 + 4 + 1 = 21
2. 25 = 16+8+1 ⇒ (11001)₂
3. (11100)₂ = 16 + 8 + 4 = 28
4. Ends with 1 ⇒ odd
5. 1 followed by 5 zeros ⇒ 2⁵ = 32
6. (111)₂ + (1)₂ = (1000)₂