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Knowledge Bank

 Introduction to Number System & Base Systems

Number-System
CAT / Number System / QUANT / Question Bank

 Introduction to Number System & Base Systems

Base System Made Easy | Digits, Conversions & Number Formation

🔢 Introduction to Number System & Base Systems

Welcome to your first lesson on the Number System, where we explore how numbers are formed using different digits in different bases. This concept is useful for exams like CAT, SSC CGL, and Bank PO—and it’s also a great way to build your number sense.

📍 What is a Base?

Think of a base as the foundation or limit for how high your digits can go before you carry over to the next place value.

In Base 10 (Decimal System)—which we use every day—digits range from 0 to 9. Once we go beyond 9, we carry over to the next place. That’s why after 9 comes 10, then 11, and so on.

🔢 Digits in Different Bases

Every base has its own set of valid digits:

  • Base 2: 0, 1
  • Base 4: 0, 1, 2, 3
  • Base 8: 0 to 7
  • Base 10: 0 to 9 (Decimal)
  • Base 16: 0 to 9, then A(10), B(11), C(12), ..., F(15)

Rule: A digit in any base system must be less than the base. For example, in base 5, valid digits are 0, 1, 2, 3, 4. Digit 5 is NOT allowed.

🧠 How Numbers are Formed in Base Systems

Let’s break this down with an easy example. The number 125 in base 10 means:

1 × 10² + 2 × 10¹ + 5 × 10⁰ = 100 + 20 + 5 = 125

So, the place values from right to left are powers of the base (here 10):

  • Units → 10⁰
  • Tens → 10¹
  • Hundreds → 10²

🔄 Converting from Any Base to Base 10

If you have a number like (abc)B, then to convert it to base 10:

a × B² + b × B¹ + c × B⁰

Example: (213)4 = 2×4² + 1×4¹ + 3×4⁰ = 32 + 4 + 3 = 39 in decimal.

💡 Decimal Numbers with Fractions

We can apply the same logic for decimals too.

125.12 = 1×10² + 2×10¹ + 5×10⁰ + 1×10⁻¹ + 2×10⁻²

Decimal digits go in negative powers of the base (10⁻¹, 10⁻², etc.).

📘 General Formula

(ABC.DE)X in decimal is:

= A × X² + B × X¹ + C × X⁰ + D / X + E / X²

Always remember:

  • Left of decimal: multiply with powers of base
  • Right of decimal: divide by increasing powers of base

📏 Comparing Numbers Across Bases

  • If two numbers are equal → their decimal equivalents are equal
  • If M > N → Base of M < Base of N
  • If M < N → Base of M > Base of N

This works when both numbers have same digits but in different bases.

⚠️ Validity of Digits in a Base

A number (ABC)x is only valid if:

Each digit A, B, C < x

If even one digit is equal to or greater than the base, the number is invalid in that base.

📚 Special Digits in Base > 10

For bases like 11, 12, 16, etc., we use letters to represent values beyond 9:

  • A = 10
  • B = 11
  • C = 12
  • D = 13, E = 14, F = 15

📊 Smallest and Largest n-digit Numbers in Base B

  • Smallest n-digit number in base B: Bn-1
  • Largest n-digit number in base B: Bn - 1

Example: In base 10:

  • Smallest 3-digit number = 10² = 100
  • Largest 3-digit number = 10³ - 1 = 999
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