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Mathable Smart Lab

CLASS VIII • NUMBER SENSETopic: Number Systems and Bases

Class VIII: A Story of Numbers

Explore how numbers can be written in different bases, why base-10 is common, and how place value changes in base-8, base-2, and base-5.

Fact Sheet: Why Different Bases?

Humans commonly use base-10, probably because we have 10 fingers. But numbers can be represented in any base.

Base-10: Places are 1, 10, 100, 1000...
Base-8: Places are 1, 8, 64, 512...
Example: 25 in base-10 becomes 31 in base-8 because 25 = 3 × 8 + 1.
8²64s place
8¹8s place
8⁰1s place

25₁₀= 3 groups of 8 + 1= 31₈

Formula Hub

Base Meaning

The base tells us how many symbols are used and when a new place starts.

Base-8 uses digits 0 to 7.

Decimal to Base-b

Divide repeatedly by b and read remainders upward.

25 ÷ 8 gives 31₈.

Base-b to Decimal

Multiply each digit by its place value.

31₈ = 3×8 + 1 = 25.

Base-2

Computer-friendly system using only 0 and 1.

13₁₀ = 1101₂.

Allowed Digits

In base-b, digits must be less than b.

8 is not allowed in base-8.

Place Value

In base-b, places are b⁰, b¹, b², b³...

Online Lab: Base Conversion Explorer

Score: 0Current Streak: 🔥 0Badge: Base Beginner


Activity Zone

🔁 Decimal to Base Converter

🔍 Base to Decimal Converter

🧱 Place-Value Block Builder

Build a base representation using groups and leftovers.

✅ Valid Digit Checker

Check whether a numeral is allowed in a given base.

⚔️ Base Comparison

31₈
25₁₀

📋 Conversion Table

Worksheet Generator

Generate practice on converting between base-10, base-8, base-5, and base-2.



Real-World Use

Computers: Computers use base-2 because circuits work with off/on states.
Time: We use base-60 ideas when measuring minutes and seconds.
History: Different civilizations used different ways to group and record numbers.

🌍 Real-Life Case Generator

Teacher Tools

Learning Outcomes

  • Explain why base-10 is a convention, not the only possible system.
  • Convert base-10 numbers to base-8 using grouping or repeated division.
  • Convert base-b numbers to base-10 using place values.
  • Identify valid and invalid digits in a given base.
  • Connect number bases to computers, clocks, and historical counting systems.

Exit Ticket Prompts

  • Convert 25₁₀ to base-8.
  • Convert 31₈ to base-10.
  • Why is 8 not a valid digit in base-8?