Lesson 4: Numeration Systems

📚 Introduction

Have you ever wondered how people count and write numbers differently around the world?
Before we had the numbers 0, 1, 2, 3, etc., different civilizations used their own ways of counting — like tally marks, Roman numerals, and even symbols on clay tablets!

In this lesson, we’ll explore how humans created numeration systems to represent numbers — and how we use them today!

By the end of this lesson, you’ll know:

• What a numeration system is
• Different types of numeration systems
• The features of our modern system (base-10 or decimal system)
• How to convert between different forms

🧠 Key Concepts and Explanations

What is a Numeration System?

A numeration system is a way of writing numbers using symbols or figures.
It allows us to record, count, and perform calculations!

Types of Numeration Systems

Our Modern System: Decimal (Base 10)

Today, we mostly use the Decimal System (also called Base-10).

It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The position of each digit shows its value — this is called the place value system.

Place Value Table Example

Number: 357 means:

• 3 hundreds → 300
• 5 tens → 50
• 7 units → 7

So, 357 = 300 + 50 + 7.

Roman Numerals Basic Guide

Roman numerals use specific letters:

• I = 1
• V = 5
• X = 10
• L = 50
• C = 100
• D = 500
• M = 1000

Examples:

• III = 3
• IX = 9 (one less than 10)
• XL = 40 (10 less than 50)
• XC = 90 (10 less than 100)
• CM = 900 (100 less than 1000)

📝 Sample Problem Walkthroughs

Example 1: Writing in Place Value

Question: Write 482 in expanded form.

Solution:

482 = (4 × 100) + (8 × 10) + (2 × 1) = 400 + 80 + 2.

Example 2: Reading Roman Numerals

Question: What is XLII?

Solution: X = 10, L = 50 → XL = 40 (10 less than 50), II = 2, so XLII = 42.

📝 Practice Exercises

Exercise 1:

Write the number 236 in expanded form.

Exercise 2:

What number does the Roman numeral XXVII represent?

Exercise 3:

Using tally marks, show the number 7.

Exercise 4:

Fill the place value table for 814.

📝 Extra Challenge Problems

Challenge 1:

Write the number 649 in expanded form and show its Roman numeral equivalent.

Challenge 2:

Explain why the Roman numeral system is not as easy to calculate with as the decimal system.

🔁 Recap

In this lesson, we learned that numeration systems are different ways of writing numbers.
We explored ancient systems like tally marks and Roman numerals, and we learned how the decimal (base-10) system works today using place values!

🪞 Reflection Prompt

Think about this: Imagine you had to create your own number system — what symbols would you use? Would it be based on 10, 5, or another number?