Lesson 1: Number Bases – Conversions and Operations

👨🏽‍🏫 Introduction

Have you ever noticed how our daily numbers use only 10 digits — 0 to 9? That’s because we use Base 10, also called the decimal system. But it’s not the only system! Computers use base 2 (binary), some ancient cultures used base 5 or base 20, and digital systems often use base 8 or base 16.

In this lesson, we’ll explore how numbers work in different bases, and how to convert and calculate with them.

You’ll learn how to:

• Understand the concept of number bases
• Convert numbers from one base to another
• Perform addition and subtraction in other bases
• Apply base arithmetic to real-world problems like computing

🧠 Key Concepts

🔢 1. What Is a Number Base?

A number base (also called a radix) is the number of different digits used to represent values.

• Base 10 (Decimal): Uses digits 0–9
• Base 2 (Binary): Uses digits 0 and 1
• Base 5: Uses digits 0 to 4
• Base 8 (Octal): Uses digits 0 to 7

Example:

In base 10:

In base 2:

🔁 2. Converting Between Bases

👉 (a) From Base 10 to Another Base:

Use repeated division by the new base, recording the remainders.

Example: Convert 25 to base 2:

• 25 ÷ 2 = 12 remainder 1
• 12 ÷ 2 = 6 remainder 0
• 6 ÷ 2 = 3 remainder 0
• 3 ÷ 2 = 1 remainder 1
• 1 ÷ 2 = 0 remainder 1

Read from bottom to top → 11001

👉 (b) From Another Base to Base 10:

Multiply each digit by the base raised to the appropriate power and sum them.

Example: Convert 203 in base 5 to base 10:

➕➖ 3. Operations in Other Bases

Just like base 10, you add, subtract, multiply — but the place value and maximum digit depend on the base.

✅ (a) Addition in Base 5:

Example: Add

• 3 + 1 = 4 → write 4
• 4 + 2 = 6 → too big for base 5 → 6 = 1 carry, write 1
• 2 + 1 + carry 1 = 4

Answer:

✅ (b) Subtraction in Base 5:

Example: Subtract

• Convert both to base 10:
• 
• 
• 117 − 39 = 78
• Now convert 78 back to base 5 →

✍🏽 Sample Problem Walkthroughs

1. 1. Convert 45 to base 5:
      • 45 ÷ 5 = 9 rem 0
      • 9 ÷ 5 = 1 rem 4
      • 1 ÷ 5 = 0 rem 1

Answer:

1. 1. Convert to base 10:

⚠️ Invalid! Digit “2” not allowed in base 2.

1. 1. Add
      • 1+1 = 10 → write 0, carry 1
      • 1+0+1 = 10 → write 0, carry 1
      • 1+1 = 10 → write 0, carry 1
      • Final carry = 1

Answer:

🧪 Practice Exercises

✏️ A. Convert to base 10:

✏️ B. Convert from base 10:

1. Convert 35 to base 2
2. Convert 88 to base 5

✏️ C. Perform operations:

✅ Answers with Explanations

• A1:
• A2:
• B1:
• B2:
• C1: Convert both to base 10: 73 + 42 = 115 →
• C2:
• C3: 116 − 47 = 69 →

📌 Recap

• ✅ Every base has its own digits and place value rules
• ✅ You can convert between base 10 and others using repeated division or powers
• ✅ Arithmetic in other bases uses same methods, but follows the rules of that base

💭 Reflection Prompt

Think about the calculator on your phone or a digital clock. What base system might they be using behind the scenes? How do number bases help machines and people communicate?