📚 Lesson: Understanding Number Systems

🎯 Learning Objectives

By the end of this lesson, students will be able to:

• Identify and describe the characteristics of decimal, binary, octal, and hexadecimal number systems.
• Convert numbers between these systems and decimal.
• Apply the Division-Remainder Method for base conversions.
• Understand binary groupings for conversions between binary, octal, and hexadecimal.

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1️⃣ Introduction to Number Systems

Key Concepts:

• Positional Number Systems: Each digit has a weighted value based on its position.
• Radix/Base: The number of unique digits, including zero, used to represent numbers.

System	Base	Digits Used	Weighting (Powers of…)
Decimal	10	0–9	Powers of 10
Binary	2	0, 1	Powers of 2
Octal	8	0–7	Powers of 8
Hexadecimal	16	0–9, A–F (A=10 to F=15)	Powers of 16

2️⃣ Converting to Decimal

Method: Weighted Sum of Powers

• Expand each digit using its base and position.
• Example (Binary):

$1101.01_2 = 1×2^3 + 1×2^2 + 0×2^1 + 1×2^0 + 0×2^{-1} + 1×2^{-2}$

Practice:

• Convert the following to decimal:

• $2306.71_{10}$
• $7206.351_8$
• $A31C.04_{16}$

3️⃣ Converting from Decimal

Method: Division-Remainder (for integers)

• Divide the number by the target base.
• Record the remainder.
• Repeat until the quotient is 0.
• Read remainders in reverse.

Example:

Convert $2458_{10}$ to Octal:

• 2458 ÷ 8 = 307, R=2
• 307 ÷ 8 = 38, R=3
• 38 ÷ 8 = 4, R=6
• 4 ÷ 8 = 0, R=4
• Result: $4632_8$

4️⃣ Conversions Between Binary, Octal, and Hexadecimal

Binary ↔ Octal:

• Group binary digits in sets of 3 (right to left).
• Example:

$101100010110_2$ → $101\ 100\ 010\ 110$ → $5426_8$

Binary ↔ Hexadecimal:

• Group binary digits in sets of 4.
• Example:

$01100010110_2$ → $1011\ 0001\ 0110$ →

$B16_{16}$

5️⃣ Exercises & Activities

In-Class Practice:

• Convert $145_{10}$ to Binary, Octal, and Hexadecimal using Division-Remainder.
• Use calculators to verify results. Group Challenge:
• Each group gets a number in one base and must convert it to the other three. Discussion Prompt:
• Why do computers use binary, and how do hexadecimal and octal help simplify binary representation?

6️⃣ Wrap-Up & Homework

Summary:

• Number systems are foundational to digital logic and computing.
• Understanding conversions helps in programming, electronics, and data representation. Homework:
• Memorize hexadecimal digit-to-decimal mappings.
• Convert five random decimal numbers to binary, octal, and hexadecimal.