Number Base System

Lesson Objectives

Convert numbers from one base to another (e.g., base 2 to base 10).
Perform addition, subtraction, multiplication, and division in number bases.
Understand the relationship between base systems.
Apply number base conversions to solve real-life problems.

Lesson Introduction

Have you ever seen a digital clock or computer screen showing 1010 instead of 10? That’s binary — a base 2 number system used by machines. In this lesson, we will learn how to convert numbers from one base to another and perform operations like addition and multiplication using number base systems.

Core Lesson Content

Let’s work through a range of examples from basic to advanced to fully understand how the number base system works.

Worked Example

Example 1 (Basic Conversion):
Convert \(1011_2\) to base 10.
\(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 0 + 2 + 1 = 11\)
Therefore, \(1011_2 = 11_{10}\)

Example 2:
Convert \(45_{10}\) to base 2.
\(45 \div 2 = 22\text{ R }1\)
\(22 \div 2 = 11\text{ R }0\)
\(11 \div 2 = 5\text{ R }1\)
\(5 \div 2 = 2\text{ R }1\)
\(2 \div 2 = 1\text{ R }0\)
\(1 \div 2 = 0\text{ R }1\)
Read from bottom up: \(45_{10} = 101101_2\)

Example 3: Add \(1011_2 + 1101_2\) in base 2.

Working:

\( \begin{array}{r} \phantom{+}1011_2 \\ +1101_2 \\ \hline 11000_2 \end{array} \)

Therefore, \(1011_2 + 1101_2 = 11000_2\)

Example 4:
Subtract \(10101_2 - 1101_2\).
Convert both to base 10: \(10101_2 = 21_{10}\), \(1101_2 = 13_{10}\)
\(21 - 13 = 8\)
Convert 8 back to base 2: \(8_{10} = 1000_2\)
Final Answer: \(1000_2\)

Example 5:
Multiply \(11_2 \times 10_2\).
Convert to base 10: \(11_2 = 3\), \(10_2 = 2\)
\(3 \times 2 = 6\) → \(6_{10} = 110_2\)
Final Answer: \(110_2\)

Example 6:
Divide \(1100_2 \div 10_2\).
\(1100_2 = 12_{10}, 10_2 = 2_{10}\)
\(12 \div 2 = 6\) → \(6 = 110_2\)
Final Answer: \(110_2\)

Example 7:
Convert \(7A_{16}\) to base 10.
\(7 \times 16^1 + 10 \times 16^0 = 112 + 10 = 122\)
Final Answer: \(122_{10}\)

Example 8:
Convert \(254_{10}\) to base 8.
\(254 \div 8 = 31 \text{ R }6\)
\(31 \div 8 = 3 \text{ R }7\)
\(3 \div 8 = 0 \text{ R }3\)
Answer: \(254_{10} = 376_8\)

Example 9:
Convert \(1001_2\) to base 10 and then to base 8.
\(1001_2 = 9_{10}\), \(9 \div 8 = 1\text{ R }1\), \(1 \div 8 = 0\text{ R }1\)
Final Answer: \(11_8\)

Example 10:
Convert \(25_{10}\) to base 5.
\(25 \div 5 = 5\text{ R }0\)
\(5 \div 5 = 1\text{ R }0\)
\(1 \div 5 = 0\text{ R }1\)
Final Answer: \(100_5\)

Exercises

\(23_{10} = ?_{2}\)
[WAEC] Convert \(1011_2\) to base 10. [Past Question]
[NECO] Add \(1101_2 + 1001_2\) in base 2. [Past Question]
\((11011_2 - 1001_2) = ?_{2}\)
\(45_{10} = ?_{8}\)
[JAMB] Multiply \(13_8 \times 5_8\) in base 8. [Past Question]
\(127_{10} = ?_{16}\)
\((3F_{16} + 1A_{16}) = ?_{16}\)
\(1001_2 \div 11_2 = ?_{2}\)
\(Convert\ 254_{10}\ to\ base\ 5\)

Conclusion / Recap

Today, we studied the Number Base System, focusing on conversions and operations across various bases. Our next topic will be on Logarithms and Indices.