Number Base II

Number Base System - Grade 12

Lesson Objectives

Convert between different number bases including base 2, base 8, base 10, and base 16.
Perform arithmetic operations in different bases.
Interpret positional values in any number base system.
Apply base conversions in computer science and digital electronics contexts.

Lesson Introduction

Every day, we use base 10, also called the decimal number system. However, computers use base 2, and sometimes base 16 is used in programming. In this lesson, we’ll explore how to understand, convert, and perform operations across different number bases.

Core Lesson Content

Worked Examples

Example 1: Convert

1011_2

to base 10.

1011_2 = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0

= 8 + 0 + 2 + 1 = 11_{10}

Example 2: Convert

45_{10}

to base 2.
Divide 45 by 2 repeatedly and write down the remainders:

45 \div 2 = 22\ R1

22 \div 2 = 11\ R0

11 \div 2 = 5\ R1

5 \div 2 = 2\ R1

2 \div 2 = 1\ R0

1 \div 2 = 0\ R1

Read remainders from bottom to top:

45_{10} = 101101_2

Example 3: Convert

7A_{16}

to base 10.

7A_{16} = 7 \times 16^1 + 10 \times 16^0 = 112 + 10 = 122_{10}

Example 4: Convert

98_{10}

to base 8.

98 \div 8 = 12\ R2

12 \div 8 = 1\ R4

1 \div 8 = 0\ R1

So,

98_{10} = 142_8

Example 5: Add

1011_2 + 1101_2

.
Align binary numbers:

1011 + 1101 = 11000_2

Example 6: Multiply

101_2 \times 11_2

101_2 = 5,\ 11_2 = 3

in decimal.

5 \times 3 = 15 = 1111_2

Example 7: Subtract

11001_2 - 1000_2

.
Convert to decimal:

25 - 8 = 17 = 10001_2

Example 8: Convert

100111_2

to hexadecimal.
Group in 4s from right:

0010\ 0111

0010 = 2,\ 0111 = 7 \Rightarrow 100111_2 = 27_{16}

Example 9: What is the value of the digit 3 in

2034_5

3 \text{ is in the } 5^1 \text{ place } \Rightarrow 3 \times 5^1 = 15

Example 10: Express

345_{10}

in base 4.
Divide successively by 4:

345 \div 4 = 86\ R1

86 \div 4 = 21\ R2

21 \div 4 = 5\ R1

5 \div 4 = 1\ R1

1 \div 4 = 0\ R1

345_{10} = 11121_4

Exercises

Convert $11001_2$ to base 10.
[NABTEC] Convert $255_{10}$ to hexadecimal. [Past Question]
Convert $789_{10}$ to base 8.
[WAEC] Add $1110_2 + 1001_2$ . [Past Question]
Multiply $1001_2 \times 11_2$ .
Convert $3C_{16}$ to base 10.
[NECO] What is the place value of 6 in $7624_8$ ? [Past Question]
Convert $240_{10}$ to base 4.
[JAMB] Subtract $11101_2 - 1011_2$ . [Past Question]
Convert $111101_2$ to hexadecimal.

Conclusion/Recap

In this lesson, you have learned how to convert between various number bases, perform arithmetic in these bases, and apply these concepts to real-world problems. In the next lesson, we will look at Binary Arithmetic and its application in computing.