Number Bases

Week 2: Number Bases

Class: JSS1 | Subject: Mathematics | Term: First Term

Learning Objectives

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

• Explain the concept of number bases
• Identify and work with different number bases (base 2, base 8, base 10)
• Convert numbers from one base to another
• Perform addition and subtraction in different bases

Introduction to Number Bases

A number base (also called a radix) is the number of different digits or combination of digits and letters that a system of counting uses to represent numbers. The base of a number system determines how many unique symbols are used before moving to the next positional digit. Our everyday counting system is base 10 (decimal), which uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Different number bases are used for various purposes. For example, computers use base 2 (binary) because electronic circuits have two states: on (1) and off (0). Base 8 (octal) and base 16 (hexadecimal) are used in programming to simplify binary representations.

Base Ten (Decimal System)

The decimal system is the most commonly used number system. It has ten digits (0-9), and each position represents a power of 10. For example, in the number 4,327:

• 4 x 1000 = 4,000
• 3 x 100 = 300
• 2 x 10 = 20
• 7 x 1 = 7

Therefore, 4,327 in base 10 = 4,000 + 300 + 20 + 7

Base Two (Binary System)

The binary system uses only two digits: 0 and 1. Each position represents a power of 2. This system is fundamental to computer science because all computer data is stored and processed in binary form.

The place values in binary are: ...64, 32, 16, 8, 4, 2, 1

Example: Convert 1101 (base 2) to base 10

1101 = 1x8 + 1x4 + 0x2 + 1x1 = 8 + 4 + 0 + 1 = 13 in base 10

Converting Base 10 to Base 2

To convert a decimal number to binary, we repeatedly divide by 2 and record the remainders.

Example: Convert 25 (base 10) 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

Reading remainders from bottom to top: 25 in base 10 = 11001 in base 2

Base Eight (Octal System)

The octal system uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. Each position represents a power of 8.

Example: Convert 375 (base 8) to base 10

375 = 3x64 + 7x8 + 5x1 = 192 + 56 + 5 = 253 in base 10

Example: Convert 156 (base 10) to base 8

156 / 8 = 19 remainder 4
19 / 8 = 2 remainder 3
2 / 8 = 0 remainder 2
Reading from bottom: 156 = 234 in base 8

Base Five (Quinary System)

The quinary system uses five digits: 0, 1, 2, 3, and 4. Each position represents a power of 5.

Example: Convert 342 (base 5) to base 10

342 = 3x25 + 4x5 + 2x1 = 75 + 20 + 2 = 97 in base 10

Addition in Base Two

Binary addition rules: 0+0=0, 0+1=1, 1+0=1, 1+1=10 (write 0 carry 1), 1+1+1=11 (write 1 carry 1)

Example: Add 1101 + 1011 in base 2

Column by column from right: 1+1=10 (write 0, carry 1), 0+1+1=10 (write 0, carry 1), 1+0+1=10 (write 0, carry 1), 1+1+1=11 (write 1, carry 1), final carry 1.
Answer: 1101 + 1011 = 11000 in base 2

Verification: 13 + 11 = 24 in base 10. 11000 in base 2 = 16+8 = 24. Correct!

Subtraction in Base Two

Example: Subtract 101 from 1100 in base 2

1100 - 101: Verification: 12 - 5 = 7 in base 10 = 111 in base 2
Answer: 1100 - 101 = 111 in base 2

Real-Life Applications of Number Bases

• Binary (Base 2): Used in all digital computers and electronic devices.
• Octal (Base 8): Used in computer programming, especially in Unix/Linux file permissions.
• Hexadecimal (Base 16): Used in programming for memory addresses and colour codes in web design.
• Base 12 (Duodecimal): Reflected in timekeeping (12 hours, 12 months) and measurement (12 inches in a foot).
• Base 60 (Sexagesimal): Used by ancient Babylonians, still seen in time (60 seconds, 60 minutes) and angles (360 degrees).

Evaluation Questions

• 1. Convert 45 (base 10) to base 2
• 2. Convert 110110 (base 2) to base 10
• 3. Convert 247 (base 8) to base 10
• 4. Convert 100 (base 10) to base 8
• 5. Add: 1011 + 1101 in base 2

Summary

Number bases are different systems of counting. While we use base 10 in daily life, computers use base 2 (binary). Understanding how to convert between bases and perform arithmetic in different bases is an important mathematical skill. The key to conversion is understanding that each digit's value depends on its position and the base of the number system.