Number Systems - SSS 1 Digital Technologies note

NUMBER SYSTEMS

Class: SSS One
Subject: Digital Technologies
Term: First Term
Week: 3
Topic: Number Systems
Duration: 80 Minutes (Double Period)

INTRODUCTION

Computers and other digital devices process information using numbers. However, the number system normally used in everyday life is not the only number system used in computing. Digital devices commonly represent and process information using the binary number system, while other number systems such as decimal, octal and hexadecimal are also important in computing.

Understanding number systems enables students to understand how numerical information is represented inside a computer and how values can be converted from one number system to another.

MEANING OF NUMBER SYSTEM

Definition: A number system is a collection of symbols or digits and a set of rules used to represent numerical values. Each number system has a base, which indicates the number of unique digits used in that system, including zero.

For example, the decimal number system has ten digits, from 0 to 9, so it has a base of 10. The binary number system has only two digits, 0 and 1, so it has a base of 2.

UNDERSTANDING BASE OR RADIX

The base, also called the radix, tells us the number of different digits or symbols available in a number system.

Number System	Base	Digits/Symbols Used
Decimal	10	0 – 9
Binary	2	0 and 1
Octal	8	0 – 7
Hexadecimal	16	0 – 9 and A – F

Important: A digit that is not allowed in a particular number system cannot be used in that system. For example, the digit 8 cannot appear in an octal number because octal uses only the digits 0 to 7.

1. DECIMAL NUMBER SYSTEM (BASE 10)

The decimal number system, also called the denary number system, is the number system commonly used in everyday life.

It has a base of 10 and uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Place Values in Decimal

The value of a digit depends on its position. Each position represents a power of 10.

Thousands	Hundreds	Tens	Units
103	102	101	100
1000	100	10	1

For example:

345₁₀ = (3 × 10²) + (4 × 10¹) + (5 × 10⁰)

= 300 + 40 + 5
= 345

2. BINARY NUMBER SYSTEM (BASE 2)

The binary number system is a number system with base 2. It uses only two digits:

0 and 1

Binary is fundamental to digital computing because electronic digital circuits can represent two basic states, commonly represented as 0 and 1.

Remember: A single binary digit is called a bit. A bit can have either a value of 0 or 1.

Binary Place Values

Binary place values are powers of 2, beginning with 2⁰ at the right-hand side.

Position	27	26	25	24	23	22	21	20
Value	128	64	32	16	8	4	2	1

3. OCTAL NUMBER SYSTEM (BASE 8)

The octal number system is a base-8 number system. It uses eight digits:

0, 1, 2, 3, 4, 5, 6 and 7

The digits 8 and 9 are not used in the octal number system.

Octal Place Values

The place values in octal are powers of 8.

Position	83	82	81	80
Value	512	64	8	1

4. HEXADECIMAL NUMBER SYSTEM (BASE 16)

The hexadecimal number system is a base-16 number system. It uses sixteen symbols. The first ten are the decimal digits 0–9, while the remaining six values are represented by the letters A–F.

Decimal	Hexadecimal
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	A
11	B
12	C
13	D
14	E
15	F

Example: In hexadecimal, A represents 10, B represents 11, C represents 12, D represents 13, E represents 14 and F represents 15 in decimal.

COMPARISON OF THE FOUR NUMBER SYSTEMS

Number System	Base	Symbols Used	Example
Decimal	10	0 – 9	34510
Binary	2	0, 1	10112
Octal	8	0 – 7	3458
Hexadecimal	16	0 – 9, A – F	A2DE16

CONVERSION BETWEEN NUMBER SYSTEMS

Converting between number systems means changing a numerical value from one base to another without changing the actual value represented.

Important: The subscript after a number indicates its base. For example, 1011₂ is binary, 25₈ is octal, 25₁₀ is decimal, and 25₁₆ is hexadecimal.

BINARY TO DECIMAL

To convert a binary number to decimal, multiply each binary digit by its corresponding power of 2 and add the results.

Example: Convert 1011₂ to decimal.

1011₂
= (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰)
= (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1)
= 8 + 0 + 2 + 1
= 11

Therefore, 1011₂ = 11₁₀

DECIMAL TO BINARY

To convert a whole decimal number to binary, repeatedly divide the number by 2 and record the remainder at each step. Continue until the quotient becomes zero. Read the remainders from bottom to top.

Example: Convert 25₁₀ to binary.

Division	Quotient	Remainder
25 ÷ 2	12	1
12 ÷ 2	6	0
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1

Reading the remainders from bottom to top:

11001

Therefore:

25₁₀ = 11001₂

BINARY TO OCTAL

Binary can be converted directly to octal by grouping the binary digits into sets of three, starting from the right. If necessary, leading zeros are added to complete the first group.

Binary-to-Octal Conversion Table

Binary	Octal
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

Example: Convert 11100101₂ to octal.

11100101₂
= 011 100 101

011 = 3
100 = 4
101 = 5

Therefore:
11100101₂ = 345₈

This is one of the useful direct conversion methods from the original number system material.

OCTAL TO BINARY

To convert octal to binary, replace each octal digit with its equivalent three-bit binary group.

Example: Convert 345₈ to binary.

3 = 011
4 = 100
5 = 101

345₈ = 011100101₂
= 11100101₂

Leading zeros may be removed from the complete number when they do not affect the value.

BINARY TO HEXADECIMAL

To convert binary to hexadecimal, group the binary digits into sets of four, starting from the right. Add leading zeros where necessary.

Binary	Hexadecimal
0000	0
0001	1
0010	2
0011	3
0100	4
0101	5
0110	6
0111	7
1000	8
1001	9
1010	A
1011	B
1100	C
1101	D
1110	E
1111	F

Example: Convert 11100101₂ to hexadecimal.

11100101₂
= 1110 0101

1110 = E
0101 = 5

Therefore:
11100101₂ = E5₁₆

The four-bit conversion relationship between binary and hexadecimal is also useful for reverse conversion.

HEXADECIMAL TO BINARY

To convert hexadecimal to binary, replace each hexadecimal digit with its four-bit binary equivalent.

Example: Convert A2DE₁₆ to binary.

A = 1010
2 = 0010
D = 1101
E = 1110

Therefore:
A2DE₁₆ = 1010001011011110₂

This direct conversion method is based on the four-bit relationship between hexadecimal and binary digits.

DECIMAL TO OCTAL

To convert a whole decimal number to octal, repeatedly divide the number by 8 and record the remainders. Read the remainders from bottom to top.

Example: Convert 1792₁₀ to octal.

Division	Quotient	Remainder
1792 ÷ 8	224	0
224 ÷ 8	28	0
28 ÷ 8	3	4
3 ÷ 8	0	3

Reading the remainders from bottom to top gives:

3400

Therefore:

1792₁₀ = 3400₈

The repeated-division method is also demonstrated in the original material.

DECIMAL TO HEXADECIMAL

To convert a whole decimal number to hexadecimal, repeatedly divide the number by 16. If a remainder is 10 or greater, replace it with the corresponding hexadecimal letter A–F.

Example: Convert 1792₁₀ to hexadecimal.

Division	Quotient	Remainder	Hexadecimal
1792 ÷ 16	112	0	0
112 ÷ 16	7	0	0
7 ÷ 16	0	7	7

Reading the hexadecimal remainders from bottom to top:

700

Therefore:

1792₁₀ = 700₁₆

OCTAL TO DECIMAL

To convert an octal number to decimal, multiply each digit by its corresponding power of 8 and add the results.

Example: Convert 345₈ to decimal.

345₈
= (3 × 8²) + (4 × 8¹) + (5 × 8⁰)
= (3 × 64) + (4 × 8) + (5 × 1)
= 192 + 32 + 5
= 229

Therefore:
345₈ = 229₁₀

This place-value method is demonstrated in the original material.

HEXADECIMAL TO DECIMAL

To convert hexadecimal to decimal, replace A–F with their decimal values, multiply each digit by its corresponding power of 16, and add the results.

Example: Convert A2DE₁₆ to decimal.

A2DE₁₆
= (A × 16³) + (2 × 16²) + (D × 16¹) + (E × 16⁰)
= (10 × 4096) + (2 × 256) + (13 × 16) + (14 × 1)
= 40960 + 512 + 208 + 14
= 41694

Therefore:
A2DE₁₆ = 41694₁₀

The original material uses this same place-value method for hexadecimal to decimal conversion.

OCTAL AND HEXADECIMAL THROUGH BINARY

When converting between octal and hexadecimal, an effective method is to use binary as an intermediate number system.

Example: Octal to Hexadecimal

Convert 345₈ to hexadecimal.

Step 1: Convert octal to binary.

3 = 011
4 = 100
5 = 101

345₈ = 011100101₂

Step 2: Group the binary digits into sets of four from the right.

011100101 = 0000 1110 0101

0000 = 0
1110 = E
0101 = 5

Therefore:
345₈ = E5₁₆

Why this works: Each octal digit corresponds to exactly three binary bits, while each hexadecimal digit corresponds to exactly four binary bits. This makes binary a useful bridge between the two systems.

IMPORTANCE OF NUMBER SYSTEMS IN COMPUTING

Number systems are important in Digital Technologies for several reasons:

1. They help computers represent and process numerical information.
2. Binary provides the basic numerical representation used by digital electronic systems.
3. Hexadecimal provides a shorter and more convenient way of writing long binary values.
4. Octal can also provide a compact representation of binary values.
5. Number-system conversion helps programmers, computer technicians and students interpret data represented in different bases.
6. Understanding number systems provides a foundation for studying computer architecture, digital electronics and programming.

COMMON ERRORS TO AVOID

• Do not use digits that are not allowed in a number system.
• Remember that binary uses only 0 and 1.
• Remember that octal uses 0 to 7 only.
• Remember that hexadecimal uses 0–9 and A–F.
• When using repeated division, read the remainders from bottom to top.
• When converting binary to octal, group the digits in threes.
• When converting binary to hexadecimal, group the digits in fours.
• Always identify the base of a number when necessary.

CLASS ACTIVITY

Activity 1: Identify the Number System

State the number system represented by each of the following:

1. 101101₂
2. 725₈
3. 348₁₀
4. AF2₁₆

Activity 2: Convert the Following

1. 1010₂ to decimal
2. 25₁₀ to binary
3. 11100101₂ to octal
4. 11100101₂ to hexadecimal
5. 345₈ to decimal

SUMMARY

A number system is a method of representing numerical values using a defined set of symbols and rules. The most common number systems studied in computing are decimal, binary, octal and hexadecimal.

Decimal has base 10 and uses 0–9. Binary has base 2 and uses 0 and 1. Octal has base 8 and uses 0–7, while hexadecimal has base 16 and uses 0–9 and A–F.

Numbers can be converted from one system to another using place-value calculations, repeated division, and direct binary grouping methods.

EVALUATION QUESTIONS

1. What is a number system?
2. What is meant by the base or radix of a number system?
3. State the base of the decimal number system.
4. List the digits used in the binary number system.
5. State the digits used in the octal number system.
6. What are the hexadecimal symbols A–F used to represent?
7. Convert 1011₂ to decimal.
8. Convert 25₁₀ to binary.
9. Convert 11100101₂ to octal.
10. Convert 345₈ to decimal.

ASSIGNMENT

1. Define number system and explain the meaning of base.
2. Write the base and symbols used by the decimal, binary, octal and hexadecimal number systems.
3. Convert 110101₂ to decimal.
4. Convert 45₁₀ to binary.
5. Convert 10101111₂ to hexadecimal.
6. Convert 572₈ to decimal.
7. Convert 255₁₀ to hexadecimal.

QUICK ONLINE QUIZ

1. What is the base of the binary number system?

A. 2 B. 8 C. 10 D. 16

2. Which digits are used in the binary number system?

A. 0 and 1 B. 1 and 2 C. 0 to 7 D. 0 to 9

3. What is 1011₂ in decimal?

A. 9 B. 10 C. 11 D. 12

4. Which of the following is a hexadecimal digit?

A. G B. H C. E D. X

5. Which number system has base 8?

A. Binary B. Decimal C. Octal D. Hexadecimal