By the end of this chapter, you will be able to:
These skills are essential because they help you work confidently with digital systems and technology in your trade.
Number systems form the foundation of all digital computing and data representation, essential knowledge for ICT technicians who design, troubleshoot, and maintain computer systems and networks. Understanding different number systems enables effective data encoding, error detection, and system programming. This chapter focuses on identifying and working with number systems commonly used in ICT, with emphasis on practical applications relevant to Kenyan ICT technicians.
Number systems provide structured ways to represent numbers using different bases. In ICT, the choice of number system affects data storage, processing, and communication. Kenyan ICT technicians must grasp these systems to handle binary data, convert between formats, and debug digital devices efficiently.
Number systems are mathematical frameworks for representing numerical values using a set of symbols and a base or radix. Key terms include:
Example 1: Identify the base and digits of the number 1011.
Given: Number = 1011
The digits are 1, 0, 1, 1.
Since digits are only 0 and 1, this is base 2 (binary).
Answer: Base 2, digits {1, 0}
Example 2: Determine if 1A3 is valid in base 10.
Given: Number = 1A3
Digits are 1, A, 3.
A is not a decimal digit (0–9), so invalid in base 10.
Answer: Not valid in base 10
Example 3: Identify the base of number 2F3.
Given: Number = 2F3
Digits include 2, F, 3.
F is valid in hexadecimal (base 16).
Answer: Base 16 (hexadecimal)
ICT technicians commonly encounter several number systems, each with unique uses in computing and communications.
Binary uses base 2 with digits 0 and 1. It is fundamental in digital electronics where two voltage levels represent 0 and 1.
$$N = \sum_{i=0}^{n-1} d_i \times 2^i$$
where \(d_i\) is the digit at position \(i\), starting from 0 at the rightmost digit.
Example 1: Convert binary 1010 to decimal.
Given: Binary = 1010
$$N = 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0$$
$$N = 1 \times 8 + 0 + 1 \times 2 + 0$$
$$N = 8 + 0 + 2 + 0 = 10$$
Answer: 10 decimal
Example 2: Convert binary 11011 to decimal.
Given: Binary = 11011
$$N = 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$$
$$N = 16 + 8 + 0 + 2 + 1 = 27$$
Answer: 27 decimal
Example 3: Convert binary 100101 to decimal.
Given: Binary = 100101
$$N = 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0$$
$$N = 32 + 0 + 0 + 4 + 0 + 1 = 37$$
Answer: 37 decimal
Decimal uses base 10 with digits 0–9. It is the default system for human counting and arithmetic.
$$N = \sum_{i=0}^{n-1} d_i \times 10^i$$
where \(d_i\) is the digit at position \(i\).
Example 1: Express decimal 345 in expanded form.
Given: Decimal = 345
$$N = 3 \times 10^2 + 4 \times 10^1 + 5 \times 10^0$$
$$N = 300 + 40 + 5 = 345$$
Answer: 345
Example 2: Convert decimal 789 to expanded form.
Given: Decimal = 789
$$N = 7 \times 10^2 + 8 \times 10^1 + 9 \times 10^0$$
$$N = 700 + 80 + 9 = 789$$
Answer: 789
Example 3: Convert decimal 1204 to expanded form.
Given: Decimal = 1204
$$N = 1 \times 10^3 + 2 \times 10^2 + 0 \times 10^1 + 4 \times 10^0$$
$$N = 1000 + 200 + 0 + 4 = 1204$$
Answer: 1204
Octal uses base 8 with digits 0–7. It is used in some computing systems as a shorthand for binary.
$$N = \sum_{i=0}^{n-1} d_i \times 8^i$$
Example 1: Convert octal 17 to decimal.
Given: Octal = 17
$$N = 1 \times 8^1 + 7 \times 8^0$$
$$N = 8 + 7 = 15$$
Answer: 15 decimal
Example 2: Convert octal 25 to decimal.
Given: Octal = 25
$$N = 2 \times 8^1 + 5 \times 8^0$$
$$N = 16 + 5 = 21$$
Answer: 21 decimal
Example 3: Convert octal 130 to decimal.
Given: Octal = 130
$$N = 1 \times 8^2 + 3 \times 8^1 + 0 \times 8^0$$
$$N = 64 + 24 + 0 = 88$$
Answer: 88 decimal
Hexadecimal uses base 16 with digits 0–9 and letters A–F representing values 10–15. It is widely used in programming and memory addressing.
$$N = \sum_{i=0}^{n-1} d_i \times 16^i$$
where \(d_i\) can be 0–9 or A–F (10–15).
Example 1: Convert hexadecimal 1A to decimal.
Given: Hex = 1A
\(A = 10\)
$$N = 1 \times 16^1 + 10 \times 16^0$$
$$N = 16 + 10 = 26$$
Answer: 26 decimal
Example 2: Convert hexadecimal 2F to decimal.
Given: Hex = 2F
\(F = 15\)
$$N = 2 \times 16^1 + 15 \times 16^0$$
$$N = 32 + 15 = 47$$
Answer: 47 decimal
Example 3: Convert hexadecimal 3B7 to decimal.
Given: Hex = 3B7
\(B = 11\)
$$N = 3 \times 16^2 + 11 \times 16^1 + 7 \times 16^0$$
$$N = 3 \times 256 + 11 \times 16 + 7$$
$$N = 768 + 176 + 7 = 951$$
Answer: 951 decimal
Create a free account to open more of this chapter.
Free: practical guides, quick cards, workplace scenarios and more.
Create a free accountThis chapter began by identifying various number systems, defining key terms and exploring types such as decimal, binary, octal, and hexadecimal. It then covered base conversion techniques, including converting from decimal to other number systems, from other number systems to decimal, from binary to other systems, and vice versa. The chapter proceeded to arithmetic operations within these number systems, detailing binary arithmetic, the use of complements, and arithmetic in octal and hexadecimal systems. It also examined different binary codes such as Binary Coded Decimal (BCD), ASCII, EBCDIC, Gray code, and Excess-3, explaining their structure and applications. Representation of binary coded decimal was discussed to highlight how decimal digits are expressed in binary form. Finally, the chapter addressed arithmetic operations involving BCD, demonstrating how calculations are performed using binary coded decimal values. This comprehensive treatment equips learners with essential skills to work confidently across various number systems and their arithmetic processes.
Convert the decimal number \(156\) to its equivalent in binary. (2 marks)
A network device uses an 8-bit binary address. Convert the binary address \(11010101_2\) to decimal. (2 marks)
At the start of this chapter we promised you would be able to:
Tick each one you can genuinely do.
So, are you there yet?
You're competent when you can confidently do 50% or more of what this chapter promised.
Sign in to record how you're doing.