Kenya’s diverse professional sectors frequently require precise numerical understanding applicable across finance, logistics, ICT, healthcare, and agriculture. Mastery of number systems equips diploma students to interpret, convert, and manipulate data accurately, ensuring effective decision-making and problem-solving in workplaces ranging from county government offices to banks and hospitals. This chapter develops calculation skills on number systems, place values, and absolute values, fundamental to discrete mathematics and crucial for all professional fields.
Number systems form the foundation for representing and manipulating numerical data in different formats. Understanding these systems enables professionals to convert data between formats, perform arithmetic operations, and apply these in practical contexts such as financial records, inventory counts, and digital data processing.
Number systems consist of a set of symbols and rules used to represent numbers. Key terms include:
Example 1: Identify the base of the number system if the digits used are 0, 1, 2, 3, 4, 5.
Given: Digits = {0,1,2,3,4,5}
Base = Number of digits = 6
Answer: Base 6
Example 2: Determine the place value of digit 3 in the number 4352 in the decimal system.
Given: Number = 4352 (base 10), Digit = 3
Place value of 3 is \(3 \times 10^2 = 300\)
Answer: 300
Example 3: What is the base of the number system if the largest digit used is 9?
Given: Largest digit = 9
Base = Largest digit + 1 = 10
Answer: Base 10
Example 4: In base-8 (octal), what is the base and the digit set?
Given: Base-8 system
Base = 8, digits = {0,1,2,3,4,5,6,7}
Answer: Base 8, digits 0 to 7
Example 5: What is the decimal equivalent of digit 5 in base 6?
Given: Digit 5 in base 6
Decimal equivalent = 5 (since digit values are same as decimal for single digits)
Answer: 5
Absolute value of a number is its distance from zero on the number line, regardless of direction, always non-negative. It is crucial in measuring magnitudes in finance (e.g., profit/loss), healthcare (e.g., temperature deviations), and other fields.
The absolute value of any number \(x\) is denoted \(|x|\) and defined as:
$$ |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases} $$
Example 1: Find the absolute value of -25 for a hospital’s temperature reading deviation.
Given: \(x = -25\)
$$|x| = -(-25) = 25$$
Answer: 25
Example 2: Calculate \(|x|\) if \(x = 0\) for a bank’s financial balance.
Given: \(x=0\)
$$|0|=0$$
Answer: 0
Example 3: Find the absolute value of 45, representing profit in Ksh.
Given: \(x=45\)
$$|45|=45$$
Answer: 45
Example 4: For a cooperative’s loss of Ksh -150, find the absolute value.
Given: \(x=-150\)
$$|x|=-(-150)=150$$
Answer: 150
Example 5: Calculate absolute value of \(-7.5\) degrees Celsius temperature deviation in a weather station.
Given: \(x=-7.5\)
$$|x|=-(-7.5)=7.5$$
**Answer: 7.5$$ ### 3.1.3 Place values Place value assigns value to a digit based on its position within a number. In any base \(b\), the place value of a digit \(d\) at position \(n\) (counting from right, starting at 0) is: $$ d \times b^n$$ Understanding place values is essential in accurate data entry, coding, and financial calculations. #### Worked Examples **Example 1:** Find the place value of digit 4 in 2431 (base 10) at position 2. Given: \(d=4\), \(b=10\), \(n=2\) $$4 \times 10^2 = 4 \times 100 = 400$$ **Answer: 400** **Example 2:** Determine place value of digit 3 in 1325 (base 5) at position 1. Given: \(d=3\), \(b=5\), \(n=1\) $$3 \times 5^1 = 3 \times 5 = 15$$ **Answer: 15** **Example 3:** Calculate place value of digit 1 in 1011 (base 2) at position 3. Given: \(d=1\), \(b=2\), \(n=3\) $$1 \times 2^3 = 1 \times 8 = 8$$ **Answer: 8** **Example 4:** Find place value of digit 5 in 2567 (base 8) at position 2. Given: \(d=5\), \(b=8\), \(n=2\) $$5 \times 8^2 = 5 \times 64 = 320$$ **Answer: 320** **Example 5:** In a retail business, find place value of digit 9 in 394 (base 10) at position 1. Given: \(d=9\), \(b=10\), \(n=1\) $$9 \times 10^1 = 9 \times 10 = 90$$ **Answer: 90** ### 3.1.4 Types of number systems Common number systems include: - **Decimal (Base 10):** Digits 0-9, used in daily transactions. - **Binary (Base 2):** Digits 0-1, used in computing and ICT. - **Octal (Base 8):** Digits 0-7, historically used in computing. - **Hexadecimal (Base 16):** Digits 0-9 and A-F, used in programming and digital systems. Each has unique applications in various sectors such as banking systems, hospital data management, and agricultural sensors. #### Worked Examples **Example 1:** Convert binary number 1011 to decimal. Given: Binary 1011 $$1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0$$ $$= 8 + 0 + 2 + 1 = 11$$ **Answer: 11** **Example 2:** Convert decimal 45 to binary. Given: Decimal 45 Divide by 2 repeatedly: 45 ÷ 2 = 22 remainder 1 22 ÷ 2 = 11 remainder 0 11 ÷ 2 = 5 remainder 1 5 ÷ 2 = 2 remainder 1 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Write remainders bottom to top: 101101 **Answer: 101101** **Example 3:** Convert octal number 127 to decimal. Given: Octal 127 $$1 \times 8^2 + 2 \times 8^1 + 7 \times 8^0$$ $$= 64 + 16 + 7 = 87$$ **Answer: 87** **Example 4:** Convert decimal 254 to hexadecimal. Given: Decimal 254 Divide by 16 repeatedly: 254 ÷ 16 = 15 remainder 14 (E) 15 ÷ 16 = 0 remainder 15 (F) Write remainders bottom to top: FE **Answer: FE** **Example 5:** Convert hexadecimal 1A3 to decimal. Given: Hexadecimal 1A3 Digits: 1, A(10), 3 $$1 \times 16^2 + 10 \times 16^1 + 3 \times 16^0$$ $$= 256 + 160 + 3 = 419$$
Answer: 419
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Create a free accountThis chapter introduced the fundamental concepts of number systems, beginning with key definitions, absolute values, and place values that form the basis for understanding different numbering schemes. It explored the main types of number systems including decimal, binary, octal, and hexadecimal, highlighting their unique characteristics and uses. The chapter then detailed methods for converting numbers between these systems, covering conversions from decimal to others, from other systems to decimal, as well as conversions involving binary to and from other bases. Arithmetic operations within these systems were examined, with a focus on binary addition, subtraction, multiplication, and division, including the use of ones and twos complement for subtraction. Octal and hexadecimal arithmetic operations were also discussed, specifically addition and subtraction. Finally, the chapter covered various binary codes such as Binary Coded Decimal (BCD), ASCII, Gray Code, and Excess-3, explaining their applications and how arithmetic operations are performed within BCD. This comprehensive coverage equips students with the necessary skills to work confidently with different number systems and their applications in discrete mathematics.