Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Rational and non-rational numbers
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Rational and non-rational numbers
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Junior Secondary 3
Term: 1st Term
Week: 6
Theme: Numbers And Numeration
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
This topic introduces students to the classification of real numbers into rational and non-rational (irrational) categories. Understanding this distinction is fundamental to number theory and forms the basis for more advanced mathematical concepts. It is crucial for appreciating the properties of numbers encountered in various calculations, measurements, and real-world problem-solving. For Nigerian learners, this knowledge is applied in practical scenarios such as accurate measurements in carpentry, tailoring, engineering design (e.g., constructing circular objects), and even in understanding financial calculations involving fractions and percentages.
This section provides in-depth explanations of rational and non-rational numbers, including definitions, properties, and examples relevant to the Nigerian context. 2.1 Number Systems Hierarchy (Brief Review) Before delving into rational and non-rational numbers, it is important to briefly recall the hierarchy of number systems leading up to them: Natural Numbers (Counting Numbers): {1, 2, 3, ...} Whole Numbers: {0, 1, 2, 3, ...} (Natural numbers plus zero)
Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...} (Whole numbers and their negatives) All these numbers are subsets of rational numbers. 2.2 Rational Numbers A rational number is any number that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not equal to zero ($q \neq 0$).
Characteristics of Rational Numbers:
1. Fraction Form: Can always be written as $\frac{p}{q}$.
2. Decimal Representation: When expressed as a decimal, a rational number will either: Terminate: The decimal representation ends after a finite number of digits (e.g., 0.5, 0.25, 1.75).
Repeat (or recur): The decimal representation continues infinitely, but a block of digits repeats itself endlessly (e.g., 0.333..., 0.142857142857...).
Examples of Rational Numbers: Integers: All integers are rational numbers because they can be written as $\frac{\text{integer}}{1}$.
Example: $5 = \frac{5}{1}$, $-3 = \frac{-3}{1}$ Fractions: All common fractions are rational numbers by definition.
Example: $\frac{1}{2}$, $\frac{3}{4}$, $- \frac{7}{8}$ Mixed Numbers: Can be converted to improper fractions.
Example: $2 \frac{1}{3} = \frac{7}{3}$ Terminating Decimals: Can be written as fractions with powers of 10 as denominators.
Example: $0.75 = \frac{75}{100} = \frac{3}{4}$
Example: $2.125 = \frac{2125}{1000} = \frac{17}{8}$ Repeating Decimals: Can be converted to fractions using algebraic methods.
Example: $0.333... = \frac{1}{3}$
Example: $0.161616... = \frac{16}{99}$ Worked Examples (Converting Decimals to Fractions): Example 1: Convert $0.75$ to a fraction in its simplest form.
Step 1: Write the decimal as a fraction with a denominator that is a power of 10. $0.75 = \frac{75}{100}$ (since there are two decimal places)
Step 2: Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD). The GCD of 75 and 100 is 25. $\frac{75 \div 25}{100 \div 25} = \frac{3}{4}$ Result: $0.75 = \frac{3}{4}$ (a rational number).
Example 2: Convert $0.666...$ (or $0.\bar{6}$) to a fraction in its simplest form.
Step 1: Let $x$ be the repeating decimal. $x = 0.666... \quad (Equation 1)$ Step 2: Multiply Equation 1 by 10 (since one digit is repeating). $10x = 6.666... \quad (Equation 2)$ Step 3: Subtract Equation 1 from Equation 2. $10x - x = 6.666... - 0.666...$ $9x = 6$ Step 4: Solve for $x$. $x = \frac{6}{9}$ Step 5: Simplify the fraction. $x = \frac{2}{3}$ Result: $0.666... = \frac{2}{3}$ (a rational number).
Example 3: Convert $1.25$ to a fraction in its simplest form.
Step 1: Separate the whole number part and the decimal part. $1.25 = 1 + 0.25$ Step 2: Convert the decimal part to a fraction. $0.25 = \frac{25}{100} = \frac{1}{4}$ Step 3: Combine the whole number and the fraction. $1 + \frac{1}{4} = 1 \frac{1}{4}$ Step 4: Convert the mixed number to an improper fraction. $1 \frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{5}{4}$ Result: $1.25 = \frac{5}{4}$ (a rational number). 2.3 Non-rational (Irrational) Numbers An irrational number (or non-rational number) is any real number that cannot be expressed as a simple fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
Characteristics of Irrational Numbers:
1. Cannot be Written as a Fraction: There is no combination of two integers whose ratio equals an irrational number.
2. Decimal Representation: When expressed as a decimal, an irrational number will always be: Non-terminating: The decimal representation continues infinitely.
Non-repeating: No block of digits repeats itself endlessly.
Examples of Irrational Numbers: Square Roots of Non-Perfect Squares: $\sqrt{2} \approx 1.41421356...$ (used in geometry for diagonals) $\sqrt{3} \approx 1.73205081...$ $\sqrt{5} \approx 2.23606798...$ $\sqrt{10} \approx 3.16227766...$ Pi (π): This is the most famous irrational number.
Definition: π is the as a Fraction: There is no combination of two integers whose ratio equals an irrational number.
2. Decimal Representation: When expressed as a decimal, an irrational number will always be: Non-terminating: The decimal representation continues infinitely.
Non-repeating: No block of digits repeats itself endlessly.
Examples of Irrational Numbers: Square Roots of Non-Perfect Squares: $\sqrt{2} \approx 1.41421356...$ (used in geometry for diagonals) $\sqrt{3} \approx 1.73205081...$ $\sqrt{5} \approx 2.23606798...$ $\sqrt{10} \approx 3.16227766...$ Pi (π): This is the most famous irrational number.
Definition: π is the ratio of a circle's circumference ($C$) to its diameter ($D$). That is, $\pi = \frac{C}{D}$.
Value: $\pi \approx 3.1415926535...$ Its decimal representation is non-terminating and non-repeating.
Common Approximations for calculations: $\frac{22}{7}$ or $3.14$ or $3.142$. It is crucial to understand that these are approximations and not the exact value of π, which is irrational.
Other transcendental numbers: For JSS3, focus on $\pi$ and square roots of non-perfect squares. (e.g., Euler's number 'e' is also irrational, but too advanced for this level). Worked Example (Understanding Irrationality): * Why is $\sqrt{2}$ irrational?** If one were to calculate $\sqrt{2}$ using a calculator, the display would show something like $1.41421356...$. The digits continue indefinitely without any discernible repeating pattern. No matter how many decimal places are calculated, it will never end or repeat a sequence. This property makes it impossible to express $\sqrt{2}$ as a fraction of two integers. --- discussion, sharing their findings and observing the consistency of the $\frac{C}{D}$ ratio across different objects. 3.5 Classification Practice (15 minutes)
Teacher Activity: Write a mixed list of numbers on the board.
Example: $0.25, \sqrt{9}, \sqrt{7}, \frac{5}{3}, -6, 0.12345..., \pi, 2\frac{1}{2}, 0.888...$ Instruct students, individually or in pairs, to classify each number as "Rational" or "Non-rational (Irrational)" and briefly explain their reasoning. Review answers as a class, discussing each number.
Student Activity: Classify the given numbers. Justify their classifications. * Correct any misconceptions during the class review. --- This section outlines detailed activities for the teacher and students to facilitate understanding of rational and non-rational numbers. 3.1 Introduction and Review (10 minutes)
Teacher Activity: Begin by asking students to recall different types of numbers they have learned (natural, whole, integers, fractions, decimals). Write a few numbers on the board (e.g., 5, -2, 1/2, 0.75, 0.333...).
Ask: "Can these numbers all be written as a fraction of two integers?" Introduce the terms "rational" and "non-rational (irrational)" numbers as the classification for all these types.
Student Activity: Respond to teacher's questions about number types. Attempt to express simple numbers as fractions. 3.2 Exploring Rational Numbers (20 minutes)
Teacher Activity: Formally define rational numbers, emphasising the $\frac{p}{q}$ form where $p, q$ are integers and $q \ne 0$.
Provide various examples: Integers: e.g., $7 = \frac{7}{1}$, $-4 = \frac{-4}{1}$ Fractions: e.g., $\frac{2}{5}$, $1\frac{3}{4} = \frac{7}{4}$ Terminating decimals: e.g., $0.6 = \frac{6}{10} = \frac{3}{5}$, $1.25 = \frac{125}{100} = \frac{5}{4}$ Repeating decimals: e.g., $0.444... = \frac{4}{9}$, $0.121212... = \frac{12}{99} = \frac{4}{33}$ Guide students through converting a few terminating and simple repeating decimals to fractions using the methods explained in the "Key Concepts" section (e.g., $0.8$ to $\frac{4}{5}$, $0.333...$ to $\frac{1}{3}$).
Student Activity: Listen to definitions and examples. Practise converting given decimals to fractions individually or in pairs. Ask clarifying questions. 3.3 Introducing Non-Rational (Irrational)
Numbers (15 minutes)
Teacher Activity: Introduce irrational numbers as numbers that cannot be expressed as $\frac{p}{q}$.
Explain their decimal representation: non-terminating and non-repeating.
Provide examples: $\sqrt{2}, \sqrt{3}, \sqrt{5}$. Briefly explain that square roots of non-perfect squares are irrational. Introduce $\pi$ as a prominent irrational number. Explain its definition as the ratio of a circle's circumference to its diameter ($\pi = \frac{C}{D}$). Emphasise its non-terminating and non-repeating nature. Mention common approximations like $\frac{22}{7}$ and $3.14$, stressing that these are approximations.
Student Activity: Listen and take notes on the definition and examples of irrational numbers. Use calculators to find decimal values of $\sqrt{2}$ or $\sqrt{3}$ to observe their non-repeating nature (optional, depending on availability). 3.4 Practical Determination of the Approximate Value of Pi (π) (30 minutes)
Teacher Activity: Divide the class into small groups (e.g., 4-5 students per group).
Distribute materials: string, rulers/tape measures, and various circular objects (e.g., lids of "Gari" buckets, tin milk cans, coins, plates, bicycle tyres, Bangles/Wristwatches from students etc.). Aim for at least 3-4 objects per group.
Instruct each group to:
1. Measure the circumference (C) of each circular object using the string and then measuring the string's length with a ruler.
2. Measure the diameter (D) of the same object by measuring across its widest point (or folding the string in half that covered the circumference and measuring the string's new length, then dividing by 2 to get diameter).
3. Record C and D in a table.
4. Calculate the ratio $\frac{C}{D}$ for each object.
5. Record the calculated ratio in the table. Guide groups as they work, ensuring accurate measurements and calculations. After calculations, lead a class discussion on the results.
Ask: "What do you notice about the values of $\frac{C}{D}$ you calculated for different objects?" Consolidate findings: The values should be approximately $3.14$. Explain that this constant ratio is known as pi ($\pi$) and reiterate its irrational nature, while their practical results are approximations.
Student Activity: Work in groups to measure and record data. Calculate the $\frac{C}{D}$ ratio for each object. Participate in the class discussion, sharing their findings and observing the consistency of the $\frac{C}{D}$ ratio across different objects. 3.5 Classification Practice (15 minutes)
Teacher Activity: Write a mixed list of numbers on the board.
Example: $0.25, \sqrt{9}, \sqrt{7}, \frac{5}{3}, -6, 0.12345..., \pi, 2\frac{1}{2}, 0.888...$ Instruct students, individually or in pairs, to classify each number as "Rational" or "Non-rational (Irrational)" and briefly explain their reasoning. Review answers as a class, discussing each number.
Student Activity: Classify the given numbers. Justify their classifications. Correct any misconceptions during the
Construction and Engineering (e.g., Bridge Design, Water Storage Tanks): Rational numbers are fundamental for all linear measurements, material quantities, and cost calculations in construction projects (e.g., $\frac{1}{2}$ bag of cement, $0.75$ cubic meters of sand). Irrational numbers, specifically $\pi$, are critical when designing and building circular structures common in Nigeria.
Examples include: Culverts and pipes: Calculating the required diameter or circumference for drainage systems in cities like Lagos or Abuja.
Water storage tanks: Determining the volume of cylindrical tanks to ensure adequate water supply for communities. The formula for the volume of a cylinder, $V = \pi r^2 h$, directly involves $\pi$.
Roundabouts and circular buildings: Architects and civil engineers use $\pi$ to calculate perimeter, area, and structural integrity.
Application: A local engineer calculating the material needed for a cylindrical concrete pillar would use the formula for circumference ($C = \pi D$) or area ($A = \pi r^2$), requiring a rational approximation of $\pi$. Tailoring and Textile Design (e.g., Circular Skirts, Traditional Attire): Nigerian tailors frequently work with fractions and decimals for fabric measurements (e.g., buying $2 \frac{1}{2}$ yards of Ankara fabric, cutting a piece $0.8$ meters long). These are rational numbers. When creating patterns for circular designs, such as the flare of a traditional "iro" (wrapper) or the base of a "gele" (headtie), or a circular skirt, the concept of $\pi$ comes into play.
Application: A seamstress designing a flowing circular skirt needs to calculate the radius from the desired waist circumference, using the formula $C = 2\pi r$. This involves working with $\pi$ and its rational approximations to ensure the pattern is cut correctly and fits well. Agriculture and Land Management (e.g., Irrigation Systems, Farm Demarcation): Farmers use rational numbers for dividing land, calculating yields per square meter, or mixing pesticide solutions (e.g., a concentration of $\frac{1}{50}$ part pesticide). For modern farming practices, especially with efficient irrigation, circular pivot irrigation systems are common.
Application: A large-scale farmer in Kano planning a circular irrigation system needs to calculate the area of the circular plot to determine water requirements and coverage. The area formula $A = \pi r^2$ will be used, making $\pi$ an indispensable value in optimising farm operations. Even the square root of non-perfect squares might appear when calculating diagonals of plots. ---