Real Numbers
Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.
First Principles
Objective: Identify, distinguish, and classify numbers into the real number hierarchy: Natural numbers (\(\mathbb{N}\)), Whole numbers (\(\mathbb{W}\)), Integers (\(\mathbb{Z}\)), Rational numbers (\(\mathbb{Q}\)), and Irrational numbers (\(\mathbb{Q}'\)).
The Boma Metaphor: Imagine nested enclosures (bomas) on a Kenyan farm. The innermost boma holds counting numbers for livestock. Larger fences enclose zero, debts/negatives, and measured fractions. Everything that can be measured or located on the continuous number line lives inside the Grand Real Enclosure (\(\mathbb{R}\)).
Interactive Number Sorter
Click any number tag below to trace its exact nesting through the number sets:
Belongs to: Rational, Real
Reason: 22/7 is already written as p/q with integers 22 and 7 (q ≠ 0). It is rational, even though it approximates π.
(a) Hierarchy of Number Sets
Understanding real numbers requires seeing how each set builds directly on the limitations of the previous one:
- Natural Numbers (\(\mathbb{N}\)): \(\{1, 2, 3, 4, \dots\}\) — used for counting indivisible objects.
- Whole Numbers (\(\mathbb{W}\)): \(\{0, 1, 2, 3, \dots\}\) — includes zero to indicate empty sets.
- Integers (\(\mathbb{Z}\)): \(\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}\) — accounts for deficits, depths, and negative quantities.
- Rational Numbers (\(\mathbb{Q}\)): Any value expressible as \(\frac{p}{q}\) where \(p, q \in \mathbb{Z}\) and \(q \neq 0\). In decimal form, rationals either terminate (e.g., \(0.75 = \frac{3}{4}\)) or repeat periodically (e.g., \(0.3636\dots = \frac{4}{11}\)).
- Irrational Numbers (\(\mathbb{Q}'\)): Real numbers that cannot be written as \(\frac{p}{q}\). Their decimal expansions are non-terminating and non-repeating (e.g., \(\sqrt{2}, \sqrt{3}, \pi, e\)).
- Real Numbers (\(\mathbb{R}\)): The complete continuum \(\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'\). Every single point on a continuous geometric line represents a unique real number.
Key Formulas
Formal Definition of Rational Numbers (\(\mathbb{Q}\)):
\[\mathbb{Q} = \left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z}, \; q \neq 0 \right\}\] A number is rational if and only if it can be represented as an integer fraction with non-zero denominator.Set Inclusion Nesting:
\[\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\] Every natural number is a whole number; every whole number is an integer; every integer is rational; every rational is real.Partition of the Real Continuum:
\[\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}' \quad \text{and} \quad \mathbb{Q} \cap \mathbb{Q}' = \emptyset\] A real number is either rational or irrational; it can never be both.Decimal Characterization:
\[x \in \mathbb{Q} \iff x \text{ is a terminating or repeating decimal}\] \[x \in \mathbb{Q}' \iff x \text{ is a non-terminating, non-repeating decimal}\]Recurring Decimal to Rational Conversion Formula:
\[x = 0.\overline{a_1 a_2 \dots a_n} \implies x = \frac{a_1 a_2 \dots a_n}{\underbrace{99\dots9}_{n \text{ nines}}}\] Example: \(0.\overline{36} = \frac{36}{99} = \frac{4}{11}\).Worked Examples
Example 1 (Easy): Classifying Terminating Decimals and Integers
Problem: Classify the numbers \(-8\) and \(0.875\) into their most specific and broad number sets.
- Analyze \(-8\):
- It is not a counting number or zero, so \(-8 \notin \mathbb{N}\) and \(-8 \notin \mathbb{W}\).
- It is a whole quantity with a negative sign: \(-8 \in \mathbb{Z}\) (Integer).
- Can be written as \(\frac{-8}{1}\), so \(-8 \in \mathbb{Q}\) (Rational) and \(-8 \in \mathbb{R}\) (Real).
- Analyze \(0.875\):
- Terminates after 3 decimal places: \(0.875 = \frac{875}{1000}\).
- Simplify fraction: \(\frac{875 \div 125}{1000 \div 125} = \frac{7}{8}\).
- Since \(7, 8 \in \mathbb{Z}\) and \(8 \neq 0\), \(0.875 \in \mathbb{Q}\) (Rational) and \(0.875 \in \mathbb{R}\) (Real).
Example 2 (Medium): Converting Repeating Decimals to Rational Fractions
Problem: A surveyor records an angular measurement containing the recurring decimal \(x = 1.2454545\dots = 1.2\overline{45}\). Show that this number is rational by converting it to a simplified fraction \(\frac{p}{q}\).
- Let \(x = 1.2454545\dots\)
- Multiply by \(10\) to isolate the non-repeating part in front of the decimal point: \[10x = 12.454545\dots \quad \text{--- (Equation 1)}\]
- The repeating block has \(2\) digits (\(45\)), so multiply Equation 1 by \(10^2 = 100\): \[1000x = 1245.454545\dots \quad \text{--- (Equation 2)}\]
- Subtract Equation 1 from Equation 2 to eliminate the infinite repeating tail: \[1000x - 10x = 1245.4545\dots - 12.4545\dots\] \[990x = 1233\]
- Solve for \(x\) and simplify by dividing numerator and denominator by their greatest common divisor (\(\gcd(1233, 990) = 9\)): \[x = \frac{1233}{990} = \frac{1233 \div 9}{990 \div 9} = \frac{137}{110}\]
- Since \(137, 110 \in \mathbb{Z}\) and \(110 \neq 0\), \(1.2\overline{45}\) is strictly Rational (\(\mathbb{Q}\)).
Example 3 (Hard): Proof of Irrationality & Finding Rationals Between Irrationals
Problem:
(a) Prove whether \(5 - 2\sqrt{3}\) is rational or irrational, given that \(\sqrt{3}\) is irrational.
(b) Construct a rational number with denominator \(50\) that lies strictly between \(\sqrt{2}\) and \(\sqrt{3}\).
- Part (a) Proof by Contradiction:
- Assume \(5 - 2\sqrt{3}\) is rational. Then \(5 - 2\sqrt{3} = \frac{a}{b}\) for integers \(a, b\) (\(b \neq 0\)).
- Rearranging algebraically: \[2\sqrt{3} = 5 - \frac{a}{b} = \frac{5b - a}{b} \implies \sqrt{3} = \frac{5b - a}{2b}\]
- Since \(a, b \in \mathbb{Z}\), \(5b - a\) is an integer and \(2b\) is a non-zero integer.
- This implies \(\sqrt{3}\) is rational, which directly contradicts the fact that \(\sqrt{3}\) is irrational.
- Therefore, the assumption was false: \(5 - 2\sqrt{3}\) is Irrational (\(\mathbb{Q}'\)).
- Part (b) Finding a Rational Between \(\sqrt{2}\) and \(\sqrt{3}\):
- We know \(\sqrt{2} \approx 1.4142\dots\) and \(\sqrt{3} \approx 1.7320\dots\)
- We need an integer \(k\) such that: \(1.4142 < \frac{k}{50} < 1.7320\)
- Multiply all sides by 50: \(1.4142 \times 50 < k < 1.7320 \times 50 \implies 70.71 < k < 86.60\)
- Choosing \(k = 75\): \[\frac{75}{50} = \frac{3}{2} = 1.50\]
- Check: \(1.4142 < 1.50 < 1.7320\). Thus, \(\frac{75}{50} = \frac{3}{2}\) is a valid rational number in the interval.
Common Mistakes
Misconception 1: "\(\frac{22}{7}\) is irrational because \(\pi\) is irrational."
Misconception 2: "Every decimal that does not stop is irrational."
Misconception 3: "Multiplying two irrational numbers always produces an irrational number."
Real World
1. Livestock & Shamba Auditing (\(\mathbb{N}\) and \(\mathbb{W}\))
A farmer in Eldoret counting dairy cattle in a boma uses natural numbers \(\mathbb{N} = \{1, 2, 3, \dots\}\). If all cattle are sold, the boma count is \(0 \in \mathbb{W}\). You cannot own \(-4\) or \(3.7\) live cows.
2. Night Temperatures & Mobile Money Deficits (\(\mathbb{Z}\))
In high-altitude areas like Nyahururu or Mount Kenya, night temperatures drop to \(-2\,^{\circ}\text{C}\). In mobile banking (e.g., Fuliza overdrafts), an account balance can read \(-\text{Ksh } 450\). These require negative integers.
3. Land Surveying & Construction Foundations (\(\mathbb{Q}'\))
When civil engineers lay out a square building foundation of \(10\text{ m} \times 10\text{ m}\), the diagonal distance by the Pythagorean theorem is exactly \(\sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}\text{ m}\). Because \(\sqrt{2}\) is irrational, this distance can never be represented by an exact fraction, necessitating laser and decimal rounding in construction.
Practice