Irrational Numbers • Topic 1 of 3

What Are Irrational Numbers?

What are Irrational Numbers? Irrational numbers are numbers that cannot be written as a simple fraction (ratio) of two integers. In other words, they cannot be expressed in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).

What is the Real Number System? The real number system is the collection of all rational and irrational numbers together. Every point on the number line represents a real number.

Key Differences between Rational and Irrational Numbers:

Rational NumbersIrrational Numbers
Can be written as \(\frac{p}{q}\)Cannot be written as \(\frac{p}{q}\)
Decimal expansion terminates or repeatsDecimal expansion never terminates and never repeats
Examples: \(\frac{1}{2}=0.5\), \(\frac{1}{3}=0.\overline{3}\)Examples: \(\sqrt{2}=1.414213...\), \(\pi=3.141592...\)

Famous Irrational Numbers:

  • \(\sqrt{2}\) (square root of 2)
  • \(\sqrt{3}\), \(\sqrt{5}\), \(\sqrt{6}\), \(\sqrt{7}\), \(\sqrt{8}\), \(\sqrt{10}\) (square roots of non-perfect squares)
  • \(\pi\) (pi) ≈ 3.14159265358979...
  • \(e\) (Euler's number) ≈ 2.71828182845904...

Properties of Irrational Numbers:

  • The sum of a rational and an irrational number is irrational.
  • The product of a non-zero rational and an irrational number is irrational.
  • The sum or product of two irrational numbers may be rational or irrational.
Real Numbers: Rational vs IrrationalRATIONAL ℚIRRATIONALCan be written as p/qDecimal terminates or repeatsExamples:1/2 = 0.5 (terminates)1/3 = 0.333… (repeats)−7, 0, 4, 2.5Cannot be written as p/qDecimal never terminatesor repeatsExamples:√2 = 1.41421356…π = 3.14159265… φ, eTogether they form the Real Number Line
Solved Examples
1
Worked Example
Identify whether the following numbers are rational or irrational: (a) \(\sqrt{25}\) (b) \(\sqrt{10}\) (c) \(\frac{22}{7}\)
Solution- \(\sqrt{25} = 5\), which can be written as \(\frac{5}{1}\) → **Rational - \(\sqrt{10}\) cannot be simplified to a fraction → **Irrational - \(\frac{22}{7}\) is already in fraction form → **Rational - **Answer:** (a) Rational, (b) Irrational, (c) Rational. *Example 2: Which of the following is NOT an irrational number? (a) \(\sqrt{8}\) (b) \(\sqrt{9}\) (c) \(\sqrt{12}\) (d) \(\sqrt{15}\) Solution: - \(\sqrt{9} = 3\), which is a rational number (can be written as \(\frac{3}{1}\)) - All other options are square roots of non-perfect squares → irrational - **Answer:** (b) \(\sqrt{9}\). *Example 3: Give an example to show that: (a) Sum of two irrational numbers can be rational (b) Product of two irrational numbers can be rational Solution: - (a) Take \(\sqrt{2}\) and \(-\sqrt{2}\). Both are irrational. Their sum = \(\sqrt{2} + (-\sqrt{2}) = 0\), which is rational. - (b) Take \(\sqrt{2}\) and \(\sqrt{2}\). Both are irrational. Their product = \(\sqrt{2} \times \sqrt{2} = 2\), which is rational. - **Answer:** (a) \(\sqrt{2} + (-\sqrt{2}) = 0\), (b) \(\sqrt{2} \times \sqrt{2} = 2\).

Key Points

  • Irrational numbers cannot be written as \(\frac{p}{q}\) where \(p\) and \(q\) are integers (\(q \neq 0\)).
  • Real numbers = Rational numbers ∪ Irrational numbers.
  • Decimal expansion of irrational numbers is non-terminating and non-repeating.
  • Square roots of non-perfect squares are irrational.
  • \(\pi\) and \(e\) are famous irrational numbers.
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.An irrational number cannot be written as:
Explanation: Not a ratio of integers.
Q2.The decimal form of an irrational number is:
Explanation: Non-terminating, non-repeating.
Q3.Which is irrational?
Explanation: $\sqrt2$.
Q4.$\sqrt{16}$ is:
Explanation: $=4$, rational.
Practice more MCQs → 📝 Assignment · 35 marks