Irrational Numbers • Topic 3 of 3

The Real Number Line

How to Represent \(\sqrt{2}\) on a Number Line: We can use the Pythagorean theorem to construct irrational numbers geometrically.

Method for \(\sqrt{2}\):

  • Draw a number line. Mark 0 and 1.
  • At point 1, draw a perpendicular line of length 1 unit.
  • Join 0 to the top of this perpendicular. The length of this line = \(\sqrt{1^2 + 1^2} = \sqrt{2}\).
  • Using a compass, transfer this length to the number line.

Method for \(\sqrt{3}\):

  • First construct \(\sqrt{2}\) as above.
  • At point \(\sqrt{2}\), draw a perpendicular of length 1.
  • Join 0 to the top. Length = \(\sqrt{(\sqrt{2})^2 + 1^2} = \sqrt{2 + 1} = \sqrt{3}\).

Approximation Methods for Irrational Numbers:

MethodDescriptionExample for \(\sqrt{2}\)
Guess and checkFind perfect squares around the number\(1^2=1\), \(2^2=4\), so \(\sqrt{2}\) is between 1 and 2
Long division methodSystematic digit-by-digit calculation1.414213...
Using calculatorQuick approximation1.414213562

Approximating \(\pi\):

  • \(\pi \approx 3.14\) (to 2 decimal places)
  • \(\pi \approx 3.142\) (to 3 decimal places)
  • \(\pi \approx \frac{22}{7} \approx 3.142857\) (fraction approximation)
  • \(\pi \approx 3.1416\) (to 4 decimal places)
The Real Number Line0123−1√2√3√5πIrrational numbers fill the 'gaps' between rational numbers on the real line.Together: Every point on the number line is a real number.
Solved Examples
1
Worked Example
Between which two consecutive integers do the following irrational numbers lie? (a) \(\sqrt{10}\) (b) \(\sqrt{20}\)
Solution- (a) \(3^2 = 9\), \(4^2 = 16\). Since \(9 < 10 < 16\), \(\sqrt{10}\) is between 3 and 4. - (b) \(4^2 = 16\), \(5^2 = 25\). Since \(16 < 20 < 25\), \(\sqrt{20}\) is between 4 and 5. - **Answer:** (a) Between 3 and 4, (b) Between 4 and 5. *Example 2: Approximate \(\sqrt{10}\) to two decimal places using guess and check. Solution: - \(3^2 = 9\) (too low), \(4^2 = 16\) (too high) → between 3 and 4 - Try \(3.1^2 = 9.61\) (low), \(3.2^2 = 10.24\) (high) → between 3.1 and 3.2 - Try \(3.16^2 = 9.9856\) (low), \(3.17^2 = 10.0489\) (high) → between 3.16 and 3.17 - \(3.162^2 = 9.998244\), \(3.163^2 = 10.004569\) - To two decimal places, \(\sqrt{10} \approx 3.16\) - **Answer:** 3.16. *Example 3: On a number line, between which two marked points would \(\sqrt{7}\) be placed if the number line is marked at integers? Solution: - \(2^2 = 4\), \(3^2 = 9\) - Since \(4 < 7 < 9\), \(\sqrt{7}\) is between \(\sqrt{4}=2\) and \(\sqrt{9}=3\) - **Answer:** Between 2 and 3.

Key Points

  • Irrational numbers can be constructed geometrically using the Pythagorean theorem.
  • \(\sqrt{n}\) for a non-perfect square \(n\) lies between \(\sqrt{a^2}\) and \(\sqrt{(a+1)^2}\) where \(a^2 < n < (a+1)^2\).
  • Approximation methods: guess and check, long division, calculator.
  • The spiral of Theodorus shows consecutive square roots geometrically.
  • \(\pi \approx 3.14\) and \(\sqrt{2} \approx 1.414\) are common approximations.
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Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Real numbers are made up of:
Explanation: Rationals + irrationals.
Q2.Every point on the number line represents a:
Explanation: A real number.
Q3.$\sqrt2$ can be located using a right triangle with legs:
Explanation: Hypotenuse $=\sqrt2$.
Q4.Is every real number rational?
Explanation: Irrationals are also real.
Practice more MCQs → 📝 Assignment · 35 marks