Exercise 1.2 Practice
Irrational Numbers
Q1: True or False
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State whether the following statements are true or false. Justify your answers.
(i) Every rational number is a real number.
(ii) Every point on the number line represents a rational number.
(iii) Every real number is either rational or irrational.
(i) Every rational number is a real number.
(ii) Every point on the number line represents a rational number.
(iii) Every real number is either rational or irrational.
(i) True: Real numbers are the collection of all rational and irrational numbers. Since rational numbers are part of this collection, every rational number is a real number.
(ii) False: The number line contains infinitely many points that are not rational (irrational numbers like $\sqrt{2}, \pi$, etc.).
(iii) True: The set of real numbers is defined as the union of the set of rational numbers and the set of irrational numbers.
(i) True, (ii) False, (iii) True
Q2: Square Roots
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Is the square root of every odd positive integer always an irrational number? Give an example to support your answer.
No. The square roots of all odd positive integers are not always irrational.
Example: Consider the odd positive integer 9.
$\sqrt{9} = 3$, which is a rational number (can be written as 3/1).
$\sqrt{9} = 3$, which is a rational number (can be written as 3/1).
Another example is 25. $\sqrt{25} = 5$, which is rational.
No, e.g., $\sqrt{9} = 3$ (Rational)
Q3: Construction
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Show how $\sqrt{2}$ can be represented on the number line.
Concept: We use the Pythagorean theorem ($h^2 = p^2 + b^2$).
We can write $2 = 1^2 + 1^2$.
We can write $2 = 1^2 + 1^2$.
Steps:
1. Draw a number line. Mark point O at 0 and A at 1. So, OA = 1 unit.
2. At A, draw a perpendicular AB of length 1 unit.
3. Join OB. By Pythagoras theorem, $OB = \sqrt{OA^2 + AB^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.
1. Draw a number line. Mark point O at 0 and A at 1. So, OA = 1 unit.
2. At A, draw a perpendicular AB of length 1 unit.
3. Join OB. By Pythagoras theorem, $OB = \sqrt{OA^2 + AB^2} = \sqrt{1^2 + 1^2} = \sqrt{2}$.
4. Using a compass with center O and radius OB, draw an arc intersecting the number line at point P.
5. Point P represents $\sqrt{2}$ on the number line.
5. Point P represents $\sqrt{2}$ on the number line.
Construct right triangle with sides 1 and 1. Hypotenuse is $\sqrt{2}$.