Samacheer Kalvi Class 9 Maths Solutions Chapter 2 Real Numbers Exercise 2.3

Question 1. Represent the following irrational numbers on the number line.

(i) \( \sqrt{3} \)
Answer:

**Steps of construction for \( \sqrt{3} \):**
1. Draw a straight line and mark point A. Then, mark point B so that the distance AB is 3 cm.
2. Mark another point C on this line. Ensure the distance BC is 1 cm.
3. Find the middle point of AC by drawing a perpendicular bisector of AC. Let this midpoint be 'O'.
4. With O as the center and OC (or OA) as the radius, draw a semicircle. This semicircle forms the upper part of the construction.
5. Draw a line from point B that is perpendicular to the main line (AB). This perpendicular line should touch the semicircle at point D.
6. Now, the length of BD will be \( \sqrt{3} \). You can transfer this length to the number line from B, and the point it reaches will be E, so BE is equal to BD, which is \( \sqrt{3} \).
In simple words: To show \( \sqrt{3} \) on a number line, you first create a special triangle where one side is \( \sqrt{3} \) long. You then use a compass to draw an arc from the point B, with the length of the triangle's side BD, to find where \( \sqrt{3} \) is on the number line.

🎯 Exam Tip: Always clearly label all points (A, B, C, D, O, E) and measurements (like 3 cm, 1 cm) on your construction diagram to get full marks. Remember that the length of BD will always represent the square root of the length AB.

(ii) \( \sqrt{4.7} \)
Answer:

**Steps of construction for \( \sqrt{4.7} \):**
1. Draw a straight line and mark point A. Then, mark point B so that the distance AB is 4.7 cm.
2. Mark another point C on this line. Ensure the distance BC is 1 cm.
3. Find the middle point of AC by drawing a perpendicular bisector of AC. Let this midpoint be 'O'.
4. With O as the center and OC (or OA) as the radius, draw a semicircle. This semicircle is key to finding the square root geometrically.
5. Draw a line from point B that is perpendicular to the main line (AB). This perpendicular line should touch the semicircle at point D.
6. Now, the length of BD will be \( \sqrt{4.7} \). You can transfer this length to the number line from B, and the point it reaches will be E, so BE is equal to BD, which is \( \sqrt{4.7} \).
In simple words: The process to represent \( \sqrt{4.7} \) is similar to \( \sqrt{3} \), but you start with a segment AB that is 4.7 cm long instead of 3 cm. The steps for drawing the semicircle and the perpendicular line remain the same to find the square root.

🎯 Exam Tip: When constructing square roots, ensure your line segment AB is accurately measured for the given value. Small errors in initial measurement can lead to inaccuracies in the final square root representation.

(iii) \( \sqrt{6.5} \)
Answer:

**Steps of construction for \( \sqrt{6.5} \):**
1. Draw a straight line and mark point A. Then, mark point B so that the distance AB is 6.5 cm.
2. Mark another point C on this line. Ensure the distance BC is 1 cm.
3. Find the middle point of AC by drawing a perpendicular bisector of AC. Let this midpoint be 'O'.
4. With O as the center and OC (or OA) as the radius, draw a semicircle. This semicircle helps in creating the geometric relationship needed for the square root.
5. Draw a line from point B that is perpendicular to the main line (AB). This perpendicular line should touch the semicircle at point D.
6. Now, the length of BD will be \( \sqrt{6.5} \). You can transfer this length to the number line from B, and the point it reaches will be E, so BE is equal to BD, which is \( \sqrt{6.5} \).
In simple words: For \( \sqrt{6.5} \), you follow the same drawing method as for \( \sqrt{3} \) and \( \sqrt{4.7} \), but you make the first segment AB exactly 6.5 cm long. This setup then naturally helps you find the length of \( \sqrt{6.5} \) on the number line.

🎯 Exam Tip: Always draw construction lines faintly and the final desired representation (e.g., the arc for \( \sqrt{x} \)) more boldly. Use a sharp pencil and proper geometric tools for accuracy.

Question 2. Find any two irrational numbers between

(i) 0.3010011000111.... and 0.3020020002....
Answer: Two irrational numbers between the given numbers are 0.301202200222....... and 0.301303300333........ There can be many such numbers, as irrational numbers fill the number line without repeating patterns.
In simple words: Look for numbers that start like the given ones but have digits that keep changing without repeating in a pattern. For example, 0.301 followed by random or increasing non-repeating digits.

🎯 Exam Tip: To find irrational numbers between two given numbers, ensure they are non-terminating and non-repeating. You can achieve this by adding non-repeating digit sequences like '010011000111...' or similar varying patterns.

(ii) \( \frac{6}{7} \) and \( \frac{12}{13} \)
Answer: First, convert the fractions to their decimal forms:
\( \frac{6}{7} = 0.\overline{857142} \)
\( \frac{12}{13} = 0.\overline{923076} \)
The two irrational numbers between these are 0.8616611666111........ and 0.8717711777111......... These numbers are non-repeating and non-terminating.
In simple words: Change the fractions into decimals first. Then, pick decimals that fall between them but have endless, mixed-up numbers after the decimal point.

🎯 Exam Tip: When dealing with fractions, always convert them to decimal form first to easily identify the range for finding irrational numbers.

(iii) \( \sqrt{2} \) and \( \sqrt{3} \)
Answer: First, find the approximate decimal values of \( \sqrt{2} \) and \( \sqrt{3} \):
\( \sqrt{2} = 1.414 \)
\( \sqrt{3} = 1.732 \)
The two irrational numbers between \( \sqrt{2} \) and \( \sqrt{3} \) are 1.515511555....... and 1.616611666........... Many other such numbers exist in this range.
In simple words: Find out what \( \sqrt{2} \) and \( \sqrt{3} \) are as decimals. Then choose any two decimals between these values that do not repeat in a pattern and go on forever.

🎯 Exam Tip: Remember that between any two distinct rational numbers, there are infinitely many irrational numbers, and vice-versa. Always show at least three decimal places for square roots in such problems.

Question 3. Find any two rational numbers between 2.2360679......... and 2.236505500..........
Answer: The two rational numbers are 2.2362 and 2.2363. These numbers terminate, meaning their decimal expansion ends. There are many possible rational answers between the given irrational numbers.
In simple words: Find simple decimal numbers that fit between the two given numbers. Rational numbers can be written as fractions or as decimals that either end or repeat.

🎯 Exam Tip: To find rational numbers between two decimals, simply pick any terminating decimal that lies within the given range. For example, if you need a rational number between 2.2360 and 2.2365, numbers like 2.2361 or 2.2364 are valid.