Case Study 1:
Riya is working on a project where she needs to plot the point √3 on the number line but finds it challenging to locate irrational numbers precisely.
Q1: How can Riya represent √3 on the number line accurately?
Answer:
Riya can use the geometric method: Draw a line segment of length 1 unit, then at one end draw a perpendicular segment of length √2 units (or construct a right triangle with sides 1 and √2), then use the Pythagoras theorem to mark the hypotenuse as √3 units from the origin.
Case Study 2:
Anjali wants to convert the repeating decimal 0.363636… into a fraction to use it in a mathematical formula.
Q2: How can Anjali express 0.363636… as a rational number?
Answer:
Let x = 0.363636…
Multiply both sides by 100: 100x = 36.363636…
Subtract original equation: 100x – x = 36.363636… – 0.363636… = 36
So, 99x = 36
x = 36/99 = 4/11
Case Study 3:
In a quiz, a student claims that all decimal expansions of rational numbers terminate or repeat. Is the student correct? Give examples.
Q3: Explain why every rational number’s decimal expansion either terminates or repeats.
Answer:
Yes, the student is correct. This is because a rational number is expressed as p/q where p and q are integers and q ≠ 0. Upon division, either the remainder becomes zero (terminating decimal) or repeats (repeating decimal).
Examples:
- 1/4 = 0.25 (terminating)
- 1/3 = 0.3333… (repeating)
Case Study 4:
Rahul notices that the decimal expansion of π goes on forever without repeating.
Q4: Why is π considered an irrational number?
Answer:
π is irrational because it cannot be expressed as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating.
Case Study 5:
A student is asked to rationalize the denominator of the expression 1/(√5 + √2).
Q5: How can the student rationalize 1/(√5 + √2)?
Answer:
Multiply numerator and denominator by the conjugate (√5 − √2):
1/(√5 + √2) × (√5 − √2)/(√5 − √2) = (√5 − √2)/(5 − 2) = (√5 − √2)/3
Case Study 6:
In a test, a student wrote that √2 is a rational number because it can be written as 1.414.
Q6: Is the student’s statement correct? Why or why not?
Answer:
No, the student’s statement is incorrect. The decimal 1.414 is an approximation; √2 is irrational because it cannot be exactly represented as a ratio of two integers.
Case Study 7:
During a math experiment, a student calculated (√3 + √2)^2.
Q7: Calculate (√3 + √2)^2 and interpret the result.
Answer:
(√3 + √2)^2 = (√3)^2 + 2√3√2 + (√2)^2 = 3 + 2√6 + 2 = 5 + 2√6.
The result combines rational (5) and irrational (2√6) parts.
Case Study 8:
For coding decimals in a program, a teacher explains the difference between rational and irrational numbers.
Q8: How would you differentiate rational and irrational numbers in terms of decimal representation?
Answer:
- Rational numbers have decimals that either terminate (e.g., 0.75) or repeat (e.g., 0.333…).
- Irrational numbers have decimals that neither terminate nor repeat (e.g., π ≈ 3.14159…).
Case Study 9:
Sam is asked to find five rational numbers between 0 and 1.
Q9: List five rational numbers between 0 and 1.
Answer:
0.1, 0.25, 0.5, 0.75, 0.9 (All fractions with denominators like 10, 4, 2, etc.)
Case Study 10:
A math teacher says that the product of two irrational numbers can sometimes be rational.
Q10: Give an example to justify the teacher’s statement.
Answer:
Example: √2 × √2 = 2, which is rational. So, the product of two irrational numbers can be rational.