Question 1

Which of the following numbers is an example of an irrational number?

Accepted Answer

\sqrt{2} — Irrational numbers are numbers that cannot be expressed as a ratio of two integers and have non-terminating, non-repeating decimal expansions. \sqrt{2} is a classic example of an irrational number, whereas 0.75 and \frac{3}{4} are rational numbers with terminating decimals and 0.333... is a rational number with a repeating decimal.

Question 2

What is the decimal expansion characteristic of rational numbers?

Accepted Answer

Rational numbers have decimal expansions that either terminate after a finite number of digits or repeat a pattern of digits infinitely. For example, 1/2 = 0.5 is a terminating decimal, and 1/3 = 0.333... is a repeating decimal. — Rational numbers can be expressed as fractions p/q where p and q are integers and q ≠ 0. When we divide p by q, the decimal either ends if the remainder becomes zero or repeats if the remainders cycle. This property helps distinguish rational numbers from irrational numbers.

Question 3

Identify the correct statement about the decimal expansion of the number 1/7.

Accepted Answer

It is a non-terminating, repeating decimal — The fraction 1/7 when expressed as a decimal results in 0.142857142857..., where the six digits 142857 repeat indefinitely. Hence, it is a non-terminating, repeating decimal, characteristic of rational numbers.

Question 4

Explain why the decimal expansion of \sqrt{2} is considered non-terminating and non-repeating.

Accepted Answer

\sqrt{2} is an irrational number whose decimal expansion goes on infinitely without terminating or repeating any pattern. This is because it cannot be expressed as a ratio of two integers, and its decimal form does not settle into a repeating cycle. — Irrational numbers like \sqrt{2} have decimal expansions that neither end nor repeat. This distinguishes them from rational numbers. The discovery of \sqrt{2}'s irrationality showed that not all numbers can be expressed as fractions, and its decimal expansion confirms this property.

Question 5

Which number cannot be expressed as a fraction of two integers?

Accepted Answer

\pi — \pi is an irrational number and cannot be expressed as a ratio of two integers. It has a non-terminating, non-repeating decimal expansion. The other options are rational numbers that can be expressed as fractions.

Question 6

Perform long division to express 3/8 as a decimal and identify whether the decimal terminates or repeats.

Accepted Answer

3/8 equals 0.375 as a decimal. The decimal terminates after three digits because the remainder becomes zero during division. — Given: 3/8
Find: Decimal expansion
Formula: Long division of numerator by denominator
Solution: Step 1: Divide 3 by 8, quotient 0, remainder 3
Step 2: Multiply remainder by 10, divide 30 by 8, quotient 3, remainder 6
Step 3: Multiply remainder by 10, divide 60 by 8, quotient 7, remainder 4
Step 4: Multiply remainder by 10, divide 40 by 8, quotient 5, remainder 0
Answer: 0.375 (terminating decimal)
Note: Students often stop division too early, missing the terminating remainder zero.

Question 7

Which of the following decimal expansions represents a rational number?

Accepted Answer

0.666666... — 0.666666... is a repeating decimal representing the rational number 2/3. The other options are examples of non-terminating, non-repeating decimals (irrational numbers).

Question 8

Describe the significance of irrational numbers in completing the number line.

Accepted Answer

Irrational numbers fill the gaps between rational numbers on the number line, making it continuous. Without irrational numbers, many points on the number line would not correspond to any number, so they complete the number system. — The number line represents all real numbers. Rational numbers alone leave gaps because some lengths and quantities cannot be expressed as fractions. Irrational numbers like \sqrt{2} fill these gaps, ensuring every point on the line corresponds to a real number, which is essential for continuous measurement.

Question 9

Plot the number \sqrt{3} on the number line using successive approximations. Which two rational numbers does it lie between?

Accepted Answer

\sqrt{3} lies between 1.7 and 1.8 on the number line. By squaring 1.7 (2.89) and 1.8 (3.24), we see that \sqrt{3} is between these two values. — Given: \sqrt{3}
Find: Interval on number line
Formula: Successive approximation by squaring
Solution: Step 1: Check 1.7^2 = 2.89 < 3
Step 2: Check 1.8^2 = 3.24 > 3
Step 3: Hence, \sqrt{3} lies between 1.7 and 1.8
Answer: Between 1.7 and 1.8
Note: Narrowing the interval further improves approximation.

Question 10

Match the following numbers with their correct decimal expansion type.

Accepted Answer

— Matching rational numbers with terminating or repeating decimals and irrational numbers with non-terminating, non-repeating decimals helps reinforce classification.

Question 11

True or False: Every real number can be expressed as a fraction of two integers.

Accepted Answer

False. Not every real number can be expressed as a fraction of two integers; irrational numbers cannot be expressed as such fractions. — Real numbers include both rational and irrational numbers. Rational numbers can be expressed as fractions, but irrational numbers like \pi and \sqrt{2} cannot be expressed as a ratio of two integers.

Question 12

Fill in the blank: The decimal expansion of an irrational number is _____ and _____, meaning it neither ends nor repeats.

Accepted Answer

non-terminating / non-repeating — Irrational numbers have decimal expansions that continue infinitely without terminating or repeating any pattern, distinguishing them from rational numbers.

Question 13

Explain the closure property of real numbers with respect to addition.

Accepted Answer

The closure property states that the sum of any two real numbers is also a real number. For example, adding \sqrt{2} (irrational) and 3 (rational) results in a real number that may be irrational. — Real numbers include both rational and irrational numbers. When two real numbers are added, the result remains within the set of real numbers. This property ensures that operations on real numbers do not produce numbers outside this set, maintaining consistency.

Question 14

Calculate the product of \pi and 2. Is the result rational or irrational?

Accepted Answer

The product of \pi and 2 is approximately 6.2832, which is an irrational number because \pi is irrational and multiplying it by a rational number (2) does not make it rational. — Given: \pi (irrational), 2 (rational)
Find: Product and nature
Formula: Product = \pi × 2
Solution: Step 1: Multiply \pi by 2
Step 2: Approximate value is 6.2832
Step 3: Since \pi is irrational, the product remains irrational
Answer: Approximately 6.2832, irrational
Note: Multiplying an irrational number by a non-zero rational number results in an irrational number.

Question 15

Why can't irrational numbers be exactly represented as fractions or decimals?

Accepted Answer

Irrational numbers cannot be exactly represented as fractions because they are not ratios of integers. Their decimal expansions are infinite and non-repeating, so they cannot be fully written as decimals either. — By definition, irrational numbers are not expressible as p/q where p and q are integers. Their decimal expansions go on forever without repeating, making exact decimal representation impossible. They can only be approximated.

A Peek Beyond

A Peek Beyond — Study Notes

Introduction

Rational Numbers and Their Decimal Expansions

Irrational Numbers

Practice Questions — A Peek Beyond

All 8 Chapters in Ganita Prakash