Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write the decimal expansion of 78.

Perfect! 7 ÷ 8 = 0.875 (terminating decimal).

(2) Is 0.142857142857... a rational number? Why?

Correct! It's a repeating decimal, so it can be expressed as pq (it equals 17).

(3) Give an example of a non-terminating repeating decimal.

Excellent! Examples include 13 = 0.333..., 23 = 0.666..., etc.

(4) What is the value of 32?

Perfect! The square root and square operations cancel each other out.

(5) Write any two irrational numbers.

First:

Second:

Great! 2 and π are irrational with non-terminating, non-repeating decimal expansions.

(6) Is 227 a rational number? Explain.

Correct! It's a fraction pq where p and q are integers, so it's rational.

(7) Find the square root of 0.81.

Perfect! 0.81 = 81100 = 9/10 = 0.9.

(8) State whether √2 + √3 is rational or irrational.

Correct! The sum of two irrational numbers is typically irrational.

(9) Express 3.125 in the form pq.

Excellent! 3.125 = 31251000 = 258 (in simplest form).

Short Answer Questions (2 Marks Each)

Answer each question in 2-3 sentences

(1) Convert 0.888... into the form p/q.

Perfect! Using the standard method for converting repeating decimals.

(2) Write two rational numbers between 1.2 and 1.3.

First number:

Second number:

Great! Any decimals between 1.2 and 1.3 work (like 1.25, 1.26, etc.).

(3) Represent 9.5 on the number line using the number line method.

9.5 ≈

Good approximation! 9.5 lies between 3 and 4, closer to 3.

(4) Classify the following numbers as rational or irrational:

(a) 18:

(b) 35:

Perfect! 18 = 32 (irrational), while 35 is a simple fraction.

(5) Represent 10 using the number line construction method.

10 ≈

Excellent! 10 lies between 3 and 4, approximately 3.162...

Long Answer Questions (4 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) Prove that 3 is irrational.

(2) Express the following recurring decimals as rational numbers:

(a) 0.121212...

(b) 0.999...

Perfect! 0.999... actually equals 1 exactly.

(3) Represent 5 on the number line using geometric construction.

5 ≈

Construction involves creating a right triangle with legs 1 and 2, giving hypotenuse 5.

(4) List three rational and three irrational numbers and compare their decimal expansions. Also explain how to differentiate between the two.

Three rational numbers:

Their decimal expansions:

Three irrational numbers:

Their decimal expansions:

How to differentiate:

Rational numbers:

Irrational numbers:

Key difference:

Excellent analysis of the fundamental difference between rational and irrational numbers!

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) Which of the following has a terminating decimal expansion?

(a) 79 (b) 13 (c) 58 (d) 411

7/9

1/3

5/8

4/11

Correct! 58 = 0.625 (terminates). Others have repeating decimals.

(2) The decimal expansion of 2/7 is:

(a) 0.285714 (b) 0.285714... (repeating) (c) 0.285714285 (d) None

0.285714

0.285714... (repeating)

0.285714285

None

Correct! 27 gives a repeating decimal 0.285714285714...

(3) Which of the following is irrational?

(a) 59 (b) 2.3 (c) 1.1 (d) 6

5/9

2.3

1.1̄

√6

Correct! 6 is irrational. Others are rational (fractions or repeating decimals).

(4) If x = 2 and y = 3, then x + y is:

(a) Rational (b) Irrational (c) Integer (d) Whole number

Rational

Irrational

Integer

Whole number

Correct! 2 + 3 is irrational (sum of two specific irrationals).

(5) Convert 2.25 into the form of pq:

(a) 94 (b) 49 (c) 52 (d) 38

9/4

4/9

5/2

3/8

Correct! 2.25 = 225100 = 94 (in simplest form).

(6) The square root of a non-perfect square is:

(a) Rational (b) Whole number (c) Irrational (d) Natural number

Rational

Whole number

Irrational

Natural number

Correct! Square roots of non-perfect squares are always irrational.

(7) Which of the following is a non-terminating repeating decimal?

(a) 17 (b) 12 (c) 34 (d) 5

1/7

1/2

3/4

Correct! 17 = 0.142857142857... (repeating). Others terminate or are whole numbers.

(8) Which number is not a real number?

(a) π (b) 0 (c) −1 (d) 5

√-1

√5

Correct! −1 is imaginary, not real. All others are real numbers.

(9) Which of the following cannot be written as a rational number?

(a) 7 (b) 0.25 (c) 13 (d) 11

0.25

1/3

√11

Correct! 11 is irrational. Others can all be written as fractions.

(10) 16 + 9 = ?

(a) 7 (b) 25 (c) 5 (d) 13

Correct! 16 + 9 = 4 + 3 = 7.

1/2

√2

√7

0.75

3.125

22/7

√3

Rational Numbers

Irrational Numbers

Real Numbers Properties Challenge

Determine whether these statements about real numbers are True or False:

Every terminating decimal is a rational number

The sum of two irrational numbers is always irrational

√16 is a rational number

22/7 is exactly equal to π

0.999... equals 1

Every repeating decimal can be expressed as p/q

Real Numbers Challenge Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Rational vs. Irrational Numbers: Understanding the fundamental difference between numbers that can and cannot be expressed as fractions

(2) Decimal Expansions: Recognizing terminating, repeating, and non-repeating decimal patterns

(3) Converting Decimals to Fractions: Mastering the technique for converting repeating decimals to p/q form

(4) Square Roots: Understanding when square roots are rational or irrational

(5) Proof Techniques: Learning to prove irrationality using contradiction method

(6) Number Line Representation: Visualizing irrational numbers on the number line

(7) Real Number Classification: Categorizing numbers into appropriate sets

(8) Properties of Operations: Understanding how operations affect rational and irrational numbers

Excellent work mastering the real number system! �_

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