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?

YesNo

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.

0.333...0.142857...

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: 2230.254

Second: π2271.59

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

(6) Is 227 a rational number? Explain.

YesNo

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.

IrrationalRational

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 pq.

Perfect! Using the standard method for converting repeating decimals.

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

First number: 1.251.311.32

Second number: 1.231.321.30

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:

IrrationalRational

(b) 35:

RationalIrrational

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)

(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: 12, 23, 582, π, 512, 3, 58π, 23, 7

Their decimal expansions: 0.5 (terminating), 0.666... (repeating), 0.625 (terminating)1.414... (non-repeating), 3.141... (non-repeating), 2.236... (non-repeating)0.5 (terminating), 1.732... (non-repeating), 0.625 (terminating)3.141... (non-repeating), 0.666... (repeating), 2.645... (non-repeating)

Three irrational numbers: 2, π, 512, 23, 582, 227, 50.5, π, 1.25

Their decimal expansions: 1.414213... (non-repeating), 3.141592... (non-repeating), 2.236067... (non-repeating)0.5 (terminating), 0.666... (repeating), 0.625 (terminating)1.414... (non-repeating), 3.142857... (repeating), 2.236... (non-repeating)0.5 (terminating), 3.141... (non-repeating), 1.25 (terminating)

How to differentiate:

Rational numbers: Have terminating or non-terminating repeating decimal expansionsHave non-terminating non-repeating decimal expansionsAre always whole numbersAre always positive

Irrational numbers: Have non-terminating non-repeating decimal expansionsHave terminating or repeating decimal expansionsAre always negativeAre always greater than 1

Key difference: Rational decimals either end or have a repeating pattern, while irrational decimals continue forever without any patternAll decimals are rational numbersIrrational numbers are always larger than rational numbersRational numbers cannot be written as decimals

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

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

(2) The decimal expansion of 27 is:

(a) 0.285714 (b) 0.285714... (repeating) (c) 0.285714285 (d) 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

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

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

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

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

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

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

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.

Real Numbers Properties Challenge

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

Real Numbers Challenge Quiz