Unit 1: Real Numbers

Exercise 1.1 Solutions

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Question 1 Rational vs Irrational
Identify each of the following as a rational or irrational number:
Solutions & Analysis
(i) 2.353535 Rational
It is a terminating decimal. Any terminating decimal is a rational number because it can be written as 23535351000000.
(ii) 0.6 Rational
It is a recurring (repeating) decimal: 0.6666... Repeating decimals can always be represented in the form pq (specifically, 23).
(iii) 2.236067... Irrational
It is a non-terminating, non-repeating decimal. The digits continue indefinitely without showing any periodic repetition.
(iv) √7 Irrational
The square root of a prime number (or any number which is not a perfect square) is always an irrational number.
(v) e Irrational
Euler's number e ≈ 2.71828... is a transcendental mathematical constant that is non-terminating and non-repeating.
(vi) π Irrational
π ≈ 3.14159... is a well-known transcendental constant which cannot be expressed exactly as a ratio of two integers.
(vii) 5 + √11 Irrational
The sum of a rational number (5) and an irrational number (√11) is always an irrational number.
(viii) √3 + √13 Irrational
The sum of two distinct square-root irrationals is also irrational because they do not cancel out.
(ix) 154 Rational
It is written explicitly in the form pq where both p and q are integers and q ≠ 0.
(x) (2 - √2)(2 + √2) Rational
Simplifying: 22 - (√2)2 = 4 - 2 = 2. Since 2 is an integer, it is a rational number (21).
Question 2 Number Line representation
Represent the following numbers on the number line:

(i) √2

Geometric Construction:

  1. Draw a standard number line. Let the origin be O(0) and point A be at 1 unit. So, OA = 1 unit.
  2. Construct a perpendicular segment AB = 1 unit at point A (so AB ⊥ OA).
  3. Join O to B. By Pythagoras theorem, the hypotenuse OB is: OB = √OA2 + AB2 = √12 + 12 = √2 units
  4. Using a compass with center O and radius OB = √2, draw an arc that cuts the number line on the positive side at point P.
  5. Point P represents the value √2 ≈ 1.414 on the number line.
-2 -1 0 (O) 1 (A) 2 3 1 unit √2 B (1, 1) P (√2)

(ii) √3

Geometric Construction:

  1. Let the segment AB = 1 unit on the number line from 0 to 1. Here, let B be at the origin 0, and A at 1.
  2. At point B(0), construct a perpendicular line segment.
  3. From point A(1), draw an arc of radius 2 units such that it intersects the perpendicular at point C.
  4. Join A to C, creating a right-angled triangle ABC with base AB = 1 unit and hypotenuse AC = 2 units.
  5. By Pythagoras theorem, the perpendicular side BC is: BC = √AC2 - AB2 = √22 - 12 = √3 units
  6. With B(0) as center and radius BC = √3, draw an arc downwards to intersect the number line at point P.
  7. Point P represents √3 ≈ 1.732 units from B(0).
-1 0 (B) 1 (A) 2 3 √3 units 2 units C P (√3)

(iii) 413

Solution:

The fraction 413 = 133 ≈ 4.333 is a positive rational number that lies between 4 and 5.

  1. Draw a number line focusing on the interval between 3, 4, and 5.
  2. Divide the segment between 4 and 5 into 3 equal subparts.
  3. Mark the first division point after 4. This point P represents the number 413.
3 4 5 4 ⅓ 4 ⅔ P (4 ⅓)

(iv) -217

Solution:

The fraction -217 = -157 ≈ -2.143 is a negative rational number that lies between -2 and -3.

  1. Draw a number line focusing on the negative side, specifically between -1, -2, and -3.
  2. Divide the segment between -2 and -3 into 7 equal parts.
  3. Since the value is negative, count 1 subdivision step to the left of -2.
  4. Mark point P. This point represents -217.
-3 -2 -1 P (-2 ⅐)

(v) 58

Solution:

The fraction 58 = 0.625 is a positive rational number that lies between 0 and 1.

  1. Draw a number line focusing on the interval between 0 and 1.
  2. Divide this interval into 8 equal parts.
  3. Mark the 5th division point to the right of 0.
  4. This point P represents the fraction 58.
0 1 P (⅝)

(vi) 234

Solution:

The fraction 234 = 114 = 2.75 is a positive rational number that lies between 2 and 3.

  1. Draw a number line focusing on the positive side, specifically between 1, 2, and 3.
  2. Divide the segment between 2 and 3 into 4 equal parts.
  3. Mark the 3th division point to the right of 2.
  4. This point P represents the fraction 234.
1 2 3 P (2 ¾)
Question 3 Recurring to p/q form
Express the following as a rational number pq where p and q are integers and q ≠ 0:

(i) 0.4

Let x = 0.4

x = 0.4444...   — (Equation 1)

Since there is only one repeating digit in the period, multiply both sides of the equation by 10:

10x = 10 × (0.4444...)

10x = 4.4444...   — (Equation 2)

Subtracting Equation 1 from Equation 2:

10x - x = (4.4444...) - (0.4444...)

9x = 4

x = 49

Thus, 0.4 = 49

(ii) 0.37

Let x = 0.37

x = 0.373737...   — (Equation 1)

Since there are two repeating digits in the period, multiply both sides of the equation by 100:

100x = 100 × (0.373737...)

100x = 37.373737...   — (Equation 2)

Subtracting Equation 1 from Equation 2:

100x - x = (37.373737...) - (0.373737...)

99x = 37

x = 3799

Thus, 0.37 = 3799

(iii) 0.21

Let x = 0.21

x = 0.212121...   — (Equation 1)

Since there are two repeating digits in the period, multiply both sides of the equation by 100:

100x = 100 × (0.212121...)

100x = 21.212121...   — (Equation 2)

Subtracting Equation 1 from Equation 2:

100x - x = (21.212121...) - (0.212121...)

99x = 21

x = 2199

Simplifying the fraction by dividing the numerator and denominator by their greatest common divisor (3):

x = 21 ÷ 399 ÷ 3 = 733

Thus, 0.21 = 733

Question 4 Properties of Real Numbers
Name the property of real numbers used in the following statements:
Sr. Statement Property Used & Explanation
(i) (a + 4) + b = a + (4 + b) Associative Property w.r.t Addition
Changing the grouping of the terms does not affect their sum.
(ii) √2 + √3 = √3 + √2 Commutative Property w.r.t Addition
Changing the order of the terms does not change their sum.
(iii) x - x = 0 Additive Inverse Property
Adding a number to its negative (inverse) equals the additive identity (0).
(iv) a(b + c) = ab + ac Left Distributive Property of Multiplication over Addition
Multiplication distributes over addition from the left side.
(v) 16 + 0 = 16 Additive Identity Property
Adding zero to any number keeps the identity of the number unchanged.
(vi) 100 × 1 = 100 Multiplicative Identity Property
Multiplying any number by one keeps the identity of the number unchanged.
(vii) 4 × (5 × 8) = (4 × 5) × 8 Associative Property w.r.t Multiplication
Changing the grouping of the factors does not affect their product.
(viii) ab = ba Commutative Property w.r.t Multiplication
Changing the order of the factors does not change their product.
Question 5 Inequality Properties
Name the property of inequalities used in the following statements:
Sr. Statement Inequality Property & Explanation
(i) -3 < -1 ⇒ 0 < 2 Additive Property of Inequality
Adding 3 to both sides maintains the inequality direction:
-3 + 3 < -1 + 3 ⇒ 0 < 2.
(ii) If a < b then 1a > 1b Reciprocal Property of Inequality
Taking the reciprocal of positive terms reverses the inequality direction (assuming a, b > 0).
(iii) If a < b then a + c < b + c Additive Property of Inequality
Adding any real number c to both sides does not change the inequality.
(iv) If ac < bc and c > 0 then a < b Cancellation Property of Inequality w.r.t Multiplication
Dividing both sides by a positive number c maintains the inequality.
(v) If ac < bc and c < 0 then a > b Cancellation Property of Inequality w.r.t Multiplication
Dividing both sides by a negative number c reverses the inequality direction.
(vi) Either a > b or a = b or a < b Trichotomy Property
For any two real numbers a and b, exactly one of these three relations holds true.
Question 6 Rational Insertion
Insert two rational numbers between the given numbers:

(i) 13 and 14

Step 1: Find the first rational number by taking the average of the two numbers:

Average r1 = 13 + 142

Average r1 = 4 + 3122 = 712 × 2 = 724

Step 2: Find the second rational number by taking the average of 13 and our new number 724:

Average r2 = 13 + 7242

Average r2 = 8 + 7242 = 1524 × 2 = 1548

Simplifying the fraction by dividing numerator and denominator by 3:

Average r2 = 516

Conclusion: Two rational numbers between 13 and 14 are 724 and 516.

(ii) 3 and 4

Step 1: Find the first rational number by taking the average of 3 and 4:

Average r1 = 3 + 42 = 72   (3.5)

Step 2: Find the second rational number by taking the average of 72 and 4:

Average r2 = 72 + 42

Average r2 = 7 + 822 = 154   (3.75)

Conclusion: Two rational numbers between 3 and 4 are 72 and 154.

(iii) 35 and 45

Step 1: Find the first rational number by taking the average of the two fractions:

Average r1 = 35 + 452

Average r1 = 752 = 710   (0.7)

Step 2: Find the second rational number by taking the average of 35 and 710:

Average r2 = 35 + 7102

Average r2 = 6 + 7102 = 1320   (0.65)

Note: You can also choose the average between 710 and 45, which is 34 (0.75). Both are correct.

Conclusion: Two rational numbers between 35 and 45 are 710 and 1320.
Interactive Sandbox Explore Fractions
Type a fraction below (e.g. 3/5, -1/3, 2 1/2) to dynamically represent it on a number line!