irrational and irrational number

 

Rational and Irrational Numbers: Easy Notes for Students

Why Do We Need Different Kinds of Numbers?

Imagine you are counting mangoes in a basket. You use 1, 2, 3, 4... Easy! But then something happens:

• You eat all the mangoes. How do you write "nothing left"? You need 0.

• You owe your friend 5 rupees. How do you write that? You need −5.

• You cut a chapati into 2 equal parts and eat one. How do you write that? You need 1/2.

• You measure the diagonal of a square tile with side 1 unit. The answer is √2 = 1.41421356... and it never ends!

Each new problem needed a new kind of number. In this note we will learn about the last two kinds: rational numbers (like 1/2) and irrational numbers (like √2). Together, they make up all the real numbers.

The Number Family Tree (Diagram)

Think of numbers as a big family. Every number has a "home" in this tree:

                                        REAL NUMBERS (R)

                                                       |

                                  +---------------+----------------+

                                  |                                         |

   RATIONAL NUMBERS (Q)                       IRRATIONAL NUMBERS

   (can be written as p/q)                              (cannot be written as p/q)

             |                                                                  Examples:

   +----------------+---------------+                               √2, √3, √5

   |                                         |                                π, e

 INTEGERS (Z)                     FRACTIONS                   0.1010010001...

 …-2,-1,0,1,2...                 like 1/2, -3/4, 7/5

              |

 WHOLE NUMBERS (W)

       0, 1, 2, 3, ...

              |

 NATURAL NUMBERS (N)

        1, 2, 3, ...

Remember the order (smallest group to biggest):

N ⊂ W ⊂ Z ⊂ Q ⊂ R

The symbol ⊂ means "is inside". So every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational number is a real number.

Important: A number can NEVER be both rational and irrational. It is one or the other. They share no members.

Rational Numbers

3.1 Definition

A number is rational if it can be written in the form

p / q, where p and q are integers and q ≠ 0.

The word "rational" comes from the word ratio. A rational number is simply a ratio of two integers.

Or we can say, Those numbers which can be written in the form of fractional value (a/b) or in decimal but the value after decimal will be repetitive number or can be a whole number are rational number.

For example 1/3, 4/9, 4, 3.171717 etc.

Let See Other Examples of Rational Numbers

Number	Written as p/q	Why it is rational
5	5/1	Every integer can be divided by 1
−7	−7/1	Negative integers are rational too
0	0/1	Zero is rational
3/4	3/4	Already a fraction
0.5	5/10 = 1/2	Terminating decimal
2.75	275/100 = 11/4	Terminating decimal
0.333...	1/3	Repeating decimal
1.272727...	14/11	Repeating decimal
√16	4/1	Because √16 = 4, a perfect square root

Decimal Form of Rational Numbers

This is the most useful idea in this chapter. When you convert a rational number into a decimal (by dividing p by q), only two things can happen:

1. Terminating decimal: the division stops. Example: 1/4 = 0.25, 3/8 = 0.375.
2. Non-terminating repeating decimal: the division never stops, but a group of digits repeats forever. Example: 1/3 = 0.3333..., 2/11 = 0.181818..., 1/7 = 0.142857142857...

              Rational number (p/q)

                                    |

                       Divide p by q

                                    |

            +-----------------+-------------------+

             |                                              |

   Terminating                               Non-terminating

      (ends)                                       REPEATING

 0.25, 0.375                                    0.333..., 0.1818...

Short way to write repeating decimals: put a bar over the repeating digits. So 0.333... = 0.3̅ and 0.181818... = 0.18̅.

A Quick Trick: Which Fractions Terminate?

Write the fraction p/q in lowest terms (no common factor). Now look at the denominator q:

• If q has only 2 and/or 5 as prime factors, the decimal terminates. Example: 7/20 (20 = 2 × 2 × 5) = 0.35.
• If q has any other prime factor (3, 7, 11 ...), the decimal is repeating. Example: 5/12 (12 = 2 × 2 × 3) = 0.41666...

Rational Numbers on the Number Line

Every rational number has an exact position on the number line.

 -3     -2     -1      0      1      2      3

  |------|------|------|------|------|------|

                    ^    ^      ^

                 -1/2   0      1/2    3/2 is between 1 and 2

                             

  Example: 3/2 = 1.5 is exactly halfway between 1 and 2

Between any two rational numbers there are infinitely many more rational numbers. For instance, between 0 and 1 we can find 1/2, 1/3, 1/4, 1/100, 99/100, and so on, forever.

Converting a Repeating Decimal into a Fraction

This is a favourite exam topic. Follow these steps.

Example 1: Convert 0.6666... to p/q

• Step 1: Let x = 0.6666...
• Step 2: One digit repeats, so multiply by 10: 10x = 6.6666...
• Step 3: Subtract: 10x − x = 6.6666... − 0.6666...
• Step 4: 9x = 6, so x = 6/9 = 2/3

Example 2: Convert 0.272727... to p/q

• Let x = 0.272727...
• Two digits repeat, so multiply by 100: 100x = 27.272727...
• Subtract: 100x − x = 27, so 99x = 27
• x = 27/99 = 3/11

Example 3: Convert 2.1666... (only the 6 repeats) to p/q

• Let x = 2.1666...
• Multiply by 10: 10x = 21.666...
• Multiply by 100: 100x = 216.666...
• Subtract: 100x − 10x = 216.666... − 21.666... = 195
• 90x = 195, so x = 195/90 = 13/6

Rule to remember: If n digits repeat, multiply by 10ⁿ (10 for one digit, 100 for two digits, 1000 for three digits).

Irrational number:-  

Those numbers which is not rational or can't be written in the form of fractional value.

Or, A number is irrational if it cannot be written in the form p/q (with integers p and q, q ≠ 0).

The word "irrational" simply means "not a ratio". It does not mean the number is crazy or unreasonable!

For example 2.34587...., π etc

Here the confusion comes on π which is an irrational number but we have seen it's value is written as 22/7 which is a rational number.

So the point here is the actual value of π is 3.14159..... which is not repetitive value and it consider as approx value in the form of ratio is 22/7.

The Decimal Form of Irrational Numbers

In decimal form, an irrational number is non-terminating AND non-repeating. The digits go on forever, and there is no pattern that repeats.

  Rational:    0.333333333...   <- digits repeat (pattern: 3)

  Rational:    0.142857142857...<- digits repeat (pattern: 142857)

  Irrational:  1.41421356237... <- no pattern, never ends  (this is √2)

  Irrational:  3.14159265358... <- no pattern, never ends  (this is π)

Famous Irrational Numbers

Number	Approximate value	Where it comes from
√2	1.41421356...	Diagonal of a square of side 1
√3	1.73205080...	Height of an equilateral triangle of side 2
√5	2.23606797...	Hypotenuse of a right triangle with sides 1 and 2
π (pi)	3.14159265...	Circumference ÷ diameter of any circle
e	2.71828182...	Growth and compound interest (Class 11+)

Constructed Irrational Numbers

We can also build irrational numbers by following a pattern that never repeats. Example:

0.101001000100001000001...

Here the number of zeros keeps increasing (1, 2, 3, 4, ...). Because the pattern never repeats exactly, this number is irrational.

A Common Confusion About π

Students often say "π = 22/7, so π is rational." This is wrong. 22/7 = 3.142857142857... is only a close approximation of π, which is 3.14159265... They are different numbers. π itself is irrational. Whenever a textbook says "take π = 22/7", it is only to make calculation easy.

Prove √5 is a irrational number.

Ans 

Let’s take it for granted that the number √5 is a rational number. 

If √5 is rational, then it may be expressed as an equation of the type a/b, here integers are a and b and be should not be equal to 0.

√5/1 = a/b

√5b = a

Bringing both sides into balance 

Now square both the side

5b² = a²

b²= a²/5 —- (i)

This indicates that a² is divided by 5.

Hence a also divided by 5.

If 

a/5 = c

a = 5c

On squaring, we get

a² = 25c²

Replace a² in the equation with its value (1).

5b² = 25c²

b² = 5c²

b²/5 = c²

This indicates that b² may be divided by 5, and hence, b can likewise be divided by 5. Since this is the case, a and b share the factor 5 in common. However, this goes against the fact that a and b are in the coprime position. 

 we can not write √5 as in fractional form.

So our assumption of √5 is a rational number is wrong. As a result, we are forced to the conclusion that √5 is irrational

Let see another example :-

How Do We Know √2 Is Irrational? (A Simple Proof)

Mathematicians use a clever method called proof by contradiction. We assume the opposite and show it leads to nonsense.

Claim: √2 is irrational.

1. Assume the opposite: suppose √2 is rational. Then √2 = p/q, where p and q have no common factor (the fraction is in lowest terms).
2. Square both sides: 2 = p²/q², so p² = 2q².
3. This means p² is even, so p must be even. Write p = 2k.
4. Substitute: (2k)² = 2q², so 4k² = 2q², so q² = 2k².
5. This means q² is even, so q must be even too.
6. But now both p and q are even, so both have a common factor 2. 
7. This contradicts our assumption that p/q was in lowest terms.
8. So our assumption was wrong. √2 cannot be rational. It is irrational. ∎

Same idea works for √3, √5, √7 and the square root of any prime number.

Rational vs Irrational: Side-by-Side

Feature	Rational numbers	Irrational numbers
Can be written as p/q?	Yes	No
Decimal form	Terminating OR non-terminating repeating	Non-terminating AND non-repeating
Examples	3, −2/5, 0.75, 0.3̅, √49	√2, √7, π, 0.1010010001...
Square root of	Perfect squares (4, 9, 16, 25...)	Non-perfect squares (2, 3, 5, 6, 7...)
On the number line	Yes, exact point	Yes, exact point (but harder to mark)
Between two numbers	Infinitely many	Infinitely many

Good news: Irrational numbers are real numbers too. They have exact positions on the number line, even though their decimals never end.

Locating √2 on the Number Line (Diagram)

We cannot write √2 exactly as a fraction, but we can mark it exactly using geometry and the Pythagoras theorem.

Steps:

1. Draw a number line. Mark 0 and 1.

1. At the point 1, draw a perpendicular line of length 1 unit.

1. Join 0 to the top of this line. By Pythagoras, this length = √(1² + 1²) = √2.

1. With centre at 0 and this length as radius, draw an arc that cuts the number line. The cutting point is √2.

        B (top)
         /|
        / |
 √2    /  |  1 unit
      /   |
     /    |
    /_____|_______
   O      A
    1 unit

  Number line view:

   O=0         A=1    P=√2 (≈1.414)     2
   |-----------|---------|-------------|
               ^         ^
        (perpendicular)  (arc from O with radius OB lands here)


To locate √3: From the point √2, draw a perpendicular of length 1. Join it back to 0. 
The length is √(2 + 1) = √3. Draw an arc to mark it. You can keep going like this to get √4, √5, √6 
and so on. This beautiful spiral is called the square root spiral (or Wheel of Theodorus).
Square root spiral idea:
   Start: right triangle with two sides of 1  -> hypotenuse = √2
   Next : add side of 1 to √2                 -> hypotenuse = √3
   Next : add side of 1 to √3                 -> hypotenuse = √4 = 2
   and so on...
Rules for Operations
What happens when we add, subtract, multiply or divide rational and irrational numbers?
Operation
Result
Example
Rational + Rational                       
Always rational
1/2 + 1/3 = 5/6
Rational × Rational
Always rational               
2/3 × 3/4 = 1/2
Rational + Irrational
Always irrational
2 + √3 is irrational
Rational − Irrational
Always irrational
5 − √2 is irrational
Non-zero Rational × Irrational
Always irrational
3√2 is irrational
Non-zero Rational ÷ Irrational
Always irrational
4/√5 is irrational
Irrational + Irrational
Can be either
See below
Irrational × Irrational
Can be either
See below

Surprising cases (irrational + irrational):

• √2 + (−√2) = 0 → rational!
• √2 + √3 = 3.146... → irrational
• (2 + √3) + (2 − √3) = 4 → rational!

Surprising cases (irrational × irrational):

• √2 × √2 = 2 → rational!
• √2 × √3 = √6 → irrational
• √3 × √12 = √36 = 6 → rational!

Special warning: 0 × √2 = 0, which is rational. That is why we say "non-zero rational times irrational" in the table above.

Finding Numbers Between Two Numbers

Rational Numbers Between Two Numbers

Method: Take the average (add them and divide by 2). Repeat as many times as you want.

Example: Find a rational number between 1/2 and 3/4.

• Average = (1/2 + 3/4) ÷ 2 = (2/4 + 3/4) ÷ 2 = (5/4) ÷ 2 = 5/8

Another method: Convert to decimals. Between 0.5 and 0.75 we can pick 0.6, 0.65, 0.7, and many more.

8.2 Irrational Numbers Between Two Numbers

Method: Write the numbers as decimals, then create a non-repeating, non-terminating decimal between them.

Example: Find an irrational number between 0.1 and 0.2.

• 0.1 = 0.10000... and 0.2 = 0.20000...
• Choose 0.1101001000100001... (the zeros keep increasing). It lies between 0.1 and 0.2 and is irrational.

Another example: Find an irrational number between 2 and 3.

Since 4 < 5 < 9, we have √4 < √5 < √9, so 2 < √5 < 3. So √5 works.

How to Decide Quickly: Rational or Irrational?

Use this simple checklist when you see a number in an exam:

START: look at the number
|
+--> Is it a whole number, fraction or integer? -> RATIONAL
|
+--> Is it a decimal that ENDS? -> RATIONAL
|
+--> Is it a decimal with a REPEATING pattern? -> RATIONAL
|
+--> Is it the square root of a PERFECT square? -> RATIONAL
| (√4, √25, √0.09, √(9/16))
|
+--> Is it the square root of a NON-perfect square? -> IRRATIONAL
| (√2, √7, √10)
|
+--> Is it π, or a decimal that never ends and never repeats? -> IRRATIONAL

Common Mistakes to Avoid

1. Thinking √ always means irrational. √9 = 3 is rational. Only the root of a non-perfect square is irrational.
2. Thinking π = 22/7 exactly. It is only an approximation.
3. Thinking 0.333... is irrational because it never ends. It repeats, so it is rational (1/3).
4. Thinking 0 is not rational. 0 = 0/1, so it is rational.
5. Thinking a long decimal like 3.14159 is irrational. It ends, so it is rational (314159/100000). Only non-terminating, non-repeating decimals are irrational.

Forgetting q ≠ 0. Division by zero is not defined, so 5/0 is not a number at all.

Solved Examples

Example 1. Is 0.75 rational or irrational?

0.75 ends after two digits, so it is terminating. 0.75 = 75/100 = 3/4. Rational.

Example 2. Is √50 rational or irrational?

√50 = √(25 × 2) = 5√2. Since √2 is irrational and 5 is a non-zero rational, 5√2 is irrational.

Example 3. Is (3 + √2)(3 − √2) rational or irrational?

Use (a + b)(a − b) = a² − b². So (3 + √2)(3 − √2) = 9 − 2 = 7. Rational.

Example 4. Show that 5 + √3 is irrational.

Suppose 5 + √3 is rational, say equal to r. Then √3 = r − 5. A rational minus a rational is rational, so √3 would be rational. But √3 is irrational. Contradiction. So 5 + √3 is irrational.

Example 5. Express 0.4̅ (0.444...) as a fraction.

Let x = 0.444... Then 10x = 4.444... Subtract: 9x = 4, so x = 4/9.

Example 6. Without dividing, say whether 13/80 has a terminating decimal.

80 = 2 × 2 × 2 × 2 × 5. Only 2 and 5 appear. So the decimal terminates. (13/80 = 0.1625)

Example 7. Simplify (√5 + √2)².

= (√5)² + 2(√5)(√2) + (√2)² = 5 + 2√10 + 2 = 7 + 2√10. This is irrational because 2√10 is irrational.

Example 8. Find two irrational numbers between 3 and 4.

Since 9 < 10 < 11 < 16, we get 3 < √10 < √11 < 4. So √10 and √11 work.

Practice Questions and Answers

Part A: Multiple Choice Questions (1 mark each)

Q1. Which of these is an irrational number?

(a) 0.25 (b) √36 (c) √7 (d) 0.6̅

Q2. Which of these is rational?

(a) π (b) √3 (c) 0.1010010001... (d) 22/7

Q3. The decimal form of an irrational number is:

(a) terminating (b) non-terminating repeating (c) non-terminating non-repeating (d) none of these

Q4. Which number is both an integer and a rational number?

(a) 1/2 (b) −4 (c) √2 (d) 0.5

Q5. The value of √2 × √8 is:

(a) √10 (b) 4 (c) 16 (d) 2√2

Q6. 0.333... can be written as:

(a) 1/3 (b) 3/10 (c) 33/100 (d) 3/9 is not equal to it

Q7. Which fraction has a terminating decimal?

(a) 1/3 (b) 2/7 (c) 3/16 (d) 5/11

Q8. The sum of a rational and an irrational number is always:

(a) rational (b) irrational (c) an integer (d) zero

Part B: Short Answer Questions (2 marks each)

Q9. Write any three rational numbers between 2 and 3.

Q10. Express 0.7̅ (0.777...) in p/q form.

Q11. Is 2 − √5 rational or irrational? Give a reason.

Q12. Is 0.12112111211112... rational or irrational? Explain.

Q13. Without long division, tell whether 7/25 has a terminating or repeating decimal.

Q14. Find one irrational number between 1/7 and 2/7.

Q15. True or false: "The product of two irrational numbers is always irrational." Give an example to support your answer.

Part C: Long Answer Questions (3–4 marks each)

Q16. Convert 0.1̅8̅ (that is, 0.181818...) into a fraction in lowest terms.

Q17. Prove that √3 is irrational.

Q18. Simplify and say whether the result is rational or irrational: (4 + √7)(4 − √7).

Q19. Show how to represent √3 on the number line using the square root spiral method.

Q20. Convert 1.2333... (only the 3 repeats) to p/q form.

Answer Key

Part A

Q	Answer	Reason
1	(c) √7	7 is not a perfect square. √36 = 6, 0.25 ends, and 0.6̅ repeats, so those are rational.
2	(d) 22/7	It is a fraction of two integers. The others are irrational.
3	(c)	Irrational decimals never end and never repeat.
4	(b) −4	−4 is an integer and also −4/1.
5	(b) 4	√2 × √8 = √16 = 4.
6	(a) 1/3	Standard repeating decimal. (Also 3/9 = 1/3, so the option (d) statement is false.)
7	(c) 3/16	16 = 2⁴, only prime factor 2.
8	(b) irrational	Rational + irrational is always irrational.

Part B

A9. Any three, for example 2.1, 2.5, 2.9 (or 21/10, 5/2, 29/10). There are infinitely many.

A10. Let x = 0.777... Then 10x = 7.777... Subtract: 9x = 7, so x = 7/9.

A11. Irrational. √5 is irrational, and (rational − irrational) is always irrational.

A12. Irrational. The number of 1s after each 2 keeps increasing, so there is no repeating block and the decimal never ends.

A13. 25 = 5 × 5. Only prime factor 5, so the decimal terminates. (7/25 = 0.28)

A14. 1/7 = 0.142857... and 2/7 = 0.285714... So any non-repeating non-terminating decimal between them works, for example 0.2010010001... (it lies between 0.1428... and 0.2857...).

A15. False. Example: √2 × √2 = 2, which is rational. (But √2 × √3 = √6 is irrational. So the product can be either.)

Part C

A16. Let x = 0.181818... Two digits repeat, so 100x = 18.181818... Subtract: 99x = 18, so x = 18/99 = 2/11.

A17. Assume √3 = p/q in lowest terms (p, q integers, q ≠ 0). Then 3 = p²/q², so p² = 3q². This means 3 divides p², so 3 divides p. Write p = 3k. Then 9k² = 3q², so q² = 3k², so 3 divides q. Now 3 divides both p and q, which contradicts "lowest terms". Hence √3 is irrational.

A18. (4 + √7)(4 − √7) = 4² − (√7)² = 16 − 7 = 9, which is rational.

A19. Draw the number line with 0 and 1. At 1, draw a perpendicular of length 1 and join its top to 0 to get a hypotenuse of √2. Draw an arc with centre 0 and radius √2 to mark √2 on the line. At the point √2, draw a perpendicular of length 1; the new hypotenuse from 0 is √(2 + 1) = √3. With centre 0 and this radius, draw an arc to cut the number line. That point is √3.

A20. Let x = 1.2333... Then 10x = 12.333... and 100x = 123.333... Subtract: 100x − 10x = 111, so 90x = 111, giving x = 111/90 = 37/30.

Quick Revision Summary

• Rational number = can be written as p/q (q ≠ 0). Decimal form ends or repeats.
• Irrational number = cannot be written as p/q. Decimal form never ends and never repeats.
• Integers, fractions, terminating decimals, repeating decimals and square roots of perfect squares are all rational.
• Square roots of non-perfect squares (like √2, √3, √5) and π are irrational.
• Real numbers = rational numbers + irrational numbers.
• N ⊂ W ⊂ Z ⊂ Q ⊂ R.
• Rational ± irrational = irrational. Non-zero rational × irrational = irrational.
• Irrational + irrational or irrational × irrational can be rational or irrational.
• 22/7 is only an approximation of π, not equal to it.
• Repeating decimal → fraction: multiply by 10ⁿ (n = number of repeating digits) and subtract.
• √2 can be shown exactly on the number line using the Pythagoras theorem.