Chapter 22.1 - Rational and Irrational numbers

In the previous section we saw how rational numbers are represented on a number line. In this section we will see irrational numbers.

We have seen examples of irrational numbers in an earlier chapter. We were able to represent √2 on the number line. We saw it in fig.16.6. For convenience it is shown again in fig.22.4(a) below.

Fig.22.4

In the same manner we can represent √3, √5 √7 etc., √3 is shown in fig.b

Now we come to the question of accuracy. We know that √2 = 1.41421356... 
Can we mark a non-terminating non-recurring decimal like 1.41421356... accurately? 
The answer is yes. The method is the same 'zooming in' that we saw in the fig.22.2 in the previous section. Let us see the details:

We know that 1.41421356... lies in between 1 and 2. In the fig.22.5 below, ‘zoom level 1’ shows the portion between 1 and 2, at an enlarged scale.

In this zoom level, we have enough space to clearly mark ten subdivisions between 1 and 2. So the fourth subdivision will be 1.4 and the fifth subdivision will be 1.5.

Then we zoom in on the region between this 1.4 and 1.5. This is shown as 'zoom level 2'. In this zoom level, we have enough space to clearly mark ten subdivisions between 1.4 and 1.5. So the first subdivision will be 1.41 and the second subdivision will be 1.42.

Then we zoom in on the region between this 1.41 and 1.42. This is shown as 'zoom level 3'. In this zoom level, we have enough space to clearly mark ten subdivisions between 1.41 and 1.42. So the fourth subdivision will be 1.414 and the fifth subdivision will be 1.415.

So, if we take the left red subdivision in zoom level 3, it will represent 1.414. It is fairly accurate. We know that 0.004 is a small quantity. But we are able to mark it by 'zooming in'. We can mark a quantity, even if it is non-recurring and non-terminating, however small it may be. All we need to do, is to zoom in to the required level.

Like √3, √7, π etc., also have decimal forms. Even though they are non-recurring and non-terminating decimals, they can be represented on a number line.

So we can conclude that:
Any irrational number can be represented on a number line.

---

Now we try to do the reverse: We are given an irrational number on the number line. Can we represent it as a fraction?

The answer is:
No irrational numbers cannot be expressed in the form ^p⁄_q. We have seen the proof for this, in the case of √2. (Details here)

---

We will now see some important differences between rational and irrational numbers:
I. Sum of two rational numbers will be a rational number.
Proof:
(a) Consider two rational numbers ^p⁄_q and ^r⁄_s. Their sum is:
^p⁄_q + ^r⁄_s = ^ps⁄_qs + ^rq⁄_sq = ^(ps+rq)⁄_qs.
(b) Let us analyse the above result. 
• ^p⁄_q and ^r⁄_s are rational numbers. So p, q, r and s are integers (... -4, -3, -2, -1, 0, 1, 2, 3, 4...)
• Also q and s not equal to zero
(c) So ps, rq and qs will be integers. (ps+rq) will also be an integer
(d) Thus, the result in (a) is an [ ^integer⁄_integer]. That means, the result in (a) is a rational number

II The sum of two irrational numbers may be rational or irrational
Example 1: Sum of √3 and √2 is irrational
Proof:

5. If 'a' is a rational number, [a + ¹⁄_a] will also be a rational number.
6. Then ¹⁄₂[a + ¹⁄_a] will also be a rational number
7. But from (4), we have '¹⁄₂[a + ¹⁄_a]' is equal to √3 which is an irrational number.
8. So 'a' is irrational. That means (√3 + √2) is irrational

Example 2: Sum of (1+√2) and (1-√2) is rational
Proof:
• This can be proved just by adding: 1 + √2 + 1 - √2 = 2
• '2' is a rational number

III. Difference of two rational numbers will be a rational number.
Proof:
(a) Consider two rational numbers ^p⁄_q and ^r⁄_s. Their difference is:
^p⁄_q - ^r⁄_s = ^ps⁄_qs - ^rq⁄_sq = ^(ps-rq)⁄_qs.
(b) From (I) above, we can write:
The result in (a) is an [ ^integer⁄_integer]. That means, the result in (a) is a rational number

IV. The difference of two irrational numbers may be rational or irrational
This can be proved in the same way as in (II)

V. Product of two rational numbers will be a rational number.
Proof:
(a) Consider two rational numbers ^p⁄_q and ^r⁄_s. Their product is:
(^p⁄_q) × (^r⁄_s) = ^pr⁄_qs. This is a rational number

VI. The product of two irrational numbers may be rational or irrational
Example 1: Consider 6√5. 
1. We have √5 = 2.236...
2. So 6√5 = 6 × 2.236... = 13.416...
3. 13.416... is an irrational number. So 6√5 is an irrational number
4. Similarly we can prove 2√5 is also an irrational number
5. Now take the products. 6√5 × 2√5 = 12 × 5 = 60
6. 60 is a rational number. So we got a rational number when two irrationals were multiplied

Example 2: 
1. √3 × √5 = √15
2. √15 is an irrational number
3. So we got an irrational number when we multiplied two irrational numbers

VII. Quotient of two rational numbers will be a rational number.
Proof:
(a) Consider two rational numbers ^p⁄_q and ^r⁄_s. Their quotient is:
(^p⁄_q) ÷ (^r⁄_s) = (^p⁄_q) × (^s⁄_r) = ^ps⁄_qr. This is a rational number

VIII. The quotient of two irrational numbers may be rational or irrational
Example 1: ^(8√3)⁄_(√3) = 8 
'8' is a rational number
Example 2: ^(8√15)⁄_(2√3) = ^(8√15)⁄_(2√3) = ^{(8×√5×√3)}⁄_(2√3) = 4√5
'4√5' is an irrational number

---

From the above discussion, we can make the following conclusions:

Conclusion 1:
From I and II, we can conclude this:
• Sum of two rational numbers will always be a rational number. 
• But sum of two irrational numbers may be rational or irrational
Conclusion 2:
From III and IV, we can conclude this:
• Difference of two rational numbers will always be a rational number. 
• But difference of two irrational numbers may be rational or irrational
Conclusion 3:
From V and VI, we can conclude this:
• Product of two rational numbers will always be a rational number. 
• But product of two irrational numbers may be rational or irrational
Conclusion 4:
From VII and VIII, we can conclude this:
• Quotient of two rational numbers will always be a rational number. 
• But quotient of two irrational numbers may be rational or irrational

---

The above 4 conclusions are applicable when the rational numbers and irrational numbers are treated separately. We will now see what will be the result when they come together in calculations.

Conclusion 5

• Sum of a rational number and an irrational number will always be an irrational number. 

Conclusion 6

• Difference of a rational number and an irrational number will always be an irrational number.

Conclusion 7

• Product of a rational number and an irrational number will always be an irrational number.

    ♦ In this case it is important to mention that, the rational number should not be zero. Other wise, the product will be zero. 

Conclusion 8

• Quotient of a rational number and an irrational number will always be an irrational number.

    ♦ In this case it is important to mention that, the rational number should not be zero. Other wise, the quotient will be zero.

The reader may write his/her own examples related to the above conclusions 5, 6, 7 and 8

---

This completes a main portion of our discussion. In the next section we will take up the topic of real numbers.

PREVIOUS      CONTENTS       NEXT



                        Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved

Chapter 22 - Real Numbers

In the previous section we completed the discussion on Circles, arcs and sectors. In a previous chapter, we have seen different types of numbers. (see fig.16.1). In this chapter, we will see how and where 'Real numbers' fit into the list of numbers.

First let us recall how numbers are represented on a number line. In fig.22.1(a), a horizontal line segment is drawn between points A and B. We will fix this distance AB as 1 unit.

Fig.22.1

• So, if we take two line segments, each equal to AB, and put them end to end, we will get a distance of 2 units. This distance is the distance from zero to 2 on the number line shown in fig.d

• If we take 3 such segments and put them end to end, we will get a distance of 3 units. This distance is the distance from zero to 3 on the number line shown in fig.d. In this way, we can find the position of any number on the number line.

Now what about fractions? Consider fig.b. The segment AB is divided into 2 equal parts at C. So AC = BC = ¹⁄₂ unit. 
• If we take two line segments, one equal to AB, and the other equal to AC, and put them end to end, we will get a distance 1¹⁄₂ = ³⁄₂ units. If we measure out this end to end distance (from zero) on the number line, we can mark ³⁄₂. It is shown in fig.d

Another example: In fig.c, the segment AB is divided into 3 equal parts at D and E. So AD = DE = BE = ¹⁄₃ unit. 
• If we take three line segments, two of them equal to AB, and the third equal to AD, and put them end to end, we will get a distance 2¹⁄₃ = ⁷⁄₃ units.  If we measure out this end to end distance (from zero) on the number line, we can mark ⁷⁄₃. It is shown in fig.d

In the above examples, we marked points using a geometrical method. That is., we measured the distance from the 'standard 1 unit', or it's fractions, and then marked it on the number line. Can we do the marking with out actual measuring? Let us analyse:

• ³⁄₂ does not give us any problems because in decimal form, it is 1.5. It is a terminating decimal. 
• But ⁷⁄₃ is a different case. We discussed about such numbers here. It is a recurring decimal. It is written as ⁷⁄₃ = 2.333... Using this decimal form, we can represent ⁷⁄₃ on the number line by the following procedure: 

2.333... has infinite number of decimal places. The digit '3' repeats forever. If more number of decimal places are taken, we will get more accuracy. If we want such a greater accuracy, we will have to 'zoom in' on the region between 2 and 3. Because 2.333... lies in between 2 and 3. In the fig.22.2 below, ‘zoom level 1’ shows the portion between 2 and 3, at an enlarged scale.

Fig.22.2

In this zoom level, we have enough space to clearly mark ten subdivisions between 2 and 3. So the third subdivision will be 2.3 and the fourth subdivision will be 2.4. They are specially marked in red because 2.333... lies between them

Next, we zoom in on the region between this 2.3 and 2.4. This is shown as 'zoom level 2'. In this zoom level, we have enough space to clearly mark ten subdivisions between 2.3 and 2.4. So the third subdivision will be 2.33 and the fourth subdivision will be 2.34. They are specially marked in red because 2.333... lies between them

Next, we zoom in on the region between this 2.33 and 2.34. This is shown as 'zoom level 3'. In this zoom level, we have enough space to clearly mark ten subdivisions between 2.33 and 2.34. So the third subdivision will be 2.333 and the fourth subdivision will be 2.334. They are specially marked in red because 2.333... lies between them

So, if we take the left red subdivision in zoom level 3, it will represent 2.333. It is fairly accurate. We know that 0.003 is a small quantity. But we are able to mark it by 'zooming in'. It is like zooming in on Google Maps. At higher zoom levels, we are able to see smaller details. But in our case of number line, there is no limit. We can mark any quantity, however small it may be. All we need to do, is to zoom in to the required level. 
So we can conclude that:
• Any natural number can be represented on a number line
• Any fraction can be represented on a number line

---

Now we try to do the reverse: We are given a number line with a mark  on it. We want to represent that mark as a number or a fraction. 
Consider the mark P in fig.22.3(a). It is exactly at 4

Fig.22.3

So we write: The mark P represents number 4. 
Consider mark Q. It is exactly at 6.5 
So we write: The mark Q represents number 6.5, which is equal to ¹³⁄₂
Consider mark R. We have seen that any recurring decimal can be represented on a number line. If it is given that R is 5.333... , can we represent it as a fraction? 
The answer is yes. The procedure is as follows:
• Let x = 5.333...
• 10x = 10 × 5.333... = 53.333... = 48 + 5.333... = 48 + x (∵ x = 2.333..)
⇒ 10x = 48 + x ⇒ 9x = 48 ⇒ x = ⁴⁸⁄₉ = 5³⁄₉ = 5¹⁄₃ = ¹⁶⁄₃

So, even if the given mark on the number line is a recurring decimal, we can convert it into a fraction. Let us see a few more examples:
■ Express 1.272727... as a fraction:
1. We can write: 1.272727... = 1.2̅7
2. A line is drawn above 2 and 7. That means the block '27' repeats for ever
3. Let x = 1.272727... Since two digits are repeating, we will multiply by 100
• 100x = 100 × 1.272727... = 127.272727... = 126 + 1.272727... = 126 + x (∵ x = 1.272727..)
⇒ 100x = 126 + x ⇒ 99x = 126 ⇒ x = ¹²⁶⁄₉₉ = 1³⁄₁₁

■ Express 0.2353535... as a fraction:
1. We can write: 0.2353535... = 0.23̅5
2. A line is drawn above 3 and 5. That means the block '35' repeats for ever
3. Let x = 0.2353535... Since two digits are repeating, we will multiply by 100
• 100x = 100 × 0.2353535... = 23.535353... =  23.3 + 0.2353535... = 23.3 + x (∵ x = 0.2353535...)
⇒ 100x = 23.3 + x ⇒ 99x = 23.3 ⇒ x = ²³³⁄₉₉₀

---

■ Based on the above discussion, we can write the following:
1. Any natural number can be represented on a number line
2. Any fraction can be represented on a number line
    ♦ The fractions  like ¹⁄₂, ¹⁄₅, ³⁄₅ etc., that give terminating decimals
    ♦ The fractions like ¹⁄₃, ¹⁄₇, ³⁄₅ etc., that give non-terminating, recurring decimals
■ Reverse of the above is also possible: 
If a point is marked on a number line, it can be represented by any one of the two categories given below:
1. A natural number
2. A fraction
    ♦ The fraction may be one  like ¹⁄₂, ¹⁄₅, ³⁄₅ etc., that give terminating decimals
    ♦ The fraction may be one like ¹⁄₃, ¹⁄₇, ³⁄₅ etc., that give non-terminating, recurring decimals

---

■ Any number that can be expressed as a fraction ^p⁄_q where q is not equal to zero is called a rational number. 

• Simple natural numbers like 2, 5 etc., can be written as ²⁄₁ and ⁵⁄₁. So they are rational numbers. 
• Decimals like 0.2, 0.35, 0.5 etc., can be easily written in the form of fractions (with non-zero denominators). So they are rational numbers.
• Non-terminating recurring decimals like 0.333..., 1.2̅7, 0.23̅5, etc., can be written in the form of fractions (with non-zero denominators). So they are rational numbers.

---

In this section we saw the details about rational numbers. In the next section we will see irrational numbers.

PREVIOUS      CONTENTS       NEXT

                        Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved