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.
• 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
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.
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
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
• 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.
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 |
• 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
• 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
• 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:
• 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 |
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
Consider the mark P in fig.22.3(a). It is exactly at 4
 |
| Fig.22.3 |
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
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.
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