In the previous sections we completed the discussion on equations in two variables'. In this section, we will learn about irrational numbers.
We will first see the types of numbers that we have learned so far. The fig.16.1 below shows a summary.
• Counting numbers 1,2,3, so on are called Natural numbers. They are denoted by the symbol N. They can be represented on a number line as shown in the fig. 16.1
• If we add '0' to the collection of natural numbers, we get the collection of Whole numbers. They are denoted by the symbol W. The '0' is at the beginning of the number line
• To the left of the zero, we have negative numbers. If we add those negative numbers also to the whole numbers, we get the collection of Integers. They are denoted by the symbol Z. 'Z' comes from the German word 'zahlen', which means 'to count'
• In between the integers, there are infinite number of fractions. A few examples are shown in the fig.
♦ We know that 3⁄4 comes in between 0 and 1
♦ 7⁄2 is equal to 31⁄2 . It comes in between 3 and 4
♦ -9⁄4 is equal to -21⁄2 . It comes in between 3 and 4
We can write any number of examples: 3978⁄3979 is a fraction which comes in between 0 and 1. If we include all such fractions also to integers, we will get the collection of Rational numbers.
■ A number ‘r’ is called a rational number, if it can be written in the form p⁄q, where p and q are integers and q ≠ 0.
Notice that, all the numbers shown in the last number line in fig.16.1 can be written in the form p⁄q, where p and q are integers, and q ≠ 0. For example, -17 can be written as -17⁄1. Here, p = -17, and q = 1. So every number that comes in the last number line in fig.16.1 is a rational number.
Rational numbers are denoted by the letter Q. The word rational comes from 'Ratio', and Q comes from the word 'Quotient'.
Let us now see some questions based on the above discussion:
Are the following statements true or false? Give reasons for your answers.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
Solution: (i) False. Because 'zero' is a whole number but not a natural number.
(ii) True. Because every integer 'm' can be expressed in the form m⁄1, and so it is a rational number.
(iii) False. Because fractions like 1⁄2, 3⁄4, 15⁄17 etc., are rational numbers, but they are not integers.
Now we will learn about some new numbers.
• Consider two identical squares as shown in fig.16.2(a) below. Their sides are 1 unit.
• In fig.b, each of them are cut along the diagonal. Thus we will get four equal triangles.
• In fig.c, these 4 triangles are arranged in such a way that, a new single square is formed. What is the peculiarity of this new larger square? Let us analyse:
1. The 4 triangles in the new square has the total area of the two squares in fig.a
2. Each of the two squares in fig.a has an area of 1 × 1 = 1 sq.unit
3. So the total area of the two squares in fig.a is 1 + 1 = 2 sq.unit
4. Thus the area of the new square in fig.c is 2 sq.unit
5. We want to calculate the side of this square. Is the side equal to 1?
No. Because, if the side is 1, it's area would be 1 × 1 = 1 sq.unit. This is less than the actual area of 2 sq.unit, which we calculated in (4)
6. Is the side equal to 2?
No. Because, if the side is 2, it's area would be 2 × 2 = 4 sq.unit. This is greater than the actual area of 2 sq.unit, which we calculated in (4)
7. So we conclude that, the side is a quantity, which is 'greater than 1', but 'less than 2'
• A quantity which is greater than 1, but less than 2 would be a fraction.
• So, if we have a square of area 2 sq.units, it's side would be a fraction.
Let us investigate more about this fraction:
8. We have seen that the fraction is greater than 1, but less than 2. That is., it lies between 1 and 2
9. Let us take the fraction which lies exactly midway between 1 and 2.
10. The fraction which lies exactly midway is 11⁄2 which is same as 3⁄2. The calculation is shown in fig.16.3(a) below:
11. The position of 3⁄2 between 1 and 2 is also shown in the number line on the right side of the calculation.
12. Is 3⁄2, our required value?
No. because 3⁄2 × 3⁄2 = 9⁄4 = 21⁄4 . This is greater than 2
13. So we conclude that the required fraction is less than 3⁄2.
14. Also from (5), it is greater than 1. That is., the required fraction lies in between 1 and 3⁄2
15. Let us take the fraction which lies exactly midway between 1 and 3⁄2. The fraction which lies exactly midway is 11⁄4 which is same as 5⁄4 . The calculation is shown in fig.16.3(b) above.
16. The position of 5⁄4 between 1 and 3⁄2 is shown in the number line in fig.16.4(b) above.
17. Is 5⁄4 our required value?
No. because 5⁄4 × 5⁄4 = 25⁄16 = 19⁄16. This is less than 2
18. So we conclude that the required fraction is greater than 5⁄4.
19. Also from (12), it is less than 3⁄2. That is., the required fraction lies in between 5⁄4 and 3⁄2
20. Let us take the fraction which lies exactly midway between 5⁄4 and 3⁄2. The fraction which lies exactly midway is 11⁄8. This is shown in fig.
21. Is 11⁄8 our required value?
No. because 11⁄8 × 11⁄8 = 121⁄64 = 157⁄64. This is less than 2
22. But 157⁄64 is 'closer to 2' than 19⁄16. So 11⁄8 is a better result than 5⁄4 .
But still, it is not the exact required value.
Instead of proceeding like this by trial and error, we will use algebra. We will see it in the next section.
We will first see the types of numbers that we have learned so far. The fig.16.1 below shows a summary.
 |
| Fig.16.1 |
• If we add '0' to the collection of natural numbers, we get the collection of Whole numbers. They are denoted by the symbol W. The '0' is at the beginning of the number line
• To the left of the zero, we have negative numbers. If we add those negative numbers also to the whole numbers, we get the collection of Integers. They are denoted by the symbol Z. 'Z' comes from the German word 'zahlen', which means 'to count'
• In between the integers, there are infinite number of fractions. A few examples are shown in the fig.
♦ We know that 3⁄4 comes in between 0 and 1
♦ 7⁄2 is equal to 31⁄2 . It comes in between 3 and 4
♦ -9⁄4 is equal to -21⁄2 . It comes in between 3 and 4
We can write any number of examples: 3978⁄3979 is a fraction which comes in between 0 and 1. If we include all such fractions also to integers, we will get the collection of Rational numbers.
■ A number ‘r’ is called a rational number, if it can be written in the form p⁄q, where p and q are integers and q ≠ 0.
Notice that, all the numbers shown in the last number line in fig.16.1 can be written in the form p⁄q, where p and q are integers, and q ≠ 0. For example, -17 can be written as -17⁄1. Here, p = -17, and q = 1. So every number that comes in the last number line in fig.16.1 is a rational number.
Rational numbers are denoted by the letter Q. The word rational comes from 'Ratio', and Q comes from the word 'Quotient'.
Let us now see some questions based on the above discussion:
Are the following statements true or false? Give reasons for your answers.
(i) Every whole number is a natural number.
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
Solution: (i) False. Because 'zero' is a whole number but not a natural number.
(ii) True. Because every integer 'm' can be expressed in the form m⁄1, and so it is a rational number.
(iii) False. Because fractions like 1⁄2, 3⁄4, 15⁄17 etc., are rational numbers, but they are not integers.
Now we will learn about some new numbers.
• Consider two identical squares as shown in fig.16.2(a) below. Their sides are 1 unit.
• In fig.b, each of them are cut along the diagonal. Thus we will get four equal triangles.
 |
| Fig.16.2 |
1. The 4 triangles in the new square has the total area of the two squares in fig.a
2. Each of the two squares in fig.a has an area of 1 × 1 = 1 sq.unit
3. So the total area of the two squares in fig.a is 1 + 1 = 2 sq.unit
4. Thus the area of the new square in fig.c is 2 sq.unit
5. We want to calculate the side of this square. Is the side equal to 1?
No. Because, if the side is 1, it's area would be 1 × 1 = 1 sq.unit. This is less than the actual area of 2 sq.unit, which we calculated in (4)
6. Is the side equal to 2?
No. Because, if the side is 2, it's area would be 2 × 2 = 4 sq.unit. This is greater than the actual area of 2 sq.unit, which we calculated in (4)
7. So we conclude that, the side is a quantity, which is 'greater than 1', but 'less than 2'
• A quantity which is greater than 1, but less than 2 would be a fraction.
• So, if we have a square of area 2 sq.units, it's side would be a fraction.
Let us investigate more about this fraction:
8. We have seen that the fraction is greater than 1, but less than 2. That is., it lies between 1 and 2
9. Let us take the fraction which lies exactly midway between 1 and 2.
10. The fraction which lies exactly midway is 11⁄2 which is same as 3⁄2. The calculation is shown in fig.16.3(a) below:
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| Fig.16.3 |
12. Is 3⁄2, our required value?
No. because 3⁄2 × 3⁄2 = 9⁄4 = 21⁄4 . This is greater than 2
13. So we conclude that the required fraction is less than 3⁄2.
14. Also from (5), it is greater than 1. That is., the required fraction lies in between 1 and 3⁄2
16. The position of 5⁄4 between 1 and 3⁄2 is shown in the number line in fig.16.4(b) above.
17. Is 5⁄4 our required value?
No. because 5⁄4 × 5⁄4 = 25⁄16 = 19⁄16. This is less than 2
18. So we conclude that the required fraction is greater than 5⁄4.
19. Also from (12), it is less than 3⁄2. That is., the required fraction lies in between 5⁄4 and 3⁄2
21. Is 11⁄8 our required value?
No. because 11⁄8 × 11⁄8 = 121⁄64 = 157⁄64. This is less than 2
22. But 157⁄64 is 'closer to 2' than 19⁄16. So 11⁄8 is a better result than 5⁄4 .
But still, it is not the exact required value.
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