In the previous sections we saw how rational numbers and irrational numbers are represented on a number line. We also saw the results when operations are performed on them. We can now write this:
Let us see some important facts related to a number line:
• In a number line, as we move towards the right from zero, the numbers get larger and larger
Example: 5 is larger than 4
• As we move towards the left from zero, the numbers get smaller and smaller
Example: -2 is smaller than -1
Instead of zero, we can start our journey from any number. We will get the same experience.
• Towards the right, the numbers become larger and larger
• Towards the left, the numbers become smaller and smaller
So if we are given any two numbers, we are able to say this:
■ On the number line, the larger number will be on the right of the smaller number
For example, if we are given -34 and 12,
12 is on the right of -34 on the number line
Another example:
-114 is on the right of -150
We know that the distance between any two adjacent integers will be the ‘standard 1 unit’. This standard is to be fixed up before drawing the number line.
An example for finding distances:
• The distance of 51⁄4 from zero is 51⁄4 units
• The distance of 21⁄2 from zero is 21⁄2 units
These are marked in fig.22.7 below:
From the fig., it is clear that, the distance between the points representing 51⁄4 and 21⁄2 is obtained as:
51⁄4 - 21⁄2 = 23⁄4 units. This is shown in blue colour.
Another case:
This time 51⁄4 and 21⁄2 are both marked on the left of zero. So the points are -51⁄4 and -21⁄2. This is shown in fig.22.8 below:
Here also the distance between the two points is 23⁄4 units.
Another case:
This time 21⁄2 is marked on the left side of zero and 51⁄4 is marked on the right side of zero. This is shown in the fig.22.9 below.
From the fig. it is clear that, the distance between the points is 73⁄4 units.
One more case:
This time 51⁄4 is marked on the left side of zero and 21⁄2 is marked on the right side of zero. This is shown in the fig.22.10 below:
From the fig., it is clear that, the distance between the points is: 73⁄4 units.
Let us write a summary of the four cases:
In the above table we have four pairs, corresponding to the four cases.
■ Each pair has two numbers.
■ Each of these numbers denotes a point.
For example, the number '-51⁄4' denotes a point which is 51⁄4 units to the left of zero.
Based on this, let us analyse the table:
Case 1:
1. The number denoting the first point is 21⁄2
2. The number denoting the second point is 51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = 21⁄2
• Subtract it from the larger. That is: 51⁄4 – 21⁄2
• The result is 23⁄4
Case 2:
1. The number denoting the first point is -21⁄2
2. The number denoting the second point is -51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -51⁄4
• Subtract it from the larger. That is: -21⁄2 - (-51⁄4)
⇒ -21⁄2 + 51⁄4 ⇒ 51⁄4 - 21⁄2
• The result is 23⁄4
Case 3:
1. The number denoting the first point is -21⁄2
2. The number denoting the second point is 51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -21⁄2
• Subtract it from the larger. That is: 51⁄4 - (-21⁄2)
⇒ 51⁄4 + 21⁄2
• The result is 73⁄4
Case 4:
1. The number denoting the first point is 21⁄2
2. The number denoting the second point is -51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -51⁄4
• Subtract it from the larger. That is: 21⁄2 - (-51⁄4)
⇒ 51⁄4 + 21⁄2
• The result is 73⁄4
So we can see that, in all four cases, the step 3 is the same. That means the procedure for finding the distance is the same. We can write it in the form of a theorem.
Theorem 22.1
1. We have two points on the number line
2. Each point is denoted by a number. Let the numbers be x1 and x2
3. Take out the smaller number. Let x1 be this smaller number
4. Subtract it from the larger
5. The difference (x2 - x1) thus obtained is the distance between the points
This is shown in fig.22.11 below:
The above procedure is applicable for any positions of the points:
• The two points may both be on the left of zero
• The two points may both be on the right of zero
• One point may be on the left and the other on the right
■ That is why, zero is not shown in the above fig.22.11
The theorem 22.1 above gives us the method to find the distance between any two points.
So we obtained ‘distance’ from ‘numbers’. Can we do the reverse? That is., can we obtain number from distance?
The answer is yes. But we can obtain only one number. The distance, and the other number should be given. Let us analyse this:
In the fig.22.12 below, A and B are two points on the number line. Let the numbers denoting them be x1 and x2. That is., A is at a distance of x1 from zero, and B is at a distance of x2 from zero.
[As mentioned earlier, to find the distance, we need only the numbers. The position of zero does not have any role to play. So it is not shown in the above fig.22.12]
In the fig., A is on the left of B. So x1 is less than x2
Then, according to theorem 22.1, the distance between A and B is (x2 - x1)
■ Suppose that we are given the following two information:
• The number (x1) corresponding to point A
• The distance between A and B
With the two information, we can find the number corresponding to point B. The steps are as follows:
1. We know that the number corresponding to B will be greater than x1. Because B is on the right side of A
2. So add the distance to x1
We will get: x1 + (x2 - x1) = x1 + x2 - x1 = x2
This is the number corresponding to point B
■ Suppose that we are given the following two information:
• The number (x2) corresponding to point B
• The distance between A and B
With the two information, we can find the number corresponding to point A. The steps are as follows:
1. We know that the number corresponding to A will be less than x2. Because B is on the right side of A
2. So subtract the distance from x2
We will get: x2 - (x2 - x1) = x2 - x2 + x1 = x1
This is the number corresponding to point A
We can write this in the form of a theorem
Theorem 22.2:
• We are given two points
• The distance between them is available
• But the number corresponding to only one of them is available
■ If the left side number is available then:
Right side number = Left side number + distance
■ If the right side number is available then:
Left side number = Right side number - distance
■ All rational numbers can be represented on a number line
♦ All rational numbers represented on a number line can be expressed as a fraction
■ All irrational numbers can be represented on a number line
♦ No irrational number can be expressed as a fraction
We can now take up the topic on real numbers.
■ All numbers which can be represented on a number line are called real numbers.
So both rational numbers and irrational numbers are real numbers
In physics we have learned that:
In physics we have learned that:
• Images which can be caught on a screen are called 'real images'.
♦ Virtual images cannot be caught on a screen.
• In a similar way, numbers which can be represented on a number line are called real numbers.
♦ Similar to virtual images, are there virtual or unreal numbers? We will learn about them in higher classes.
Operations on Real numbers
We have seen that real numbers can be represented on a number line. The number line is a geometrical representation of numbers. Positive numbers are marked on the right of zero and negative numbers are marked on the left of zero. This is shown in the fig.22.6 below: |
| Fig.22.6 |
Example: 5 is larger than 4
• As we move towards the left from zero, the numbers get smaller and smaller
Example: -2 is smaller than -1
Instead of zero, we can start our journey from any number. We will get the same experience.
• Towards the right, the numbers become larger and larger
• Towards the left, the numbers become smaller and smaller
So if we are given any two numbers, we are able to say this:
■ On the number line, the larger number will be on the right of the smaller number
For example, if we are given -34 and 12,
12 is on the right of -34 on the number line
Another example:
-114 is on the right of -150
Distance between numbers on a number line
Number line is a geometrical representation of numbers. So we will be able to measure the distances between points. But we do not have to make actual measurements. We will be able to find distances by simple addition or subtraction.We know that the distance between any two adjacent integers will be the ‘standard 1 unit’. This standard is to be fixed up before drawing the number line.
An example for finding distances:
• The distance of 51⁄4 from zero is 51⁄4 units
• The distance of 21⁄2 from zero is 21⁄2 units
These are marked in fig.22.7 below:
 |
| Fig.22.7 |
51⁄4 - 21⁄2 = 23⁄4 units. This is shown in blue colour.
Another case:
This time 51⁄4 and 21⁄2 are both marked on the left of zero. So the points are -51⁄4 and -21⁄2. This is shown in fig.22.8 below:
 |
| Fig.22.8 |
Another case:
This time 21⁄2 is marked on the left side of zero and 51⁄4 is marked on the right side of zero. This is shown in the fig.22.9 below.
 |
| Fig.22.9 |
One more case:
This time 51⁄4 is marked on the left side of zero and 21⁄2 is marked on the right side of zero. This is shown in the fig.22.10 below:
 |
| Fig.22.10 |
Let us write a summary of the four cases:
In the above table we have four pairs, corresponding to the four cases.
■ Each pair has two numbers.
■ Each of these numbers denotes a point.
For example, the number '-51⁄4' denotes a point which is 51⁄4 units to the left of zero.
Based on this, let us analyse the table:
Case 1:
1. The number denoting the first point is 21⁄2
2. The number denoting the second point is 51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = 21⁄2
• Subtract it from the larger. That is: 51⁄4 – 21⁄2
• The result is 23⁄4
Case 2:
1. The number denoting the first point is -21⁄2
2. The number denoting the second point is -51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -51⁄4
• Subtract it from the larger. That is: -21⁄2 - (-51⁄4)
⇒ -21⁄2 + 51⁄4 ⇒ 51⁄4 - 21⁄2
• The result is 23⁄4
Case 3:
1. The number denoting the first point is -21⁄2
2. The number denoting the second point is 51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -21⁄2
• Subtract it from the larger. That is: 51⁄4 - (-21⁄2)
⇒ 51⁄4 + 21⁄2
• The result is 73⁄4
Case 4:
1. The number denoting the first point is 21⁄2
2. The number denoting the second point is -51⁄4
3. Distance between the two points is obtained as follows:
• Smaller of the two = -51⁄4
• Subtract it from the larger. That is: 21⁄2 - (-51⁄4)
⇒ 51⁄4 + 21⁄2
• The result is 73⁄4
Theorem 22.1
1. We have two points on the number line
2. Each point is denoted by a number. Let the numbers be x1 and x2
3. Take out the smaller number. Let x1 be this smaller number
4. Subtract it from the larger
5. The difference (x2 - x1) thus obtained is the distance between the points
This is shown in fig.22.11 below:
 |
| Fig.22.11 |
• The two points may both be on the left of zero
• The two points may both be on the right of zero
• One point may be on the left and the other on the right
■ That is why, zero is not shown in the above fig.22.11
So we obtained ‘distance’ from ‘numbers’. Can we do the reverse? That is., can we obtain number from distance?
The answer is yes. But we can obtain only one number. The distance, and the other number should be given. Let us analyse this:
In the fig.22.12 below, A and B are two points on the number line. Let the numbers denoting them be x1 and x2. That is., A is at a distance of x1 from zero, and B is at a distance of x2 from zero.
 |
| Fig.22.12 |
In the fig., A is on the left of B. So x1 is less than x2
Then, according to theorem 22.1, the distance between A and B is (x2 - x1)
■ Suppose that we are given the following two information:
• The number (x1) corresponding to point A
• The distance between A and B
With the two information, we can find the number corresponding to point B. The steps are as follows:
1. We know that the number corresponding to B will be greater than x1. Because B is on the right side of A
2. So add the distance to x1
We will get: x1 + (x2 - x1) = x1 + x2 - x1 = x2
This is the number corresponding to point B
■ Suppose that we are given the following two information:
• The number (x2) corresponding to point B
• The distance between A and B
With the two information, we can find the number corresponding to point A. The steps are as follows:
1. We know that the number corresponding to A will be less than x2. Because B is on the right side of A
2. So subtract the distance from x2
We will get: x2 - (x2 - x1) = x2 - x2 + x1 = x1
This is the number corresponding to point A
We can write this in the form of a theorem
Theorem 22.2:
• We are given two points
• The distance between them is available
• But the number corresponding to only one of them is available
■ If the left side number is available then:
Right side number = Left side number + distance
■ If the right side number is available then:
Left side number = Right side number - distance
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