In the previous sections we saw some situations where negative numbers come into play. In this section we will see some more examples.
The fig.4.3 below shows a multi storey building. It has 3 floors below ground level and several floors above ground level.
• The floor which is exactly at the ground level is named as 'floor 0'.
• The floor just above it is 'floor 1'.
• The next upper floor is 'floor 2'.
• Similarly, the floor just below floor 0 is 'floor -1'.
• The one just below floor -1 is 'floor -2 ... and so on.
• This procedure of naming the floors is used in the lifts also.
♦ When the lift is in the lower most floor of the building, the indicator in the lift shows '-3'.
♦ When it climbs one floor, the indicator shows '-2'...and so on. So higher the negative number, lower is the lift
♦ When the lift reaches the ground floor, '0' is shown.
Another situation is the expression of temperature. During cold weather the temperature begins to decrease. When the temperature decreases further, water begins to solidify. The exact temperature at which the water solidifies and becomes ice, is taken as 0o centigrade. But the temperature can decrease even further. It is important to know how much below zero is the temperature in a locality. For example, this knowledge is important in the design of heating systems. A heating system which is capable to provide normal living conditions at zero degree centigrade will not be sufficient if the temperature is much lower than zero. Measurement of the exact temperature below zero is of vital importance in many Scientific and Engineering experiments and procedures.
So we indeed need a method to measure temperatures lower than zero. Negative numbers are an easy method to do it. If the whether department says that the temperature is -20o centigrade, it means, the temperature is 20 degrees below zero. If it is -25o, it is even colder than -20o by 5 degrees. So higher the negative number, colder is the temperature.
Thus we find that negative numbers are an important part of our life. We must include them in the family of other numbers. Let us see how this is done:
• Numbers 1, 2, 3, . . ., are the Natural numbers.
• If we add '0' to this, the list becomes 0, 1, 2, 3, . . . The name of this list is Whole numbers
• Now we just met negative numbers. We must add them also to the list of whole numbers. Then we get: . . . -5, -4, -3, -2, -1, 0, 1, 2, 3, . . ., This list is known as the list of Integers.
On a number line, the integers can be given appropriate places as shown in fig.4.4 below. The arrows at the ends indicate that the line can extend to any distance on either side. Thus it can accommodate up to any large number, be it positive or negative. Also the spacing between the numbers should be uniform. This is a rule which have to be strictly followed while drawing number lines.
Let us see how the above number line works. Suppose we are standing at +5. If we begin to move towards the left, the value begins to decrease. It becomes +4, then +3, then +2 and so on. When we reach zero, there is no value. If we continue moving towards the left, the value will fall even below zero. It becomes -1, then -2, then -3 and so on. Similarly, from any where on the number line, if we move towards the right, the value increases.
• Zero can be taken as a 'bench mark'.
• The further left we are from zero, the lesser is our value. Also, as we are on the left of zero, the value will be negative
• The further right we are from zero, the greater is our value. Also, as we are on the right of zero, the value will be positive
Thus, the number line shows the position of each integer in the real world. Take for example -6 and -5. The 'numerical value' of -6 is greater than -5 (because, if we remove the '-' sign, 6 is greater than 5). But when the '-' sign is attached, things are just the opposite. -6 has a lower value than -5.
We can think of it in this way:
There is a greater distance to travel from -6 than from -5 to get to the 'boundary zero' and then to the positive side. So any body would prefer -5. So, as we move further towards the left, the value gets lower and lower.
Based on the above discussion, we can now see some solved examples:
Solved example 4.1
In the fig.4.5 given below, fill up the boxes using ‘<’ OR ‘>’
Solution:
(a) -20 > -35 (b) -9 < 2 (c) 60 > -62 (d) 0 > -1 (e) -70 > -71 (f) -73 < -7
Solved example 4.2
Mark the following points on a number line: (a) -7 (b) -1 (c) +4 (d) +8 (e) -4
Solution:
The fig.4.6 shows the required markings
Solved example 4.3
Given below are a few pairs of numbers.
(a) 4,12 (b) -21, -22 (c) 1, -100, (d) -5, -11
In each pair, pick out the number which is to the right of the other on the number line
Solution:
(a) 12. It is to the right of 4
(b) -21. It is to the right of -22
(c) 1. It is to the right of -100
(d) -5. It is to the right of -11
Solved example 4.4
Write all the integers between 4 and -7 in the increasing order of value.
Solution:
-6, -5, -4, -3, -2, -1, 0, 1, 2 and 3
Solved example 4.5
If we move from -20 to -10, will the values of the integers increase or decrease?
Solution:
Values will increase.
So now we have a good understanding about the positive and negative numbers, and their positions in the real world. In the next section we will see how addition between these numbers can be carried out.
The fig.4.3 below shows a multi storey building. It has 3 floors below ground level and several floors above ground level.
 |
| Fig.4.3 |
• The floor which is exactly at the ground level is named as 'floor 0'.
• The floor just above it is 'floor 1'.
• The next upper floor is 'floor 2'.
• Similarly, the floor just below floor 0 is 'floor -1'.
• The one just below floor -1 is 'floor -2 ... and so on.
• This procedure of naming the floors is used in the lifts also.
♦ When the lift is in the lower most floor of the building, the indicator in the lift shows '-3'.
♦ When it climbs one floor, the indicator shows '-2'...and so on. So higher the negative number, lower is the lift
♦ When the lift reaches the ground floor, '0' is shown.
Another situation is the expression of temperature. During cold weather the temperature begins to decrease. When the temperature decreases further, water begins to solidify. The exact temperature at which the water solidifies and becomes ice, is taken as 0o centigrade. But the temperature can decrease even further. It is important to know how much below zero is the temperature in a locality. For example, this knowledge is important in the design of heating systems. A heating system which is capable to provide normal living conditions at zero degree centigrade will not be sufficient if the temperature is much lower than zero. Measurement of the exact temperature below zero is of vital importance in many Scientific and Engineering experiments and procedures.
So we indeed need a method to measure temperatures lower than zero. Negative numbers are an easy method to do it. If the whether department says that the temperature is -20o centigrade, it means, the temperature is 20 degrees below zero. If it is -25o, it is even colder than -20o by 5 degrees. So higher the negative number, colder is the temperature.
Thus we find that negative numbers are an important part of our life. We must include them in the family of other numbers. Let us see how this is done:
• Numbers 1, 2, 3, . . ., are the Natural numbers.
• If we add '0' to this, the list becomes 0, 1, 2, 3, . . . The name of this list is Whole numbers
• Now we just met negative numbers. We must add them also to the list of whole numbers. Then we get: . . . -5, -4, -3, -2, -1, 0, 1, 2, 3, . . ., This list is known as the list of Integers.
On a number line, the integers can be given appropriate places as shown in fig.4.4 below. The arrows at the ends indicate that the line can extend to any distance on either side. Thus it can accommodate up to any large number, be it positive or negative. Also the spacing between the numbers should be uniform. This is a rule which have to be strictly followed while drawing number lines.
 |
| Fig.4.4 The number line |
• Zero can be taken as a 'bench mark'.
• The further left we are from zero, the lesser is our value. Also, as we are on the left of zero, the value will be negative
• The further right we are from zero, the greater is our value. Also, as we are on the right of zero, the value will be positive
Thus, the number line shows the position of each integer in the real world. Take for example -6 and -5. The 'numerical value' of -6 is greater than -5 (because, if we remove the '-' sign, 6 is greater than 5). But when the '-' sign is attached, things are just the opposite. -6 has a lower value than -5.
We can think of it in this way:
There is a greater distance to travel from -6 than from -5 to get to the 'boundary zero' and then to the positive side. So any body would prefer -5. So, as we move further towards the left, the value gets lower and lower.
Based on the above discussion, we can now see some solved examples:
Solved example 4.1
In the fig.4.5 given below, fill up the boxes using ‘<’ OR ‘>’
 |
| Fig.4.5 |
(a) -20 > -35 (b) -9 < 2 (c) 60 > -62 (d) 0 > -1 (e) -70 > -71 (f) -73 < -7
Solved example 4.2
Mark the following points on a number line: (a) -7 (b) -1 (c) +4 (d) +8 (e) -4
Solution:
The fig.4.6 shows the required markings
 |
| Fig.4.6 |
Given below are a few pairs of numbers.
(a) 4,12 (b) -21, -22 (c) 1, -100, (d) -5, -11
In each pair, pick out the number which is to the right of the other on the number line
Solution:
(a) 12. It is to the right of 4
(b) -21. It is to the right of -22
(c) 1. It is to the right of -100
(d) -5. It is to the right of -11
Solved example 4.4
Write all the integers between 4 and -7 in the increasing order of value.
Solution:
-6, -5, -4, -3, -2, -1, 0, 1, 2 and 3
Solved example 4.5
If we move from -20 to -10, will the values of the integers increase or decrease?
Solution:
Values will increase.
So now we have a good understanding about the positive and negative numbers, and their positions in the real world. In the next section we will see how addition between these numbers can be carried out.
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