In the previous section we completed the discussion on fractions. In this section we will learn about decimals.
Decimals are a 'next stage of development' from fractions. Consider two fractions, 1⁄2 and 3⁄4. They are unlike fractions. We use both of them quite often in our day to day life: 1⁄2 is one half, and 3⁄4 is three quarters.
But just by looking at them, we are not able to say which is greater. This is because their denominators are different. We have learned how to solve such problems: Convert each into suitable equivalent fractions, in such a way that they have the same denominators.
We can think of converting each into equivalent fractions with denominator 100. Let us do it:
• 1⁄2 = x⁄100 Taking cross products we get 100 = 2x ∴ x = 50. So the required equivalent fraction = 50⁄100
• 3⁄4 = x⁄100 Taking cross products we get 300 = 4x ∴ x = 75. So the required equivalent fraction = 75⁄100
75 > 50. So we get 3⁄4 > 1⁄2
Decimals are related to fractions. Not just fractions. The denominator must be any one of the following: 10, 100, 1000, 10000 etc., We can say: The denominators must be powers of 10. That is., 101, 102, 103, 104 .. etc., Let us see some examples:
2⁄10, 32⁄100, 893⁄100, 63⁄1000, 42⁄1000, 795⁄1000, 84⁄10000 ... etc., are examples of fractions which are 'related to decimals'. (In day to day life, we mostly encounter decimals which are related upto a maximum of 103 that is., 1000)
But they are still in 'fraction form'. They have an upper numerator, a lower denominator, and a line in between. Then what is the advantage of writing a fraction with denominators as powers of 10? It is true that comparison becomes a little more easier. As we compared 1⁄2 and 3⁄4 above. Apart from that, is there any other advantage? Let us find out:
When the denominator is 10, or 100, or 1000 etc., there is a rule that is commonly accepted. This rule allows us to discard the fraction form, and write every thing on a single line. That is., there will be no more numerator or denominator. Every thing will fall in a line. Let us see this rule:
Take for example 35⁄100.
• Count the number of zeros in the denominator.
• It has 100 as the denominator.
• So the number of zeros is 2.
• For each zero in the denominator, mark off a digit in the numerator from the extreme right.
This is shown in fig.6.1 below:
It is important to start the markings from the extreme right, and proceed towards the left. When the markings are complete, put a '.' sign just before the last digit which is marked off. In our case, when we reach 3, the markings are complete. So put a '.' sign before 3. This is shown in fig.6.2 below:
The '.' sign is called decimal point. So 35/100 has become .35. It is read as 'Point Three Five'
Let us see another example: 789⁄100
• Count the number of zeros in the denominator.
• It has 100 as the denominator.
• So the number of zeros is 2.
• For each zero in the denominator, mark off a digit in the numerator from the extreme right.
This is shown in fig.6.3 below:
When the markings are complete, put a decimal point just before the last digit which is marked off. In our case, when we reach 8, the markings are complete. So put the decimal point before 8. This is shown in fig.6.4 below:
So 789⁄100 has become 7.89. It is read as 'Seven Point Eight Nine'
Note that in this case we have 789, which is a 3 digit number, in the numerator. But there are only two zeros in the denominator. So while marking from the extreme right, 7 was left out. In other words, there is a digit in front of the decimal point. Compare this with the previous .35. There are no digits in front of the decimal point. In such cases, we put a zero there. Thus we write 0.35 (read as 'Zero Point Three Five') rather than just .35
Another situation that we might encounter is:
There are not enough digits in the numerator to be marked off. For example consider 7⁄100. We must mark off 2 digits, because there are 2 zeros in the denominator. But there is only one digit in the numerator. In such cases, we put extra zeros to compensate. Thus 7⁄100 becomes 0.07
A few examples based on the above discussion is given below:
(i) 39⁄10 = 3.9 (ii) 78⁄100 = 0.78 (iii) 419⁄1000 = 0.419 (iv) 293⁄100 = 2.93 (v) 36⁄1000 = 0.036 (vi) 4⁄100 = 0.04 (vii) 783⁄10 = 78.3 (viii) 29⁄10000 = 0.0029
So now we know how to write in decimal form when:
Decimals are a 'next stage of development' from fractions. Consider two fractions, 1⁄2 and 3⁄4. They are unlike fractions. We use both of them quite often in our day to day life: 1⁄2 is one half, and 3⁄4 is three quarters.
But just by looking at them, we are not able to say which is greater. This is because their denominators are different. We have learned how to solve such problems: Convert each into suitable equivalent fractions, in such a way that they have the same denominators.
We can think of converting each into equivalent fractions with denominator 100. Let us do it:
• 1⁄2 = x⁄100 Taking cross products we get 100 = 2x ∴ x = 50. So the required equivalent fraction = 50⁄100
• 3⁄4 = x⁄100 Taking cross products we get 300 = 4x ∴ x = 75. So the required equivalent fraction = 75⁄100
75 > 50. So we get 3⁄4 > 1⁄2
Decimals are related to fractions. Not just fractions. The denominator must be any one of the following: 10, 100, 1000, 10000 etc., We can say: The denominators must be powers of 10. That is., 101, 102, 103, 104 .. etc., Let us see some examples:
2⁄10, 32⁄100, 893⁄100, 63⁄1000, 42⁄1000, 795⁄1000, 84⁄10000 ... etc., are examples of fractions which are 'related to decimals'. (In day to day life, we mostly encounter decimals which are related upto a maximum of 103 that is., 1000)
But they are still in 'fraction form'. They have an upper numerator, a lower denominator, and a line in between. Then what is the advantage of writing a fraction with denominators as powers of 10? It is true that comparison becomes a little more easier. As we compared 1⁄2 and 3⁄4 above. Apart from that, is there any other advantage? Let us find out:
When the denominator is 10, or 100, or 1000 etc., there is a rule that is commonly accepted. This rule allows us to discard the fraction form, and write every thing on a single line. That is., there will be no more numerator or denominator. Every thing will fall in a line. Let us see this rule:
Take for example 35⁄100.
• Count the number of zeros in the denominator.
• It has 100 as the denominator.
• So the number of zeros is 2.
• For each zero in the denominator, mark off a digit in the numerator from the extreme right.
This is shown in fig.6.1 below:
 |
| Fig.6.1 |
 |
| Fig.6.2 |
Let us see another example: 789⁄100
• Count the number of zeros in the denominator.
• It has 100 as the denominator.
• So the number of zeros is 2.
• For each zero in the denominator, mark off a digit in the numerator from the extreme right.
This is shown in fig.6.3 below:
 |
| Fig.6.3 |
 |
| Fig.6.4 |
Note that in this case we have 789, which is a 3 digit number, in the numerator. But there are only two zeros in the denominator. So while marking from the extreme right, 7 was left out. In other words, there is a digit in front of the decimal point. Compare this with the previous .35. There are no digits in front of the decimal point. In such cases, we put a zero there. Thus we write 0.35 (read as 'Zero Point Three Five') rather than just .35
Another situation that we might encounter is:
There are not enough digits in the numerator to be marked off. For example consider 7⁄100. We must mark off 2 digits, because there are 2 zeros in the denominator. But there is only one digit in the numerator. In such cases, we put extra zeros to compensate. Thus 7⁄100 becomes 0.07
A few examples based on the above discussion is given below:
(i) 39⁄10 = 3.9 (ii) 78⁄100 = 0.78 (iii) 419⁄1000 = 0.419 (iv) 293⁄100 = 2.93 (v) 36⁄1000 = 0.036 (vi) 4⁄100 = 0.04 (vii) 783⁄10 = 78.3 (viii) 29⁄10000 = 0.0029
So now we know how to write in decimal form when:
A fraction with a denominator as powers of 10 is given.
Obviously, we must be able to do the reverse. That is., if we are given a decimal, we must be able to convert it into a fraction form, with denominator as powers of 10. Let us see an example:
Consider 0.26. We want it in the 'fraction form'. So we want the numerator and the denominator. The numerator will obviously be 26. And we know that the denominator will be a power of 10. All that we want to know is which power? Is it the 1st power 10, 2nd power 100, 3rd power 1000, or any other? Let us analyse:
• If it was 10, the decimal point would have fallen just before 6, which is the 1st digit from extreme right
• If it was 100, the decimal point would have fallen just before 2, which is the 2nd digit from extreme right. But this is exactly what we are having. The decimal point is indeed before 2. So it is 100 that we should write in the denominator.
Thus we have: 0.26 = 26⁄100. Note that:
• In the decimal form: There are two 2 digits after the decimal point
• In the corresponding fraction form, there are 2 zeros in the denominator
We can write 26⁄100 in the lowest form so that we can work with smaller numbers:
26⁄100 = 13⁄50
Another example:
Consider 0.704. We want it in the 'fraction form'. So we want the numerator and the denominator. The numerator will obviously be 704. And we know that the denominator will be a power of 10. All that we want to know is which power? Is it the 1st power 10, 2nd power 100, 3rd power 1000, or any other? Let us analyse:
• If it was 10, the decimal point would have fallen just before 4, which is the 1st digit from extreme right
• If it was 100, the decimal point would have fallen just before 0, which is the 2nd digit from extreme right
• If it was 1000, the decimal point would have fallen just before 7, which is the 3rd digit from extreme right. But this is exactly what we are having. The decimal point is indeed before 7. So it is 1000 that we should write in the denominator.
Thus we have: 0.704 = 704⁄1000. Note that:
• In the decimal form: There are two 3 digits after the decimal point
• In the corresponding fraction form, there are 3 zeros in the denominator
We can write 704⁄1000 in the lowest form so that we can work with smaller numbers:
704⁄1000 = 352⁄500 = 176⁄250 = 88⁄125
So we can write a general rule to convert any decimal to fraction form:
• Numerator of the fraction will be same as the decimal number without the 'decimal point'
• Denominator will have as many zeros (after '1') as there are number of digits after the decimal point'
(i) 0.2 = 2⁄10 = 1⁄5 (ii) 0.55 = 55⁄100 = 11⁄20 (iii) 0.893 = 893⁄1000 (iv) 0.05 = 05⁄100 = 5⁄100 = 1⁄20 (v) 0.079 = 079⁄1000 = 79⁄1000
We know that 36⁄100 = 30⁄100 + 6⁄100
But 30⁄100 can be simplified further (dividing both numerator and denominator by 10) in such a way that, even after simplification, the denominator is a power of 10. That is 30⁄100 = 3⁄10. Here, even after simplification, the denominator is a power (the 1st power) of 10
• So we can write 36⁄100 = 3⁄10 + 6⁄100
• 3⁄10 = 0.3 and 6⁄100 = 0.06.
• So 3⁄10 + 6⁄100 = 0.3 + 0.06. This addition can be done as usual. It is shown in fig.6.5 below:
 |
| Fig.6.5 |
0.36 = 36⁄100 = │30⁄100 + 6⁄100│ = │3⁄10 + 6⁄100│ = │0.3 + 0.06│ = 0.36
By going through the above cycle, we notice one thing:
• 3, which is at the 1st position after the decimal point, comes from a fraction, whose denominator is 101
• 6, which is at the 2nd position after the decimal point, comes from a fraction, whose denominator is 102
Proceeding like this we can write:
• The 3rd digit after the decimal point, comes from a fraction, whose denominator is 103
• The 4th digit after the decimal point, comes from a fraction, whose denominator is 104
And so on..
In the next section, we will see a pictorial representation of the above facts
No comments:
Post a Comment