 In the last video, we saw how we can determine whether the decimal form of a given rational number will it be terminating or non-terminating and recurring. For example, in this case, here we can see that in the simplest form of this rational number, the denominator can be factorized into 2s and 5s only. Hence, the decimal form of this rational number would be terminating. While in this case, we have an extra factor other than 2s and 5s, which would make the decimal form of this rational number non-terminating and recurring. In this video, let's actually see why do we need only 2s and 5s and nothing else in order to make the decimal form terminating. So let's start with couple of examples of the first case of terminating decimals. Let's say that we have, let's say 2.4 or maybe 9.86 or let's say that we have 0.127. So let's first try to convert these terminating decimals into their rational forms and then let's see what's so special about these rational numbers. So let's start with 2.4. If we need to convert this into its rational form, we need an integer in the numerator. 2.4 I can write this as 2.4 divided by 1. Now in order to get an integer in the numerator, I need to multiply this with 10 since we just have only one digit after the decimal. And since we don't want the value of this number to change, we need to do the same thing in the denominator. So I would multiply the denominator with 10 as well. 2.4 times 10 is equal to 24. And in the denominator, we would have 10. Similarly, if we would like to convert 9.86 into its rational form, we can write this as 9.86 divided by 1. And now on multiplying both numerator and denominator with 100, since we have two digits after the decimal value, this would be written as 986 divided by 100. Similarly, for 0.127, we can write this as 127 divided by 1000 after writing 0.127 as 0.127 divided by 1. And then on multiplying both numerator and denominator with 1000. So basically, we can convert any terminating decimal into its rational form by multiplying the numerator and denominator by 10 or powers of 10, depending on the number of digits after the decimal value. And this would always give us a 10 or a power of 10 in the denominator of its rational form. So the key takeaway of this entire discussion till now is that the rational form of any terminating decimal would always have a 10 or a power of 10 in its denominator. You can definitely convert this into its simplest form, but at least one of the forms of its rational number would always have a 10 or a power of 10 in the denominator. Now, how is that related to having 2s and 5s in the denominator? Well, let's see. Let's try to actually see what this denominator is made up of. For example, 10. 10 we can prime factorize as 2 times 5. What about 100? 100 is nothing but 10 square, which can be again prime factorized as 2 square times 5 square. 1000 is 10 cube. So this can be prime factorized as 2 cube times 5 cube. So having equal numbers of 2s and 5s in the denominator always gives us a 10 or a power of 10 in the denominator, which means that the decimal form of this rational number would be terminating. Now you might ask, what about unequal numbers of 2s and 5s in the denominator or maybe having some other factor other than 2s and 5s? Would the rational number still be terminating? Well, as we just saw, the key idea here is to get a 10 or a power of 10 in the denominator. If we can do that, we can definitely see that it's a rational form would be terminating. And if not, then it won't be terminating. It would be non-terminating and recurring. Let's see this through some examples. So let me erase this first. Let's say that the simplest form of a rational number is in the numerator we have 3 divided by in the denominator, let's say that we have 2 cube times 5 to the power 4. Now, is there a way to get a 10 or a power of 10 in the denominator? Let's see. As we just saw that we need equal numbers of 2s and 5s to get a 10 or a power of 10. Here we have 3 2s and 4 5s. What we can do here is that in order to get equal numbers of 2s and 5s, we can multiply both numerator and denominator with an extra 2. How would that make a difference? This would give us 3 times 2 would be equal to 6 divided by 2 cube times 2 would give us 2 to the power of 4. And we already have 4 5s. So this would be equal to 5 to the power 4. Now, this we can write it as 10 to the power 4, this denominator. So finally, this rational number 3 upon 2 to the power 3 times 5 to the power 4, we changed this into 6 divided by 6 divided by 10 to the power 4, which is equal to 4 0s after 1, which would finally give us 0.006, which is indeed a terminating decimal. So we can say that even if we have unequal numbers of 2s and 5s, we can always get equal numbers of 2s and 5s in the denominator by multiplying with 2 or 5, whatever we require. Similarly, let's say that we had 7 upon 2 to the power 3 times 5 to the power 1. Now we know that in order to get a 10 or a power of 10 in the denominator, we need equal numbers of 2s and 5s. And how we can get that by multiplying both numerator and denominator by two more 5s, that is 5 square and 5 square in the denominator. So this would give us 7 times 5 square and that would still be an integer divided by 2 to the power 3 times 5 to the power 3. Now this denominator would be equal to 10 to the power 3, which would make this rational number terminating. Now let's see the last case when we have any other factor other than 2s and 5s. So let me erase this first. Okay, let's say that we have 9 divided by 2 to the power 3 times 5 to the power 6 times 3 square. Now at the first glance, we would say that we have an extra factor other than 2s and 5s. So the decimal expansion of this rational number should be non-terminating and recurring. But as we discussed previously in this video and even in the last video that we always look for the simplest form of the rational number. Because like in this case, if we divide both numerator and denominator by 9, this rational number would change into 1 upon 2 to the power 3 times 5 to the power 6. And now in this form, we only have 2s and 5s in the denominator, which we can convert into powers of 10. And hence this would be terminating in its decimal form. Now there might be a possibility that on dividing both numerator and denominator by a same common number, the extra prime factor could go away. That's why we always look for its simplest form. But let's say even in the simplest form, even in the simplest form, we had an extra factor. For example, let's say that instead of 9, we had 1. Now let's first focus on 2s and 5s. We have 3 2s and 6 5s. Now if we want equal numbers of 2s and 5s, we can multiply the numerator and denominator by same quantity, that is 3 more 2s, 2 to the power 3 and 2 to the power 3, which would finally give us 2 to the power 3, that is 8 in the numerator, this 2 cube and this 2 cube would combine and give us 2 to the power 6 times 5 to the power 6 times 3 square. Now this 2 to the power 6 and 5 to the power 6 would combine and this would give us 2 times 5 is 10. So we would have 10 to the power 6. So this rational number would finally change into 8 divided by 10 to the power 6, 10 to the power 6 times 3 square. Now there is no integer on earth that we can multiply with this 3 square to get a power of 10. And if we don't have a power of 10 in the denominator, we can never change this rational number into terminating decimal expansion. And if it's not a terminating decimal expansion, the only other possibility is that its decimal expansion would be non-terminating and recurring or repeating. For example in this case 10 to the power 6 times 3 square, 10 to the power 6 times 3 square would give us 9 followed by 6 zeros, which is not a power of 10. We cannot write this into the form of 10 square, 10 cube, 10 to the power 4, 10 to the power 5, etc. So that is why its decimal form would not be terminating. And this is not only true for 3. Any other prime number other than 2s and 5s, may be 7, may be 11, 19, 23, 13, etc. would never ever give us a power of 10 in the denominator, which would never ever make the decimal form of the rational number terminating. And hence if we have any other factor other than 2 and 5, the decimal form of the rational number would be non-terminating and recurring. So now I hope this idea of having only 2s and 5s in the rational form of terminating decimal expansion is clear now.