Previously, we saw that a couple of thoughts were puzzling Nora. And we mentioned that Calculus was the key to answer them. She wanted to know that when we drop a ball, how we can find its speed at an INSTANT… And while thinking about this, she had another strange thought. She was convinced that mathematically the ball should NEVER reach the floor. This thought has puzzled many people for centuries and it is called the Zeno’s Dichotomy paradox. What are your thoughts about these problems? There were 2 more interesting problems we had discussed. Let’s say we are standing on a cliff. Then what is the best way to throw a stone such that it covers maximum distance? And the other problem was how we can find the area of any shape, especially whose boundary is made up of curved lines. We know that Calculus gives us the answers to all these problems! The question is ‘How?’ First, we will focus our attention on these three problems. And for this problem, the one for which we need to find the angle at which to throw the stone, we will have to wait till we gain some additional knowledge of calculus. Let us move on to a new page and start with this interesting thought puzzling Nora. She thinks that mathematically, the ball should never reach the floor. As the ball falls, it has to first cover half of the distance between its starting position and the floor. Then it has to cover half of the remaining distance. Then it has to cover the next half, and then the next half and so on. So the number of steps the ball has to perform never ends. Another way of saying it is that there are INFINITE number of steps the ball has to perform. Beware of the term Infinite. Always remember that Infinite is not a number. Infinite just means something which NEVER ends, or something that is limitless. Now to perform each step, the ball takes some time. Since the number of steps the ball has to cover never ends that is there are INFINTE number of steps, so Nora thinks the ball should never reach the floor. But, we know that as we drop the ball, it reaches the floor in a few seconds. So where is the mistake in Nora’s logic? In order to know the time the ball takes to reach the floor, we have to add all the ‘individual time periods’ taken to perform the individual steps. But to find this, we will have to add an infinite number of terms. Let’s say we take our calculator out and start adding these numbers. Then our addition will go on and on, and will never finish. And as we go on adding the number, the sum will get bigger and bigger. That is, as the ball continues to cover the successive steps; the TOTAL time taken by the ball will keep on increasing. Another way to say this is the ball will take an infinite amount of time to reach the floor. But we know that this isn’t true as its common knowledge that the ball will reach the floor in a few seconds. It will be a well-defined number! That means if we add all the time periods taken to perform the individual steps, we should get a number as our answer. Yes, even though we can’t mechanically add the numbers, we can show that the sum of these infinite numbers is equal to a number . To show this, we carry out a process known as TAKING THE LIMIT. And this is the central idea that Calculus uses to solve the problems we mentioned. To understand calculus, we need to understand the concept of Limits first, which we will cover in the upcoming parts! To understand this paradox, let’s keep things simple. We know that as the ball is falling, its speed increases. But for simplicity, let’s assume the speed of the ball to be constant. Of course we will get the answer even if we take the ball’s speed to be increasing, but it will be a little bit complicated. So, let’s say the ball falls with a CONSTANT speed of half a meter per second. In reality the ball’s speed will be more than this, but we want to keep it simple. With this speed, we know that the ball will take just ‘two’ seconds to cover the distance of one meter and reach the floor. So let’s see how much time the ball takes to cover these individual steps. The time required to cover the first half of a meter will be one second. Then to cover the next one fourth of the meter, it will take half a second. Then to cover the next one eighth of the meter, it will be one fourth of a second and so on. So did you observed an interesting thing? After some steps, the time taken to complete the subsequent steps will be very very small. So as the ball completes the successive steps, the time taken by it to perform the next step will get closer and closer to ZERO. This is an important concept. It is getting CLOSER to a definite number! But remember it will not be zero… Each step requires some time, however negligible it may be. We know that the ball takes 2 seconds to reach the floor. Now we can show that if we add all these values of time taken by the ball to perform each step, the sum will NEVER EXCEED two and will actually be equal to two. Before we prove this, let me give you another example. Consider this square. Let’s say the length of each of its sides is one meter. Then we know that its area will be one square meter. Now let’s divide or split this area as we did with the motion of the ball. First we divide the area into two halves. Then we take one of the two halves and further divide it into two halves. So what we get is two quarters of the square. Now we take one of the quarters and again divide it into two halves. And we keep repeating this process. What is happening here? We will realise that this is also a never ending process. But even though it is a never ending process, we know that the area of this square, which is one square meter, will be equal to the sum of the areas of these parts. So we see that the sum of these INFINITE terms is equal to one, which is a definite number. We can’t manually add all these terms, but their sum turns out to be one. As we go on dividing this square, the area of its parts get smaller and smaller. They get so small that their area is very very close to zero, but not zero. But because these parts as a whole make up the square, the sum of their areas cannot be greater than one. So if we keep on adding the areas of these parts, their sum will APPROACH one. The word ‘approach’ is very important here. Anyway, I am not sure if you realised… but you almost have an answer for the sum of the time periods taken by the ball to reach the floor. Look at the expression we had for the time taken by the ball to reach the floor. And look at this expression we have for the parts of a square. Now if we add one to this set of numbers, it becomes the same as this expression. So if we add 1 to both sides here, we get the sum of these values as 2. Let me recap! On the right hand side of this equality, we have these numbers which are same as the time taken by the ball to cover each steps. And we get the addition of these infinite terms to be two , which is exactly the time taken by the ball to reach the floor. So , Nora understood where she went wrong with her logic! She assumed that the total time the ball will take to cover these infinite steps will be infinite. Remember what we mean by infinite here. It means something that never ends. Let’s look at a quick demonstration so that you understand better. This is the expression we found for the total time taken by the ball to reach the floor right? Now let us try to find the cumulative time taken by the ball to reach the floor, for each step! For example, after 2 steps, the time taken by the ball will be ‘1 second plus half a second’, and that is equal to one point five seconds. Similarly, after 3 steps, the total time taken by the ball will be one point seven five seconds. If we continue this for other steps, we see that the total time taken will get closer and closer to two seconds. As the ball covers the successive steps, the time taken by it to cover each step is very very small. So even if we continue adding many time periods, the total time will never exceed two seconds. The total time taken will just get closer and closer to two seconds. This is an example to understand the idea of the Limit process . We can’t find the answer to the sum of infinite terms of numbers. But if we keep on adding successive steps we see that the sum approaches a number. With this, Nora is not confused anymore! We see that if we divide the motion of the ball into an infinite number of steps, it seems that it will take infinite time to cover these steps. But we can show that as we add the time periods taken in each step, the total time taken by the ball gets closer and closer to a definite number. It’s similar for the distance the ball covered too. We had seen how the distance was split into infinite parts. It doesn’t mean the distance is infinite. The sum of these distances will APPROACH 1 meter. And guess what, through this limit process we can also find the instantaneous speed of the ball. If you remember this was another question that Nora had. We will discuss it in the next part. See you there.