Week 5: Zeno of Elea: The crazy world of infinities, infinitesimals and limits – Birth of Calculus

Some infinities are bigger than other infinites
John Green

This week we take a detour from geometry. I was telling you the other kind of geometry that is non-Euclidean and has the feature that given a line and a point not in it, there are many lines passing through the point and parallel to the given line. But that geometry is not quite natural to develop as was the geometry of a sphere. It will require some tools and some familiarity with coordinates and numbers and some calculus. As calculus is an unavoidable toolkit in a physicist’s toolbox, it is better confronted sooner than later. Also historically, the new geometry was invented after the invention of calculus. So this week shall be purely about numbers and functions.

Once again the story starts in ancient Greece! This time it is not with Euclid but another philosopher called Zeno. He put forth several paradoxes (not really paradoxes) which can be explained only when one has familiarity with the infinite.

First paradox of Zeno – Achilles and the tortoise:

The number line

The first paradox concerns this: Imagine that Achilles is initially at x=0 of the number line shown below and travels at a uniform speed of 1 metres every 1 second. The tortoise on the other hand is initially at x=1 in the number line (it gets a headstart) but has a speed of only half a metres every one second. Now any child will know that even though the tortoise gets a headstart, since it is running slower than Achilles, Achilles eventually will overtake the tortoise in some time

Let A(t) denote the position of Achilles at time ‘t’. Since Achilles travels at 1 metres every 1 second, we have $\frac{A(t)-A(0)}{t-0}=1$ . But since Achilles starts at 0 initially, A(0)=0. So Achilles motion is given by $a(t)=t$ . On the other hand, let the tortoise position be given by T(t). Tortoise speed is half and initially starts at 1. So we have $\frac{T(t)-T(0)}{t-0}=\frac{1}{2}$ and putting T(0)=1, we have $T(t)=1+0.5t$ . Now, we see clearly that for t<2,

$t<2 \Longleftrightarrow 0.5t<1 \Longleftrightarrow t - 0.5 t < 1 \Longleftrightarrow t < 1 + 0.5t \Longleftrightarrow A(t) < T(t)$

And similarly,

$t>2 \Longleftrightarrow 0.5t>1 \Longleftrightarrow t - 0.5 t > 1 \Longleftrightarrow t > 1 + 0.5t \Longleftrightarrow A(t) > T(t)$

So, we have that for the first two seconds (t<2), Achilles is behind the tortoise (A(t)<T(t)), and after two seconds, Achilles beats the tortoise and keeps moving ahead. i.e. A(t)>T(t). This is common sense. A fast person will eventually beat a slow person even if the slow person has an initial headstart. It is only a matter of two seconds or whatever depending on the exact speeds and the initial headstart.

But Zeno argues that this will not happen and it is called the Zeno’s paradox.

He argues as follows:

1. Initially, the tortoise is farther – T(0)=1, A(0)=1. So it will take some time $t_1$ for Achilles to reach where the tortoise was initially.

2. But in this time $t_1$ , the tortoise even though however slow, would have moved ahead by some distance apart, to some other point – $T(t_1)$ .

3. Achilles now should has to reach $T(t_1)$ now from the old position, $T(0)$ . This will take some time $t_2$ .

4. But again in this time $t_2$ , the tortoise would have moved some smaller distance no matter however slow, to a new position $T(t_1+t_2)$ .

5. Now Achilles has to start afresh. From the current position $T(t_1)$ he has to reach this new position $T(t_1+t_2)$ and this will take him some non-zero time $t_3$ no matter however low.

6. And so on…….

This continues forever

So Zeno argues that Achilles can never reach the tortoise as he has to keep on chasing the tortoise “infinite” times where the tortoise is going. Achilles will take $t_1+t_2+t_3+t_4+.....$ seconds to reach the tortoise and each of the $t_i$ are positive and nonzero, however small. So Zeno argues that it is not possible that Achilles can catch up with the tortoise in two seconds, because Achilles will have to keep following the tortoise. So, he argues that logically, the possibility of Achilles overtaking the tortoise is impossible.

But we know there is something fishy in Zeno’s argument as we intuitively know and also from the equation of motion for A(t) and T(t) that within two seconds, Achilles should overtake the tortoise. It looks like if Achilles has to think about covering the distance that the tortoise traverses, it looks like it will take forever as he has to keep on covering the tortoise’s old positions again and again. But looks like if he does not care about it, things are normal.

Second paradox of Zeno – motion is impossible logically and is an illusion!

According to Zeno, by similar arguments, he argues that motion is impossible logically and it is hence an illusion. He argues that it is impossible to move from a point A to another point B in a line (say).

1. Then first I have to reach the point that is halfway between A and B, say A1. It will take time t1. (non zero time – as I cannot be in two different places at once)

2. Having reached A1, I now have to cross the point halfway between A1 and B – say A2. Again I will take some non zero time t2 to reach from A1 to A2.

3. Having reached A2, I now have to reach the point that is half way between A2 and B and I will take some time t3 to reach that.

4. And on and on. So I have to keep chasing infinite number of half way points A1,A2,… in times t1,t2,…. and hence I can never reach from A to B.

If you continue traveling half the distance that is left, you’ll never get to point B because you can always slice the remaining distance in half again

So according to Zeno, motion is an illusion and I can never reach from any point A to any other point B.

But everyday experience shows that I can reach from A to B in finite time. If B is 1 metres to the right of A, then travelling at a constant speed of 1 metres every one second, I can reach B in 1 second.

What is wrong in these flow of arguments by Zeno although they seem logically solid!

This is where the delicate issue of limits come into picture. Normally, the concept of limits is introduced only in high school that too in a highly unmotivated manner but with these examples, we can introduce it to even small children who would be more curious about these things.

Let us now prove deeper into this. First let us take the first problem. Achilles and the tortoise. The first logic that involved calculating A(t) and T(t) (where we did not think about Achilles covering the tortoise’s old way points), it was found that Achilles would beat the tortoise in exactly two seconds.

Let us now calculate the explicit time involved:

Initially, Achilles is at 0 and tortoise at 1. Since Achilles is moving 1 units of distance every 1 second, it will take 1 second for Achilles to cover from x=0 to x=1

So, t1=1

Now, in this this 1 second, the tortoise, moving at 0.5 units per second, would have covered a distance 0.5. So it will be at T(1)=1+0.5 (1: initial point, 0.5: additional distance covered from there)

Now, Achilles has to move from his present position (x=1) to tortoise’s new position (x=1.5)

So, he has to cover 0.5 units. Since he travels at unit speed, it would take 0.5 sec.

So, t2=0.5

Now in this 0.5 seconds, the tortoise would have covered 0.25 metres as it is going half metres every second.

Now to cover this 0.25 metres , Achilles would take t3=0.25

Repeating this, we get $t_n=(\frac{1}{2})^{n-1}$

Achilles, to cover n waypoints of the tortoise, will take time $t_1+t_2+....+t_n$ seconds which is

$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...(\frac{1}{2})^{n-1}$ seconds

I think if you had studied high school, you might have seen this somewhere! It is called the geometric series.

If you struggle with it, a little bit, you can evaluate this sum $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...(\frac{1}{2})^{n-1}$ .

Let $S_n= 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...(\frac{1}{2})^{n-1}$ . Then

$\frac{1}{2}S_n=(1-\frac{1}{2})S_n=S_n-\frac{1}{2}S_n=(1+0.5+0.25+...+\frac{1}{2}^{n-1})$

$-(0.5+0.25+0.125+...+\frac{1}{2}(\frac{1}{2})^{n-2}+\frac{1}{2}(\frac{1}{2})^{n-1})$

We now see that the terms cancel nicely and we are left with

$\frac{1}{2}S_n=1-(\frac{1}{2})^n$

So, $S_n=2(1-[\frac{1}{2}]^n))=2-(\frac{1}{2}^{n-1})$

So, it will take $2-(0.5)^{n-1}$ seconds for Achilless to overtake the tortoise $n$ times. Now let us put a table of how this varies with $n$ .

Plot:

Now here comes the cool part:

Although Achilles has to chase the tortoise infinite times, looks like he will cove rall these infinite steps in just two seconds (see the graph or data above)

As $n$ becomes very large, $S_n$ becomes closer and closer to 2. Although it never does become $2$ , we observe that it keeps getting closer and closer to 2.

Another way to convince yourself that the “infinite sum” $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...=2$ , I give a diagrammatic proof below:

We can see that we can divide a unit square into infinitely many pieces, each of area $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$ and we realize that although we can never reach the full unit square by pasting any finite number of those pieces all together, we can get how much ever closer we want to $1$ because the residual area $\frac{1}{2}^{n}$ goes arbitrarily small as $n$ goes arbitrarily large.

We see that although we are pasting together infinite number of triangles, the area resulting in it can still be finite.

What Zeno has done is divide the 2 seconds gap where Achilles beats the tortoise into infinitely many gaps where in, at each gap, he chases the previous position of the tortoise.

Dividing Achilles’ 2 seconds into infinite segments of journeys

SO a resolution of Zeno’s paradox lies in the fact that a finite part of length/time/real number can be broken into infinite pieces of segments and eventhough we do not get the original piece back by merging how much ever finitely large number of pieces back, we somehow get the original segment back when we merge all the infinite pieces!!

**Dividing two seconds of Achilles’ beating** **time into infinite segments of way points**

The resolution of Zeno‘s second paradox is also exactly similar. We have divided the segment of the journey between A and B into infinite number of way points by continuous halving of the segments. So we cannot expect to get the original segment back by merging/adding finite number of steps. Following this we are tempted to believe that “we can add infinitely many numbers” .

But it is at this stage, we must be careful. We can repeat the same paradox with Achilles running slower than the tortoise and this time, truly, Achilles cannot catch up with him. So when we add the infinite segments together, we cannot expect to get a finite answer.

Example: If $t_1+t_2+t_3+t_4+...t_n+...=1+2+4+8+16+....2^{n-1}+...$ then, we keep getting ever larger and larger numbers as we keep adding more and more terms (this is what happens when the tortoise is twice as fast as Achilles) and hence in this case adding infinite sums won’t make sense and is not expected to make sense as in this time, Achilles can never catch up with tortoise in any finite time.

Now, based on this, we give the following definition:

Definition 1: A sequence of real numbers $a_1,a_2,...a_n,....$ is said to converge to a number “a”, if given any tolerance $\epsilon>0$ , there exists a large enough number N such that for all $n\geq N$ , we have that $a_n \in (a-\epsilon,a+\epsilon)$

where $a_n \in (a-\epsilon,a+\epsilon)$ means that $a-\epsilon<a_n<a+\epsilon$

**Given any degree of closeness $\epsilon$ to a, after a certain stage, the sequence gets that close to a**

It captures intuitively that even though the sequence may not become the number $a$ after any large number of steps, we have that, after a large enough number of steps, the elements of the sequence comes very close to the number $a$ , and given any degree of closeness we demand from the function, by choosing $\epsilon>0$ very small, it is just a matter of large enough $N$ after which the elements of the sequence all come and fall into that $\epsilon$ neighbourhood of $a$ .

So, we have that the sequence $t_1+t_2+.....t_n$ converges to 2 seconds. When a sequence of sums $a_1+a_2+...+a_n$ converges to a number $a$ , we say that the infinite sum of $a_1,a_2,...$ is $a$ . The fact that a sequence converges to $a$ is also stated as: The limit of the sequence is $a$ Using this concept of limits, we can get how much ever close to a number $a$ we want, by using a sequence that converges to $a$ , without actually getting to $a$ . A joke to lighten the mood below!

**The mathematician might be called as Zeno who thinks that nothing is gonna ever happen between him and the girl!**

So the moral so far is:

THE BIRTH OF CALCULUS:

PIC: Founding fathers of calculus – Newton and Leibniz

This spirit of limits is what gave birth to calculus. Newton was watching the motion of a cart in a road and he observed the behavior of the position of the cart at each instant of time. Assuming the road to be a line and making it into a numbered line, we can capture its movement by having a table of position vs time as follows:

In other words, motion in a line can be specified by a function from a number “time” to another number “position”, denoted by $x(t)$

In other words, for every time $t$ , we need to keep track of $x$, that depends on $t$ and hence the notation $x(t)$

Now, Newton asked, what is the speed of an object moving arbitrarily in a line as $x(t)$ . Of course if it were moving with a constant velocity such that it covers equal intervals of length in equal intervals of time, then we define speed as a single number $\frac{x(t_2)-x(t_1)}{t_2-t_1}$ and this number will be independent of $t_1,t_2$ chosen (as long as $t_1\neq t_2$ ) since we are given that it always covers equal intervals of distance in equal intervals of time.

Now Newton asked, what if the object is moving in a completely arbitrary way such that the speed is not constant in any segment of the journey? Lets say the object travels in a line according to the relation $x(t)=t^2$ . Definitely, the speed depends on $t_1,t_2$ chosen. So one is tempted to then say in this case that “the speed varies from point to point“. But what is meant by the word “speed at a point – or speed at an instant – say speed at time t=3sec?

To define speed at say time ‘t’, we run into a paradox. To define speed, we need two distinct instants of time ( $t_1\neq t_2$ ) to calculate change in position with respect to change in time.

But we have only one instant given to us, which is time t.

We can take one of the instants (say $t_1$ ) as t. But the other instant has to be different from t. $t_2$ has to be $t_2=t+\Delta t$ where $\Delta t \neq 0$

So, we run into a conundrum. We want to make $\Delta t=0$ so that we do not use any other instant to calculate speed. But we cannot make $\Delta t=0$ .

But if we take $t_2=t+\Delta t$ for any non zero $\Delta t$ , Newton objects by saying that you are not calculating speed at time “t” but the average speed between $t, t+\Delta t$

Now the below cartoon signifies the thought process that led to the birth of calculus – or notion of speed at an instant of time , even though we need two instants of time to talk about speed!

**$\Delta t$ can never be made** **to 0 but can be made to converge to 0!!**

Now the cunning mind of Newton thought: I cant make $\Delta t$ zero but I can make it as close to 0 a I want by choosing a sequence converging to 0.

Now as we choose a sequence of $\Delta t$ s that converges to 0, the corresponding speeds calculated, if they converge to some number, independent of the sequence chosen, then that limit is very important.

Such a limit if exists, is called instantaneous speed at time ‘t’.

Instantaneous speed is defined as:

The limit of the sequence of the average speeds around $t$ as they become smaller and smaller…. (if exists) is called the instantaneous speed $v(t)$

The limiting speed that is calculated, called the instantaneous speed at time t is not the speed between time t and any other instant. Because for every imaginable separation $\delta t$ from $t$ , the sequence $\Delta t$ continued from then on and got closer to $t$ than $\delta t$ .

The sequence gets closer to ‘t’ than any other point that we choose, eventually and hence is actually a state of affairs at ‘t’ indeed – not between t and t+ $t+\delta t$ for any $\delta t$

Such a limit, if exists, independent of the particular sequence of $\Delta t$ chosen, is called the instantaneous speed $v(t)$ of the particle at $t$ . Note that it is the speed at $t$ as we have that it goes to zero in the limit, even though it actually does not go to zero!

It is not the speed between t and any other point as we eventually will make $t+\Delta t$ closer to $t$ than any given point. Eventually, it is just a matter of time.

Another perspective to convince you that it is the speed at an instant really, what we have essentially done is this:

I found the speed $\frac{x(t+\Delta t) -x(t)}{\Delta t}$ . But some one complained that it is not the speed at “t” but an average state of affairs between “t, t+ $\Delta t$ “. So let’s say I halve the time interval. I now calculate this average speed between t and $0.5\Delta t$ . But again someone else complained that it also isn’t the instantaneous speed. But an average state of affairs between “t, t+0.5 $\Delta t$ “. So I was sad. To satisfy that person I again halved my interval into $0.25\Delta t$ . But even then someone else complained. So I calculated for $0.125\Delta t$ . I kept on doing this. I realized that I was just wasting my time satisfying the whims and fancies of every person. I realized that how much ever I reduced my interval $\Delta t$ , someone else is gonna ask me to reduce even further. So I realised that at no stage I can satisfy anybody. But I prayed to Zeno and he made me realize that even though at no finite stage can I satisfy anyone, if I keep on evaluating the speeds for lower and lower $\Delta t$ s, then if these approach a limiting value, then in that limit I satisfied anyone. Because that limiting value of $0.5\Delta t$ is zero. I realized that by reaching out halfway to zero everytime, even though I never reached zero actually, I reached zero in the limit. So the instantaneous speed is indeed evaluated by putting $0.5\Delta t$ =0 but as I was not allowed to make it exactly zero in finitely many iterations, I followed Zeno and reached 0 in the limit by going halfway to zero everytime or by choosing any sequence that converges to zero!

Let us evaluate this for the function $x(t)=t^2$

$x(t+\Delta t) - x(t)=(t+\Delta t)^2-t^2=2t\Delta t+ \Delta t^2$

Hence $\frac{x(t+\Delta t)-x(t)}{\Delta t}= 2t + \Delta t$

Now, since $\Delta t \rightarrow 0$ , putting $\Delta t=0$ above, we get

$v(t)=2t$

This says that the instantaneous speed increases linearly with time. Another notation for $v(t)$ is $\frac{dx}{dt}$ !

So, we have proved that $\frac{d(t^2)}{dt}=2t$ , the good old formula for differentiation in high school!

Exercise: Prove that $\frac{d(t^n)}{dt}=nt^{n-1}$ , $\frac{d}{dt}(u(t).v(t))=\frac{du}{dt}v(t)+\frac{dv}{dt}u(t)$

So we have learnt two things so far:

1. A finite number can be divided into infinite number of pieces – which when all of them put together will give the original number – but if only finitely many of the pieces are put together, will get arbitrarily close to the number!

2. I can get close enough to 0 arbitrarily without becoming zero by choosing a sequence converging to zero and this enables me to talk of an instantaneous speed at time t even though I need another nearby instant $t+\Delta t$ to talk about speed. (I am making that nearby instant come arbitrarily close to the given instant, as close as I like ,closer than anyone can ever come to, so that I can talk of a limiting speed at $\Delta t=0$ by choosing a sequence converging to 0. That is called the instantaneous speed.

We will return to geometry next time. See you 🙂

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