Infinitesimals

In the previous post, I wrote about division by zero. In this post I want to talk about one particular case when such division, and its definition are important. As you probably guessed from the title, this post is about calculus. (In this post I am talking only about one variable calculus).

One of the most basic questions in calculus is finding slopes of functions. The simplest example of such a problem is to find the slope of a linear function, f(x)=mx+b.
In this case we get a straight line, so the slope is uniform. To find is we need to calculate the difference in y divided by the difference in x:

(f(x+h)-f(x))/h=(m(x+h)+b-mx-b)/h=m(h)/h=m

For a less friendlier functions, we cannot talk about globe slope, but only about slope of a certain part of the function, or even only in one single point. And this is were the problem appeared. We want to know the slope in every point, but how can we calculate it? Firstly we need to define what such a slope is. For example lets look on the function f(x)=x^2 and on the point x=3. Lets look on the points f(3.5) and f(3). If we connect those two points by a straight line, we can calculate the slope of that line. If we draw this on a paper, you will see that the function and the line are really close to each other on a small area around 3. Therefore we can thing about the slope of this line as an approximation to the slope of the function. But, obviously if instead of 3.5 we will take 3.1, we will get a better approximation. In the end we can think about the slope as:

slope=(f(3+h)-f(3))/h , h=0

And here we have it - devision by zero. It is obvious that there is no way around it. If h is not zero we get only an approximation and one which can be improved easily enough. The solution to this problem was found by Newton and Leibniz. Their idea was to define a non negative "number" that is smaller that any positive number. Such a number is called infinitesimal. The only real infinitesimal is zero, but if we agree to imagine that there is another such number, they we get many such numbers. This is because if dx is an infinitisimal that 0.5dx is also an infinitisimal. Since dx is not zero, we can divide by it. And becasue it is smaller than any positive number we can disregard it as if it was zero. It is simple to find the slope of x^2 usinig dx:

slope=(f(3+dx)-f(3))/dx=(9+6dx+dx^2-9)/dx=(6dx+dx^2)/dx=6+dx=6

Although the result is correct, we no longer use infinitesimals, but use limits instead. The reason for this is that infinitesimals are problematic. The problem lies in the very definition - it is not clear what do we mean by a number that is smaller than any positive number, but not negative or zero. We also treat it as both zero and as not zero. However, they are still in use in physics. The reasons for this is that while they are not rigorous enough for mathematicians to use, they give good intuition and they appear rather naturally in physical problems.

Division by Zero

We all know that division by zero is undefined. But what does it mean? Firstly, it is not "completely" undefined. For example, it is possible to say that:

1/0=lim1/x, x->0

This particular definition is not very good, because the limit doesn't exist. But we can always suppose that we look only on positive x and the the limit is defined to be infinity. Alternatively, it is possible to use geometric series. The formula for the sum of the geometric series says that:

1+q+q^2=q^3+...=1/1-q,

If we will take q to be 1 we will get: 1/0=1+1+1+1+....=infinity

As you can see I just wrote two different definitions to division be zero. However, this doesn't solve anything. The problem is that infinity is not a number. We could just as well say that 1/0=watermelon. Mathematically it is basically the same. Therefore when we divide by zero we no longer deal with numbers, and this what makes such division undefined.
Now, what about other definitions? A very long time ago, when "zero" appeared in mathematics there was an attempt to define division be zero by simply stating that division by zero gives a fraction whose denominator is zero. But this is again not a number.
We also cannot define 1/0=a where a is a number. The reason for this is that in this case we get:

1=0*1/0=0*a=0

And if this happens we get that the only number that we have is 0, because all the "other" numbers are equal to it. Obviously this is not an interesting situation. Because of this it is necessary that the devision by zero is not defined.

Lets look on some other example of undefined identities. The first one is 0^0. The reason this one is undefined is very simple:

0^0=0^(1-1)=0^1*0^-1=0/0=0*1/0

And we are back to division by zero. It is important to note that it is sometimes assumed that 0^0=1, but this is mostly done as an alternative to saying that in a polynomial x^n in which n can be zero x is not zero.
A more complex example is (-3)^x where is a real number (that is, x is not rational). Obviously, there is nothing special in number (-3), this expression is undefined for all negative numbers. For positive numbers the definition uses limits. For example:

3^x=lim(3^q), limq=x, q- rational

This definition works well for positive numbers, but for negarive numbers we have the problem that the square root is not real. Because of this if we don't allow complex numbers we get many undefined points in the series, and if we allow such numbers the series doesn't converge.

As a bonus, here is a little proof that 2=1. Can you see where is the error?

a=b.
a^2=ab ,
a^2+a^2=a^2+ab,
2a^2=a^2+ab,
2a^2-2ab=a^2-ab,
2(a^2-ab)=1(a^2-ab)
2(a^2-ab)/(a^2-ab)=1(a^2-ab)/(a^2-ab)
2=1

The reliability of wikipedia

As you all know there are many articles on wikipedia, and their quality varies greatly. However the main problem is that in many cases it is impossible to tell if a particular article is accurate or not. For example, I just found the following article on wikipedia:

Beard-second
The beard-second is a unit of length inspired by the light year, but used for extremely short distances such as those in nuclear physics. The beard-second is defined as the length an average physicist's beard grows in a second, or about 5 nanometers[1].

One beard-second equals 50 Ångströms (10^-10 m). 20000 Beard seconds equal 1 RCH. 2000 Beard seconds = 1 RBC.

Google search supports the beard-second for unit conversions.[2]

Unsurprisingly, this article is being considered for deletion in accordance with Wikipedia's deletion policy. Suprisingly, the part about Google supporting beard second in unit conversion is actually true. After a quick search on the internet, it seems that the beard second was invented as a teaching exercise a few years ago.

In this case, the article seems to be a joke, but in reality it is about a funny teaching exercise....