Boxing Pythagoras

Infinitesimal Calculus 1: The Numbers Between Numbers

If I were to ask a person to name a number which comes between 1 and 3, everyone from a three-year-old child to a white-bearded great-grandfather is likely to respond by saying, “2.” If I rephrase the question to ask about a number between 1 and 2, then the young child might be confused, but a fourth-grader might be able to respond with . We have to extend our understanding of what we mean by “number” to include some concepts which are not quite so intuitive. That is to say, in between the Integers, there are other numbers which are known as Rational numbers. In fact, given any Integer, , there are an infinite number of Rational numbers which are greater than and yet less than any other Integer which is greater than .

There are numbers in between the Rational numbers, too. We can define some number, , which is not equal to any Rational number. There are Rational numbers which are greater than , and those which are less than , but somehow our number squeezes itself into a gap in between the Rational numbers. In order to find such a number, we need to further extend our understanding of “number” to include the Real numbers. This should all be very familiar to the average high-school student.

Now, what happens if we extend this idea one step further? Are there more numbers which are in between the Real numbers?

For thousands of years, mathematicians have had heated debates about this question. There is a well-known concept in number theory called the Archimedean property, named after the famous mathematician Archimedes (though he, himself, had attributed the idea to his friend and mentor, Eudoxus). Euclid described the notion by saying, “Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another.” In short, this means that given any two numbers, and such that , then we should be able to add to itself a finite number of times in order to find a number which is larger than . For example, given the numbers 5 and 34, I can add 5 to itself seven times in order to get a number greater than 34– that is to say, . Given the numbers and , we can find that . Given the numbers and , we can find that .

However, this property led to a very curious problem when mathematicians began trying to discuss the number of points contained in a given line– particularly when those mathematicians attempted to compare the number of points in one line to the number of points in another line. The eminent philosopher, Aristotle, came to the conclusion that such discussions could be nothing but nonsense, and that any attempt to quantify the number of points in a given line would simply lead to confusion and folly. As an example of this, Aristotle discussed what has come to be known as his Paradox of the Wheel. Take a look at the following figure:

Figure 1: Aristotle’s Wheel Paradox

Ancient Greek mathematicians, while studying circles, wanted to find some way to discuss the circumference of the circle in the same way in which they talk about other magnitudes. So, they began “unrolling” circles to create a straight line equal in length to the circumference of the circle. Aristotle noticed that, given a larger and a smaller circle which share a centerpoint, rolling out the wheel to produce a straight line equal in length to the circumference of the larger circle causes the smaller circle to produce an equally long line. But how can this be? The smaller circle obviously has a smaller circumference, but rolling it out at the same rotational rate as the larger circle makes it seem to have an equal circumference to the big one!

Galileo Galilei, two millennia after Aristotle, attempted to resolve this paradox by arguing that there must be gaps in the continuum– that is to say, there must be empty spaces between the points in any line or figure– and that these gaps account for how the smaller circle’s circumference can be stretched to equal that of the larger circle. However, other mathematicians were quick to note that this would run afoul of the Archimedean principle. If such gaps existed, it should be possible to continue to stretch them until they became noticeably large. We should be able to magnify a line until we literally see it rend apart into pieces.

In the latter half of the 17th Century, Gottfried Wilhelm Leibniz began to argue that there are numbers which are infinitely small. To the absolute shock of the mathematical community, Leibniz was claiming that a 2000-year-old immutable law of number theory was, in fact, incorrect. There existed numbers, Leibniz claimed, which violate the Archimedean principle; numbers which are greater than zero, but which are nonetheless so much smaller than any Real number that it is impossible to find a finite ratio between that infinitely small number and any Real. You could add the number to itself a thousand times, a million, a quintillion, a googolplex– even Graham’s number of times– and that number would still remain smaller than any Real number which you could possibly imagine.

Not only did Leibniz believe that such numbers exist, he utilized them in order to create an entirely new method of mathematics: Calculus. However, the idea was so incredibly controversial that even other proponents of Calculus– like Isaac Newton, who independently developed that field of mathematics– railed against Leibniz for his reliance upon such an insane concept. Still, Leibniz’s results were indisputable, and a number of mathematicians joined with him in an attempt to find some rigorous and logical means of discussing these infinitely small numbers. However, after a great deal of failure, other avenues began to be explored in order to place Calculus on a rigorous footing. Particularly, the notion of the Limit was put forth, expanded, and eventually made rigorous in the 19th Century by Karl Weierstrass. With a rigorous and logical footing finally established for Calculus, the infinitely small numbers of Leibniz’s devising were abandoned and Calculus classes began being taught based on the idea of the Limit.

Thankfully, this was not the end for our strange and tiny numbers. One-hundred years after Weierstrass, and three-hundred after Leibniz, a model theorist named Abraham Robinson began to attack the problem. He was fascinated by Leibniz, and wanted to gain a better understanding of the mind which had invented the calculus. Robinson’s work led to his development of a new number system: one which did not adhere to the Archimedean principle, but which otherwise behaved in exactly the same manner as did the Real numbers. He called this new system the Hyperreal numbers. Just as the mathematicians had extended the Integers to find the Rationals, and then extended the Rationals to find the Reals, Robinson extended the Real numbers in order to find the Hyperreals.

The Hyperreals contain all of the Real numbers, so any number on the Real line is also on the Hyperreal line. However, the Hyperreal number line also contains two very special types of numbers which are not contained in the Reals. The first of these are Infinite numbers– numbers which have an absolute value greater than that of any Real number. That is to say, we can define some number such that, for any given Real number , it is true that . The second type refers to Infinitesimal numbers. Infinitesimals are the reciprocal of Infinite numbers, and as such, have an absolute value which is smaller than that of any Real number (except 0, which is considered to be Infinitesimal): .

Any number which is not Infinite is called a Finite number– including the Infinitesimals. Once the system is in place, it becomes quite easy to prove some simple, but powerful, properties of the Hyperreals. Given any positive Infinite numbers, and ; any positive Real numbers, and ; and any positive, non-zero Infinitesimals, and ; we can derive the following:

1. , , and are Infinite
2. and are Infinite
3. , , and are Infinite
4. is Finite (and possibly Infinitesimal, in the case )
5. is Finite and non-Infinitesimal
6. and are Finite and non-Infinitesimal
7. , , and are Infinitesimal

You may notice that there are several cases missing from the above list. These cases are indeterminate forms– that is to say, without knowing more about the particular numbers involved, it is impossible to tell whether the result will be Infinitesimal or Finite or Infinite. The indeterminate forms are:

We can also derive another very important notion:

For any Finite Hyperreal number, , there is exactly one Real number, , such that is Infinitesimal. In such a case, we call the standard part of , denoted as .

Any two numbers which are only separated by an Infinitesimal are said to be infinitely close to one another. As such, another way of wording the above is that the standard part of any Finite Hyperreal number is the Real number which is infinitely close to it.

So, now, we can answer the question with which our article started. Are there numbers between the Real numbers? We find that the Hyperreals allow us to answer this with a resounding, “Yes!” Given any Real number, , and any Infinitesimal number, , we can be absolutely certain that there are no Real numbers which come between and . This concept is the absolute foundation of Infinitesimal Calculus.