Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
Previous Section: 0.2 Complex Numbers
The concept of infinity is fundamental to calculus. Infinity and its symbol, ∞ (the lemniscate), will be used very often. In this text, and generally, infinity represents an incomprehensibly large number. For example, if we take a 10 and multiply by 10 we get 100, then if we multiply by 10 again we get 1,000. Repeat this process and we can get the following number:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
This number is massive. However, it is still insignificant compared to infinity. We say that if we continued the process of multiplying by 10 forever, then the number would approach infinity.
But what is infinity and how do we wield it?
History and Paradox
In the fifth century there was a Greek mathematician and scholar named Zeno. The Greeks had wrestled with many intellectual problems like the concept of an irrational number, negative roots, and even the infinite during the early days of mathematics and science. A problematic question for them was to define how or if it is possible to split something into tinier and tinier pieces until it could not be divided further. Different schools of thought (The Pythagoreans, The Atomists, The Eleatics, etc.) had different ideas on how to deal with this issue. Zeno belonged to the Eleatics and is known for creating Zeno’s Paradoxes.
The particular paradox that will be mentioned here is the Paradox of Infinite Divisibility. The idea is that if we take an object and cut it into two equal parts, then cut these parts into more equal parts similarly, we reach what Zeno refers to as “the elements.” There are three problems with reassembling these parts to recreate the whole:
1) the elements are nothing so the whole is made of nothing and is nothing.
2) The elements are something but have zero size, and an infinite number of zeros is zero, so the whole is made of nothing again.
3) The elements are something and are not zero in size, so they can be further divided. Thus, the process of division was incomplete.
This brings up the important question of whether or not an infinite number of infinitesimal (infinitely small) pieces makes anything at all as well as many other philosophical and mathematical problems.
Infinite Sets and Multiple Infinites
Over the course of a thousand years many mathematicians made an attempt at understanding the infinite and the infinitesimal. Then, in the 19th century, Georg Cantor found a way to describe infinity compared to other infinites. He stated that if there exists a one-to-one (and onto) function between two sets, then the sets are equinumerous. That is to say, if we can pair up terms in two lists, then the lists are the same size. This seems elementary but proves difficult with infinitely long sets.
As an example, we can show the set of all integers is equinumerous with only the positive integers using the following function:
Here is an example of some values and how they pair up:
| n | f(n) |
|---|---|
| 1 | 0 |
| 2 | 1 |
| 3 | -1 |
| 4 | 2 |
| 5 | -2 |
| 6 | 3 |
It should be clear that the left side (positive integers, or natural numbers) pairs up evenly with the right side (positive and negative integers). That is to say, the size of these sets (which is infinity) is equal!The difficult trick is to show that there is another infinitely large set which is larger. A more complicated argument can be used to show that the set of Real Numbers is larger than the set of Natural Numbers, which means that there are at least two infinities. To look into this, you can research countable and uncountable sets.
Review
Infinity is a non-numerical symbol used to represent something that is incredibly large. Something is infinitesimal if it is incredibly small (or close to 0). There are different types of infinity, and some infinities are larger than others!