Reasoning about the infinite
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Reasoning about the infinite

Zigzagging achieves an ordered list of the rational numbers, leaving no fraction out.

Zigzagging achieves an ordered list of the rational numbers, leaving no fraction out.

Infinity. What is it? In a sense, we all have a basic understanding of the concept of infinity: the idea that some things go on forever, without end. The counting numbers 1,2,3,..., for example, have no end; in other words, there is no biggest counting number, because if there were, we could add one to that number to get a larger counting number.

But if we really think about it, what ‘infinity’ is serving as, here, is a way of describing how many objects there are in some set. Early humans observed that three and two apples make five apples, and also that three and two sheep make five sheep; and then they were able to make the important leap: that the ideas of threeness and twoness, whenever combined, result in fiveness.

 The numbers 3, 2 and 5 themselves do not physically exist in the real world, but they describe how many objects we may have in some set. The important thing is that we can also reason about them, independently of apples or sheep. Why can’t we do this with infinities? In modern mathematics, infinities can be reasoned about just as numbers can. Notice I am saying infinities, implying there are different kinds – which indeed, there are. In particular, we have that some infinities are larger than others.

Consider this: how may positive even numbers (2,4,6,...) are there? Infinitely many. But are there fewer than there are counting numbers? Surprisingly, the answer is no. The way we reason about it is as follows: suppose there are two boxes, box A and box B, each containing any number of objects. Now, if we attach each item in box A to the end of a piece of string, and then attach an item in box B to the other end of the string, doing this for each item in box A, then the boxes contain the same number of objects if every object in each box is tied to precisely one object of the other box.

We say there is a bijection between the two sets (boxes). Now if we let box A be the set of counting numbers, and box B be the set of even numbers, notice that we can indeed do this – simply tie each counting number in box A to its double in box B (1↔2, 2↔4, 3↔6, and so on). If we do this for each item in A, then each item in both boxes is tied to some other item (if we look from box B, each item in box B is tied to half its value). It is not hard to see that what we are actually doing here is ordering the even numbers: by tying 1 in box A to 2 in box B, we are saying that 2 is the ‘first’ even number. Similarly 4 is the ‘second’ even number, since it is tied to 2. We are just making a numbered list of them. If a shopping list uses the numbers 1 to 10, then there are 10 items on that list; and similarly, since we use precisely all of the counting numbers on our ‘list’ of the even numbers, then they must be equal in number.

Using this notion of bijection, one can show, more surprisingly, that there are also the same number of fractions as there are counting numbers! By traversing the infinite rectangle illustrated, in a zig-zag fashion, no fraction is left out.

A larger set of numbers exists: the real numbers. In addition to the fractions, the real numbers include irrational numbers such as π and √2. The mathematician Georg Cantor showed that no bijection exists between the counting numbers and the real numbers. Therefore the infinity which describes the size of the set of real numbers is larger than that of the counting numbers!

Luke Collins is a fourth year BSc Mathematics and Computer Science (Hons) student and researcher at the Department of Mathematics at the University of Malta.

Did you know?

• Japan is making all of its medals for the Tokyo 2020 Olympics out of discarded electronics.

• In 1962 two US scientists discovered Peru’s highest mountain was in danger of collapsing. When this was made public, the government threatened the scientists and banned civilians from speaking of it. In 1970, during a major earthquake, it collapsed on the town of Yangoy killing 20,000.

• According to the scientific definition of the term, bananas, pumpkins and watermelons are all considered berries, whereas blackberries, strawberries and raspberries are not.

• Only five nations have the capability to launch a missile to any place on earth, and they happen to be the exact same five countries that are designated as permanent members of the United Nations Security Council (United States, China, France, Russia, Britain).

For more trivia see: www.um.edu.mt/think

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https://www.sciencedaily.com/releases/2019/04/190412094728.htm

• The new type of gene editing CRISPR system, called CRISPR-Cas3, can efficiently erase long stretches of DNA from a targeted site in the human genome, a capability not easily attainable in more traditional systems. Though robust applications may be well in the future, the new system has the potential to seek out and erase such ectopic viruses as herpes simplex, Epstein-Barr, and hepatitis B, each of which is a major threat to public health.

https://www.sciencedaily.com/releases/2019/04/190411172519.htm

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