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**Hi, Mathnasium! What is infinity?**

**~ Saakshi K.**

**Grade 3**

In our previous “What is infinity?” blog post (part 1), we looked at the *concept* of infinity. We established that infinity is not a number and does not behave the same way numbers do; it is greater than any real number and is the *idea* of real numbers being endless.

**Infinity Applied to the Real World**

As we navigate the real world, opportunities arise to observe the properties of infinity. The following are some scenarios of infinity in action.

Question:

Infinitely many students wait in line outside a classroom. Another student joins the line. How many students are now waiting in line?Answer: One plus an infinite amount of students is still an infinite amount of students. Any number added to infinity still leaves us with infinity!

Question:

Infinitely many students join the line of already infinitely many students waiting outside a classroom. How many students are now waiting in line?Answer: We still have infinitely many students in line!

Question:

Three siblings decide to leave the line of infinitely many students. How many students are left in line?Answer: Taking a finite amount from an infinite amount still leaves us with infinite students.

One popular demonstration of infinity is the *Hilbert’s Hotel Paradox*, which illustrates that a hotel with infinite rooms, even if they’re all occupied, can still accommodate an endless number of guests.

Many misconceptions arise about infinity because it does not behave as real numbers do. Remember that:

Infinity is endless. We cannot travel far enough to get there since there is no end. We do not need to define any end to infinity, which makes it simpler to work with than things with an end.

Infinity just IS — it is not growing or getting larger. It already is.

**Infinite Sets**

Once understanding of this concept grows, infinity can become captivating. For example, did you know there are *types* of infinity?

Not all infinities are the same. The two main types of infinity are *countable* and *uncountable*. *Countable* means we could count all the numbers in the set in an infinite amount of time, whereas *uncountable* means we could not.

An infinite set is any set that has an endless number of elements, for example, all integers {…, -2, -1, 0, 1, 2, …}, all even numbers {…, -4, -2, 0, 2, 4, …}, all odd numbers {…, -5, -3 -1, 1, 3, 5…}, and all square numbers {0, 1, 4, 9, 16, …}. These are all *countable *infinite sets because we can pair each number in the set with a natural number (known as a *bijection*).

Interesting fact: There are exactly the same amount of numbers between 0 and 1 as there are between 0 and 2. Isn’t that incredible?

Despite containing infinite numbers, the sizes of infinite sets can differ. Uncountable infinite sets, however, contain so many elements that we cannot make a pairing (bijection) with the set of natural numbers.

Interesting fact:

Cantor’s Diagonaldemonstrates the idea of uncountable infinity, where there are infinitely many real numbers and infinitely many positive integers, but the number of items in the set of real numbers is bigger than the number of items in the set of positive integers.

We know this idea of something having no end can be challenging to process. It remains unproven whether the universe is infinite or finite in size. Our human intuition about magnitude makes it hard to fathom the enormity of infinity, let alone its properties. But the beauty of mathematics and our universe is that they open us to think beyond the limits of our minds.

If you would like to learn more about infinity or discuss any other mathematical concepts, reach out to your nearest Mathnasium Learning Centre. They would be happy to talk to you!

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