"I love you to infinity!"
"I love you to infinity plus one!"
"I love you to infinity plus infinity!"
Let's look at this mathematically, together, as you were just about to do on your own.
Mathematicians are to mathematics as geologists are to rocks.
Not many people gain pleasure from observing an ordinary rock, just like nobody ever gets excited when seeing an integral. Rocks have been in our hands as tools for a very long time, and we have developed many great things from them. Similarly, mathematics has been the flashlight we've used to discover new things, from the Pythagorean theorem to Einstein's theory of general relativity. But it's not mathematics that's fascinating but the results that we find beauty in. Similarly, geologists look very carefully at rocks and listen to the stories that they have to tell.
So let's listen carefully to the sweet sounds of the whisper of mathematics -- to infinity and beyond.
Yes, infinity. The idea of infinitude is as beautiful as it is vast. Let's put aside the "you're infinitely beautiful," the "no, you're infinity plus infinity times more beautiful" and the "no, no, you're infinity multiplied by infinity times more beautiful." None of these statements is any more than complimentary than the one before it. As heartbreaking as it was when Andy had to leave for college in Toy Story 3, there are so far only two (sadly not infinite) types of infinity, namely countable and uncountable infinity.
Countable infinity is a set of numbers that you could count if you lived an infinite number of years. Take, for example, the set of odd numbers {1, 3, 5, ...}. If you count an odd number every day for an infinite number of days, you can count all the odd numbers. Let's take a closer look at this after we define "uncountable infinity."
Uncountable infinity is a set of numbers that you could never count even if you lived an infinite number of years. An example is the set of real numbers, which is all the numbers that can be represented by decimals. You can spend an infinite amount of time counting all the numbers between 0 and 1, but you'll still have the numbers between 1 and 2, 2 and 3 and infinitely many remaining intervals.
So, back to countable infinity. Here is a question: Which set of numbers do you think is bigger -- the set of prime numbers {2, 3, 5, 7, 11, ...} or the set of natural numbers {1, 2, 3, 4, 5, ...)? Euclid proved that there is an infinite number of prime numbers, which is a fun, cool little fact, but would you believe me if I told you that there is exactly the same number of prime numbers as natural numbers?
Let's say that there is a set of boys, each representing a natural number, and, likewise, a set of girls, each representing a prime number. We are going to form couples and have them go on a blind date. If there are more boys than girls (in other words, if there are more natural numbers than prime numbers), we'll know that by seeing that at one point we've run out of all the girls and have a (fairly sad) situation of only boys. But that's not the case, because even when 20 couples or 100 couples or 1 million couples are matched, there is still an infinite number of boys and girls who can be happily matched.
This is called the one-to-one correspondence: We will never find a boy or a girl (that is, a natural number or a prime number) who cannot "find a date." That may sound unrealistically perfect for the real world, of course, but the world of mathematics is as perfect as it gets.
So, to recap: Infinity = infinity + 1 = infinity + infinity = infinity × infinity. Those lovebirds were just repeating the same thing to each other (assuming that these infinities being referred to are countable infinity).
Another fun fact is that the number of rational numbers -- the set of all the numbers that can be written as a fraction (e.g., /, /, /) -- is countable infinity as well. This means that there is the same number of rational numbers as prime numbers or even numbers.
Now that we have a little taste of what infinity is, do you think there is another infinity that we don't know of? Maybe something that's in between uncountable and countable infinity? I have no idea what that infinity might look like, but if you have an idea, perhaps this branch of mathematics -- called number theory -- might be for you.
This blog post originally appeared on McGill Science Undergraduate Research Journal.