On Infinity

Infinity is not within our reach. After all, we are finite beings and even the earth and the sun will one day not here anymore. But wait, though infinity is such abstract concept, we actually deal with it quite frequently. My first experience with infinity is when my sister hold up
two mirrors facing each other and told me when I was about 6: "look, there is mirror within mirror and it goes on forever!" I was fascinated by this simple pathway to infinity.

Young kid's encounter with infinity
When I was in third grade I also experienced infinity. It is fairly easy to turn some fraction into a decimal. For example, 1/2 = 0.5. 1/4 = 0.25. The trouble comes in when trying to use long division to turn 1/3 into a decimal. All the paper in the world cannot represent 0.3333... I run
into infinity! Infinity is reached whenever there is a repeat. Same thing happens when I tried to convert 2/3 (a more fascinating repeat is discovered when I tried to convert 1/7).

Infinite joke
An old computer science student joke: Why does the programmer died in the shower? That's because the shampoo bottle say: "Wash. Rinse. Repeat."

Bad BASIC program
Old school programmers know this BASIC program

10 PRINT "HELLO"
20 GOTO 10

Ooh it writes HELLO indfinitely. You get to infinity easily when you had to always repeat (and has no mechanism to break). This demonstrates the power of programming: computers will follow your instructions even if it is stupid. Now it is considered bad practice because all programs should have a graceful exit.

We not only deal with infinitely large, we also deal with infinitely small.

Zeno's Paradox
You may have heard of a paradox known as "Zeno's paradox". A guy wonders: if I have to walk 1 mile. I first have to walk half mile.If I have to run half mile, I had to run 1/4 of a mile. And so on. So there are infinitely many tasks to do! But yet I have sucessfully done it. So this is a paradox. Zeno was a bit confused.
This is a classic introduction to calculus concept of the limit.
The derivative (slope of the tangent line) deals with the infinite small. It is a slope, of a very small (infinitely small) value dy divided by infinitely small dx. Strange concept maybe, but we now know calculus work well. Though mathematically we do not work with infinity directly. We can talk about approaching to it using the limit.
We also add up infinitely small rectangles to determine area under a curve with an integral.

Infinite Hotel
Here is a famous thought experiment and puzzle concerning infinity, known as the Infinite Hotel. So suppose there is an Infinite Hotel that can host infinite number of guests. One day there is an Infinte Convention nearby and the Infinite Hotel can accomodate all the guests, no problem. But on the same day there is another Infinite Convention and they want to come to the Infinite Hotel. You are the manager. Are you able to accomodate 2 conventions?

The answer is this: you tell the guests from the first Infinite Convention to move to odd number rooms, and tell the guests from the second Convention to go to the even number rooms.

Some math students may enjoy such thought experiment. Just about everyone else will think this is a stupid problem as there is no such thing as Infinite Hotel and Infinite Conventions.