Quick Post

Why is it always at 12am that I am suddenly filled with the desire to do something productive?

Anyway, this is a post regarding a class of functions that are fun (subjective), of which I might make an elaborated post about after I’m done with my Analytic Number Theory post.

I’ve yet to look into this thoroughly since I only started doing so today 1150pm so here is a special case:




Yeah, does not look so impressive… YET.

Here’s the cool part: Take out your calculator (preferably a scientific one), and ensure that it is set to Radians.

Now, find “F(1)”. Did you get “F(1) = 1”?

How about “F(2)”? Did you get “F(2)=2”?

Continue to do this until “F(4)”, do you see the pattern?

What about “F(5)”? Oh? It loops back to “F(5)=1″…

How about “F(6)”? How about “F(2015)”?

So here’s it: “F(n)” outputs the remainder when “n” is divided by “4”!

So “F(2015)=3” since “2015” divided by “4” gives a remainder of “3”!


Now how did the godly Julia derive such an amazing (actually not so) function?

Sorry not going through how goodnite.

How the function looks like when graphed out:



Black Holes- Part 2

Hi Guys,

As mentioned before, i will be writing about black holes this week.

The previous post from this series was done by my co-writer, Jackson, who mysteriously disappeared after posting it. Don’t worry, he’s still alive and well helping me solve my darn math and physics problems.

Okay so on to black holes.

In the previous post, Jackson covered the formation and structure of black holes.

To make up for the previous post’s lack of graphics, here’s a helpful diagram of how a black hole looks like:

black hole

Yes. It is black. Or rather, dark. As mentioned before, the gravity from the region beyond the event horizon is so strong that even light cannot escape. As such, it is well, dark.

Some of you might be thinking: “So as time passes, all the matter in the universe will end up in black holes. Since nothing can escape a black hole, whatever goes in, doesn’t come out. So we’ll all end up as black holes one day!”

If you seriously thought that, then you were thinking what i thought when i first learned about black holes.

However, that (in reality) is incorrect.

That brings me to what this post is about. This post is about the thing that kills black holes. How all the black holes will meet their end.It’s a story about the weirdness of the universe, and also how cool it can be.

This post is about Hawking radiation.

So what is Hawking radiation?

Simply put, Hawking radiation is radiation that is emitted by black holes. Over time, the black hole loses mass as it emits more Hawking radiation.

According to Einstein’s famous equation, E=mc^2, the energy from the radiation emitted is related to the mass loss by a factor of c^2 as such, more energy lost, more mass lost.

And what happens after the black hole loses all its mass? It basically disappears. It evaporates.

That’s also why scientists can make micro black holes in particle accelerators without getting the entire earth sucked into oblivion. The micro black holes evaporate in an extremely short span of time, thus we are all safe!

Also, it gets its name from the famous physicist Stephen Hawking (the cool guy who helped us understand black holes a lot more and is also famous from the awesome move Theory of Everything)


At this point you’re probably wondering how can Hawking radiation escape from black holes when basically nothing can escape from black holes?

Well, to answer that, we need to look into quantum physics, where things coming from nothing and teleportation make it possible for the Hawking radiation escape.

The first theory of how Hawking radiation escapes from black holes is by the separation of particle-antiparticle pairs that form in the vacuum of space.

So some of you might be familiar with Heisenberg’s Uncertainty Principle. Most know it for its ‘position-momentum’ uncertainty version. The theory states that the more certain an observer is of an object’s position, the less certain he will be about the object’s momentum. However, this uncertainty is extremely minute and undetectable by most of us in our daily lives. But it is indeed experimentally proven and consistent with many other theories and experimental results.

Apart from ‘position-momentum’ uncertainty, there is also ‘energy-time’ uncertainty. And due to this, in an extremely short span of time, there will be an extremely huge energy uncertainty, causing the energy to possibly go high enough that random particles may be generated. Once again, Einstein’s equation: E=mc^2 comes into play here as mass in the particles is a result of the energy uncertainty. However, as we all know, energy has to be conserved (or else evil scientists would have created an infinite energy producing machine and taken over the world with crazy weapons or maybe just selling us power at ridiculous prices) as such, the particles soon disappear as they are annihilated by the antiparticle that forms with it. As such, the universe remains in equilibrium and evil scientists don’t destroy the Earth.

This also occurs at the edges of the black holes. At the event horizon.

Sometimes, a pair might form with one particle forming within the event horizon with the other particle forming outside.

antiparticle-particle event horizon

As such the antiparticle will fall into the black hole while the particle goes out into the unexplored wilderness that is the universe. Since the antiparticle must fall in as it can’t escape, all pairs that form this way will probably end up releasing a particle outside the black hole. Thus, we observe it as Hawking radiation.

The second way Hawking radiation could possibly form is by quantum tunnelling.

This effect occurs when a particle bypasses an energy barrier without reaching the required energy. For example, certain chemical reactions require a certain temperature (which is proportionate to the average energy of the particles) for the reaction to occur. As such, quantum tunnelling would be when a particle in the solution reacts without reaching the required energy.

This is possible in black holes too. The energy barrier is the event horizon, the particle is a photon.

Regarding how this effect actually works, it relies on the fact that particles can be described as waves (something that is pretty complex and something that I’m leaving out in this post).

The idea of Hawking radiation is extremely important as we can learn a lot from observing Hawking radiation. Since it’s basically the only thing that we can see coming out of a black hole (actually not exactly but hey who cares) it serves an important role in our understanding of these immensely powerful balls of uncertainty.

There still is much more to cover on black holes. Being such an extreme object, it provides insight into what happens under extreme conditions.

Well, that’s it for this fortnight’s post, look forward to the next one!


Thanks for reading!

Clyde Lhui 🙂