Mathematics Made Simple: How Math can Shape one's Mindset

Last week, I gave an Instagram Poll on whether I should continue blogging on current affairs, or blog on personal thoughts. The poll largely tilted for personal thoughts, hence I will write on a more personal tone this time.

I took in suggestions from my viewers, some suggestions (and my responses) are below

Suggestion:"Something kids these days won't understand/relate to"

Response: Uh, okay but I'm not that old...

S: "How you did not qualify for Cambridge"

R: I may blog on my uni applications to aid my juniors some time later this year (in December), but Cambridge may not be something I want to discuss in detail. To whoever suggested this though, you may enjoy today's post.

S: "Your life in OETI/TTMSB"

R: This would be a good topic, but it would not be professional of me to publicly air my dirty laundry against colleagues or my superiors. 

S: "Your weight/love life/girls"

R: This would also be a good topic, and I may write on it in the future, but for now I'd like to keep this blog more relatable and professional rather than letting my heart rule my mind."

S: "Driving"

R: Yeap, may write on this next week, if viewers agree.

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Many who know me from Army recognize me as the stuttering dude who somehow qualified for NUS and SMU Law School. Many hence theorize that during school, I must have been active in English and Current Affairs related activities. This, while partially true, does not picture the whole story of my NUS High School life.

Those who have known me from my NUS High School days would know that this is far from the truth. In fact, I was an active member of the school's Math Interest Group, having risen to Vice President in Year 5 and President in Year 6 and helmed the Interest Group till my graduation. During my time in the ExCo, my team and I organized and brought new initiatives to the school, such as the MIG Magazine and the MIG T-Shirt, and the revamped Inter Class Math Challenge.

I had, and still have, a deep interest in math. I also planned to pursue it in University. As a loyal reader pointed out, I applied for Cambridge Math, and also applied to NIE for a career in math education. Unfortunately, I was not offered either course, but that did not deter my interest. Even though I ended up not choosing to do math in University, I still regularly solve problems and help my juniors with math problems.

"Wait, then if you like math so much? How did law come about?"

You may refer to my post on why I chose Law, but I would say it is a bunch of circumstances that resulted in my choice of Law. I also may blog on this in the future, if the opportunity and need presents itself.

I understand that not many people like doing math, to the point some detest it. Don't worry, even in NUSHS, a supposed "Math and Science school", I had friends who'd rather drop mathematics and take all 3 sciences if they could. 

But to some extent, the real beauty about teaching math is making someone who doesn't understand grasp the concept. The real beauty of math is making a difficult problem seem simple after some hard work. The real beauty of math is knowing that every problem, no matter how difficult, can be solved, or at least be simplified. Doing math actually trains the mind to be more resilient, and to have a positive mindset towards all hardships in life.

A question that I have been thinking of for a while is the following.

Prove or disprove that every integer greater than 1 has a partition that consists of prime numbers only.

For the uninitiated, a partition of an integer means a few positive integers adding up to give that integer. For example,

4

= 3 + 1 (is a partition of 4)

= 2 + 1 + 1 (is a partition of 4)

= 2 + 2 (is a partition of 4)

= 1 + 1 + 1 + 1 (is also a partition of 4)

By default, 4 is a partition of 4 too.

Back to my problem, some example of integers greater than 1 having a partition consisting of only prime numbers include

2 = 2

3 = 3

4 = 2 + 2

5 = 5

6 = 3 + 3

7 = 7

8 = 5 + 3

9 = 5 + 2 + 2

10 = 5 + 5

However, I cannot continue listing on as I need to prove the statement holds true for ALL cases, not just for some.

After thinking of many approaches, for example thinking in bases (binary, base 3, base 4, etc.), I realized that I was overthinking the problem. The solution is actually very simple, as I illustrate below.

All even positive integers n can be represented as n = 2k, where k is a positive integer (k > 0).

n = 2k = 2(k) 

= 2 + 2 + 2 + ... (k times)

Since 2 is a prime number, we have shown that any even positive integer can be expressed as the sum of prime numbers. In this case, the solution is adding '2' k number of times.

All odd positive integers m greater than 1 (m > 1) can be represented as m = 2k + 1, whereby again k is a positive integer (k > 1)

m = 2k + 1 

= 2k - 2 + 3 = (2k - 2) + 3 

= 2(k - 1) + 3 = 2 + 2 + 2 + ... (k - 1 times) +3

Since 2 and 3 are a prime numbers, we have shown that any odd positive integer (greater than 1) can be expressed as the sum of prime numbers. In this case, the solution is adding '2' (k - 1) number of times and then adding 3.

Putting the even and odd cases together, we can conclude that all integers greater than 1 have a partition of only prime numbers. In other words, all integers greater than 1 can be derived from the sum of prime numbers. We can even further generalize this to saying that all integers greater than 1 can expressed as a sum of '2's and '3's.

You see, that is the magic of mathematics. You can take a difficult problem but make it seem easy in the end.

An extension of this problem (of which I haven't thought of the solution yet) is:

Prove or disprove that every integer greater than 1 has a partition that consists of distinct prime numbers.

Again to the uninitiated, distinct means the components of the partition are not the same. For example, 3 + 1 is a partition of 4 with distinct components, whereas 2 + 2 is a partition of 4 but does NOT have distinct components.

I leave the above proof as an "exercise to the reader". XD

A further extension of this problem is an open/unsolved problem. As suggested by an old friend, Lim Li, is the Goldbach's conjecture:

Every integer greater than 2 can be expressed as the sum of two prime numbers.

One can view the problem at the Wiki page https://en.m.wikipedia.org/wiki/Goldbach%27s_conjecture

While significant progress has been made on the conjecture, it has not been proved yet. Even though it has been centuries with the problem unsolved, mathematicians still work hard at cracking the problem, because even though not all problems can be solved, we can still do our best to make things simpler for those who come after us.

I hope this article helps to explain my love for math, which many have found perplexed over the last few years. I also hope that juniors and friends alike will be inspired and consider doing some math again, be it taking a Year 1 math module in university, looking up some math problems, or even flipping open old math notes. Who knows? You may be amazed at what you learn, even after you learnt it.

Also, as a shameless plug, if anyone (juniors or same batch) is applying for math in UK or local universities, do feel free to drop me a message. I can try to help you through. :)