My Math Journal
I didn’t agree with the question in my Exam.
Relax, it was just Mathematics.
So, today was my UMAT-102 (Undergrad Mathematics) Exam that is based on the Foundations of Mathematics, and I’ve got a small story to tell you.
Last day was a bit hectic, because I wasn’t very smooth with the proofs yet, I had this thing in mind that
I won’t write a single thing in the exam that I don’t understand. Whatever I write I shall write with 100% understanding.
A cool thing indeed to say. But during my exam today, this question Struck me — Struck me hard.
I was wondering what to do, since I wasn’t able to prove it right. No matter which way I thought. Finally, having done that — wasted 20 mins and two pages, I moved on to the next question. The reason for this “stickiness” was that, I was committed to write what I understood perfectly. Yet, this didn’t makes sense to me.
Finally I finished my paper (except for that one question) then I came back to it, Q. 6(b) again. I just wasn’t going to leave a simple proof for nothing.
🚧The problem I faced
The real problem was that, when I tried to prove the statement, I considered the “proof by contradiction” approach and assumed that
∃ a, b ∊ Z, a² + 2b² = m
And this was supposed to end with a contradiction that 4|m.
But, it didn’t cause in the case when b was odd, 4 didn’t divide m, which didn’t lead to a contradiction and hence, making the proof futile.
I was feeling the pressure as if I hadn’t decided to skip that particular proof. I would have to pay the price with the next question. But I answered the final question in my paper Q.8.
And then went back to the solution to think about where I went wrong. Finally I remembered two things. The first was something that I read last night.
📖What did I read Last night?
A book about Mathematics obviously
It was a book by Jordan Ellenberg titled, “How Not To Be Wrong”. Truly a title worth reading.
The way he started the story itself was mind blowing, considering that its about Mathematics. It gave a truly different perspective to the subject.
Mathematics is to a Mathematician applying logic, what the Iron Man suit is to Tony Stark breaking through a wall.
That’s it! I just needed to express my common sense using the tools provided to me, symbols in mathematics. No way to get it wrong if I truly understood it.
So, this critical thinking had been passed on to me since my last night read. The cool thing is that when I had the last 5 minutes remaining. I realized that on taking some cases like m = 18, we can actually get a = 4 and b = 1 as results that satisfy the constraints. This made sense, since in the proof that failed, I had got a similar result.
That means…
💡So, where was I wrong?
I was trying to do was what the question asked me to: Prove the statement. This is what directed me wrong. Since, the statement itself is false, clearly illustrated with the counter-example I mentioned above.
Even my instructor had told us in class,
Don’t assume a statement to be correct, take cases and see whether it really holds and you are intuitively able to determine whether its true or not.
Had I remembered it, not only I’d save paper but also time. 🙄
Anyway, so this is what I finally wrote after striking out my previous attempts:
So, I actually disproved the statement, contrary to what the question was. (Hence, the title is not clickbait)
If I were in school, I’d have thought that since its a 6 mark question, I wouldn’t be able to write and escape with just this less. But its not about the marks or writing more wordy or complicated proofs. Its about the scrutiny of the solution. The solution is right and that’s what matters.
🧠Lesson Learned
A mathematician is always asking, “What assumptions are you making? Are they justified?”
— How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg
