Impossibility

Doing math is not the most easiest thing on earth obviously, but it seems that every year I have to digest something that is more and more abstract. There are times when I can totally picture in my head a diagram or a graph that helps me understand the topic, the concept, but I have stopped thinking about some real life examples in some cases, the ones that make the difference between just math and beautiful math.I am very sure that you know what I mean by all this. I think a lot of you had a though period understanding infinity, or irrational numbers and things like that.

impossible-triangle-with-dice

Firstly, one of the first surprises for me was 1/3. For me this was an exact quantity, I was always thinking about a big orange that can easily be divided into 3 parts, it was crystal clear. But when it started with 1/3 = 0.3333333… the thing started to look weird. Did they really wanted to tell me that I cannot measure that exactly? The division was exactly in front of my face, but it was still strange. From the same specter of things there is square root of 2, which was kind of a harder thing to understand at that point. I could kind of leave with the idea that it was hard to measure that 1/3 exactly because I was thinking basically at circles and spheres, but with square root of 2 was a different story. I have drawn tens of squares with vertices of 1 cm and tried to measure the diagonal, every time I was kind of sure everyone was making a huge mistake, or that my ruler was broken or something similarly. The next step was to do it for 10 cm, but it still was kind of strange to understand. Did you ever thought about how strange is this thing? You can really construct and see that square root of 2, but you cannot properly measure it; you cannot calculate it; it’s just something without end.

Secondly, what do you think about infinity? I am pretty sure that you know know that there are an infinity of natural numbers, but how surprising (shocking??) is the fact that there are as many even numbers as natural numbers? Those infinities are kind of ‘the same’. Kind of hard to properly understand, even if you have the proof just under your eyes. And if one tries to think about integers, they are as many as natural numbers, and rational numbers the same. The proofs are kind of hard, but they show us something that for a long period of time we consider to be impossible. And the most elegant proof concerning this was the one regarding the real numbers. There is another infinity, a different one , a ‘bigger’ one. The proof that shows that in the set (0,1) there are more numbers than in the set of natural numbers was absolutely fascinating. This was one of the most interesting things I have discovered so far doing undergraduate math. I cannot wait to see what will come…

What is your most ‘impossible’ thing math has taught you? I am very curious to find out what you think about this, leave a comment bellow. I hope you enjoyed this one.

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