So you have a number line, from 0 to 1.
Now say you have a ruler, you hold it up to these, and measure. The length from 0 to 1 is one unit. Right? Right.
So now. What is the length of 0? Not 0 to 1 just 0. Well, it’s 0 of course. What’s the length of 1? 0.
Okay what if we split it up?
What’s the length of 1/4? 0. 1/2? 0. 3/4? 0.
All of these, these rational numbers and integers, each of them have a length of 0. So where does the one unit measurement from 0 to 1 come from? Irrational numbers.
Between 0 and 1 there are more than infinite irrational numbers. More than infinite.
Say for instance you have these irrational numbers.
Sure it looks like with this you could count them all, but the thing is you can have
0.A21A11A13… You can have infinite combinations of numbers with infinite length.
They are by definition uncountable. And that’s where you get 1 unit. Each number in and of itself is of 0 length but you have so many more than infinite irrational numbers crammed between there that you get a length.
To put it into perspective if you had a box filled with all of the Real numbers, meaning all integers and rational numbers and irrational numbers, you would have 100% chance of pulling out an irrational number. Not 99.99999%, 100%.
It’s all so interesting, and such a fundamental concept of numbers. So why have I not learned until now, until I reached college and happened to have a mathematician for a teacher who cared about going and filling the gaps in my education? As he wrote all of this and more up on the board he told us that it really wouldn’t be that long before this kind of knowledge simply disappeared. Until it was gone, until no one knew about it anymore. Because no one would teach it. Concepts like this should be introduced in elementary school. Mathematical concepts like this are so fascinating. It depresses me to imagine that I was told I was getting a “superior” education with the International Baccalaureate program in high school. It didn’t even scrape the kinds of concepts I’m learning now.
I don’t know about other countries. But public education in the US is suffering. You shouldn’t have to be going into more directly mathematical fields to learn these kinds of things. You would think they’d be more fundamental.
Not to mention the things that are being taught are being taught so poorly. At the University of Utah one of the most consistently failed classes is College Algebra. Algebra. We shouldn’t even need this class! But no, not even that is getting the concepts in most people’s heads correctly. The first few weeks of my calculus class is, unavoidably, being spent learning simple algebra concepts that we’ve never heard of or hardly absorbed because we just can’t go onto these more complicated things without that foundational knowledge.
If ever I was to leave my ambitions in computers, it would be to become a teacher for this exact reason. To teach the things that aren’t on the simplified, dumbed down curriculum. What sucks too though is that I have had teachers that tried to do this, that went out of their way to teach us the cool things we should have been learning but weren’t. 2 were punished for it and ended up leaving to entirely other school districts because of the crap they were putting up with from administrators. Another got complaints from students and parents because the kids didn’t want to learn. They were so used to the simple packets, “read this textbook and regurgitate it then get the grade, forget everything, and move on” method of learning that they didn’t want to try. They didn’t want to explore and discover new topics, they didn’t want to expand their thinking and philosophy, they didn’t want to use their brains and think critically. They wanted to be told step by step how to get the grade and leave the class, gaining nothing from it but a GPA.
In the end of everything my GPA was crap in high school. My grades were crap. Why? I didn’t want to waste my time throwing up memorized crap onto the packets that were used in place of teaching and interaction. I didn’t want to write papers on studies and works that other people wrote while I could have been learning and discovering the things they wrote about and truly explore them for myself.
If college weren’t so expensive, if I wasn’t already drowning in student loans, I would never want to graduate. I would stay in school and keep learning everything I could because for the first time I have the chance to learn the things my teachers didn’t know or didn’t care to take the time to teach us. If i had (or as I like to think, once I do have) the money, I’d go back and get as many degrees as I can learning about as many subjects and areas as I can.
The thought that I could end up doing one thing, working in one field, for the rest of my life, terrifies me, no matter how interesting or how well paid that field is. Because there is so much out there. So many things I’m missing, so many things I want to see and at least begin to understand. That was the one question I could never answer: what do you want to do when you grow up? Because they want one answer. They want that one career you want to do for the rest of forever. I refuse to limit myself. Yes I want to make a living because you can’t get anywhere without being able to support yourself in something. But I don’t want to know everything there is to know about that one thing, that’s not me. “Narrow your scope” “narrow it down” “pick something” that’s what I was told growing up. Heck. No.