4 ms·

This sadly is how the world works. In school, even in high school and college, the students who guess the teacher's password are rewarded. The (few) students

• boredguy8 18y ago
  "The (few) students who make an effort to discover and think through something on their own are criticized for wandering from the beaten path, even on the occasions when they're actually right." I remember I just -couldn't- accept that we can't take square roots of negative numbers. I mean, sure, two negatives multiplied together is a positive, but...there HAS to be a way! Numbers, to my 6th grade mind, weren't arbitrarily going to deny your ability to manipulate them. Fortunately for me, I happened to be reading a book on Mandlebrot and in the course of discussing his work the author introduces the concept of imaginary numbers. So the next day in class I asked: "Are you suuuure?" "Yes, I'm sure." "Well, I was reading in this book and so-and-so says you can take the square root of negative numbers, you get 'i' for the root of -1 and..." "Why don't you go wait outside." Where I got told that although I move to the beat of a different drummer, when I'm in her class, I really just need to be quiet. That's when public education and I had a falling out from which our relationship has never recovered. All of that to say: I think part of the 'problem' is that kids who do try to think on their own are often in a situation where they're challenging accepted rules without the social grace to do so constructively. But rather than being taught the social skills, they're just shut down. Very sad, actually. They could be taught how to 'wander well'.
  • Prrometheus 18y ago
    Isn't sqrt(-1) = i an axiom though, and not a theorem? In other words, if you just started with the integers and the basic rules of arithmetic, you will never run into the need for imaginary numbers. It's only after you posit their existence that a lot of cool results flow from it. Am I wrong there?
    • boredguy8 18y ago
      I don't really know how to interpret your sentence. Complex numbers are a field in which you do 'basic rules of arithmetic'. You can't do the 'basic rules of arithmetic' on integers. i is defined as sqrt(-1), if that answers your question.
      • Prrometheus 18y ago
        I should have said "Real Numbers" instead of "Integers", sorry. My point is that sqrt(-1) = i is an extra assumption that you don't need to make in order to derive the math you do in school. If you don't make that assumption, you can still derive a lot of math. If you do make that assumption, then you can also derive the theorems of complex analysis. I think that's true, anyway. I could be wrong. It just seems strange to me that someone would be bothered by the lack of negative square roots, since their existence is never derived, only assumed. But then again, people's minds work very differently, especially in Mathematics. I was the opposite in Math class. I fought accepting "i" when they tried to teach it to us since it seemed so arbitrary and contrived to me. This was before I discovered that all Math is arbitrary and that there is no "real math", anyway.
        • ashu 18y ago
          As the "Road to Reality" explains, if you study Quantum mechanics, complex numbers become every bit as "real" as the integers or real numbers. You just can't explain some of the quantum mechanical phenomena or concepts without using complex numbers at all.
        • tokipin 18y ago
          it's as you say. it is contrived. but no more contrived than the basic integers. these are abstract systems that we have contrived in ways that are most useful to us. the natural numbers just seem more "real" because everyone in most cultures deals with "quantity" and "magnitude"
      • matstc 18y ago
        When I was learning complex numbers, the teacher didn't want us to assume sqrt(-1) = i, we had to go like this: sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.