All of kaler's Comments + Replies

Okay, those are all - well, I think I can kind of see some relation to complex numbers in there, but it's very vague. So, let me describe how I understand complex numbers. To do that, we'll have to go right back to the very basics of mathematics; numbers. Imagine, for a moment, an infinite piece of paper. (Or you can get a piece of paper and draw this, if you like; you won't need to draw the whole, infinite thing, just enough to get the idea) Take a point, nice and central. Mark it "zero". Select a second point (traditionally, this point is chosen to the right of zero, but the location doesn't matter). Mark it "one". Now, let us call the distance between zero and one a "jump". You start from zero, you move a jump in a particular direction, you get to "one". You move another jump in the same direction, you get to "two". Another jump, "three". Another jump, "four". And so on, to infinity. These are the positive integers. Now, consider an operation; addition. If I apply addition to any pair of positive integers, I get another positive integer. Any of these numbers that I add gives me a number I already have; I can add no new numbers with addition. However, I can also invert the addition operation, to get subtraction. If I want to find X+Y, I hop X jumps from the zero point,then Y more jumps. But if I want to find X-Y, I must jump X jumps to the right, then Y jumps to the left; and this gives me the negative integers. Add them to the mental numberline. At this point, multiplication gives us no new numbers. Division , however, does. You will now notice, there are still gaps between the numbers. To fill these gaps, we turn to division; X/Y gives us a plethora of new numbers (1/2, 2/3, 3/4, 4/5, so on and so forth), hundreds and millions and billions of little dots between each point on the numberline. These are the rational numbers. Is the numberline full yet? Hardly; it turns out that the rational numbers are so small a proportion of the numberline that it's st

Accepting feedback and directly applying it is great :)

It does; complex numbers are just another type of number. We'll get to them shortly. To be fair, sometimes the intuitive answer is wrong; one does have to take care. But sometimes, as in these cases, the intuitive model does work. Exactly. Perfect. You could do it that way, and it leads to the correct answers, but I think it's fundamentally problematic to see complex numbers as intrinsically different to real numbers. (For one thing, real numbers are a subset of complex numbers in any case). Right. There's only one that I can think of off the top of my head; if x^z=y^z, this does not mean that x=y (i.e. we can't just take the z'th root on both sides of the equation). This can be clearly demonstrated with x=2, y=-2 and z=2. Two squared is four, which is equal to (negative two) squared, but two is not equal to negative two. ---------------------------------------- Now, as to complex numbers. Let me start by asking you to define a "complex number".

I recommend chapter 22 ("Algebra") of volume 1 of The Feynman Lectures on Physics. Here's a PDF. My summary (intended as an incentive to read the Feynman, not a replacement for reading it): 1. We start with addition of discrete objects ("I have two apples; you have three apples. How many apples do we have between us?"). No fractions, no negative numbers, no problem. 2. We get other operations by repetition -- multiplication is repeated addition, exponentiation is repeated multiplication. 3. We get yet more operations by reversal -- subtraction is reversed addition, division is reversed multiplication, roots and logarithms are reversed exponentiation. These operations also let us define new kinds of numbers (fractions, negative numbers, reals, complex numbers) that are not necessarily useful for counting apples or sheep or pebbles but are useful in other contexts. Rules for how to work with these new kinds of numbers are motivated by keeping things as consistent as possible with already-existing rules.

Which implies that I can, tentatively, estimate you to be in the top 10% of people who are accepted for a degree. That's really good. ...I think we've found the start of the problem. Your foundations have a few holes. Dividing X by Y, at its core, means that I have X objects, I want to place them in Y exactly equal piles, how many objects do I place per pile? (At least, that's the definition I'd use). In this way, the usefulness of the operation is immediately apparent; if I have six apples, and I want to divide them among three people, I can give each person two apples. I can use the same definition if I have five apples and three people; then I give each person one and two-thirds apples. This also works for negative numbers; if I have negative-six apples (i.e. a debt of six apples) I can divide that into three piles by placing negative-two apples in each pile. Division by zero then becomes a matter of taking (say) six apples, and trying to put them into zero piles. (I hope that makes the problem with division by zero clear). And yes, there is a fancy algorithm that I can put X and Y in and get the quotient out... but that algorithm is not a particularly good basic definition of division. (Interestingly, I note that your definition jumps straight to setting out separate cases and then trying to apply a different algorithm to each individual case. This would make it very hard to work with in practice; I've worked with division algorithms on computers, and they're far simpler, conceptually, than what you had there. If that's what you've been working with, then I am really not surprised that you've been having trouble with maths). Now let's see how far this goes... Define "multiplication", "addition", and "subtraction".

A very easy way to improve your writing would be to separate your text into paragraphs. It doesn't take any intelligence but just awareness of norms. Math.stackexchange exists for that purpose. Not everybody is good at math. That's okay. Scott Alexander who's an influential person in this community writes on his blog: Math is about abstract thinking. That means "common sense" often doesn't work. One has to let go of naive assumptions and accept answers that don't seem obvious. In many cases the ability to trust that established mathematical finding are correct even if you can't follow the proof that establishes them is an useful ability. It makes life easier. In addition to what CCC wrote http://math.stackexchange.com/questions/26445/division-by-0 is a good explanation of the case.

Scholastic math is a different beast. I can say that a lot of professors have issues with the "standard" math curriculum. I have taught university calculus myself and I don't think that the curriculum and textbook I had to work with had much to do with "fluid intelligence". Sounds like one source for your troubles. It's a lot harder to succeed at school math and go through the motions if you have unanswered questions about why the method works (and aren't willing to blindly follow formulas). By all means bring your questions up to the professor. If he's teaching, there's probably some university policy that he be available to students for a certain amount of hours outside of class (i.e. it's part of his job). You lose nothing by trying. Even an e-mail wouldn't be a bad idea in the last resort. In my experience, professors tend to complain about students who never seek help until they show up the day before the final at their wits' end (or, worse still, after the final to ask why they failed). By that point it's too late. We like our multiplication rules to work nicely and division by zero causes problems. There's no consistent way to define something like 0/0 (you could say that since 1 x 0 = 0, 0/0 should be 1, but this argument works for any number). With something like 1/0, you could say "infinity", but does that then mean 0 x infinity = 1? What's 2/0 then?

This seems normal to me. What is intended is very often not an easy question to answer. The mere fact that you have been accepted for and expect to pass a double degree tells me that you are really not too stupid. (I'm not actually sure what the difference between Second Upper and First Class Honours is - I assume that's because you're referring to the education system of a country with which I am not familiar). Theory: You had a poor teacher in primary-school level maths, and failed to learn something integral to the subject way back there. Something really basic and fundamental. Despite this severe handicap, you have managed to get to the point where you're going to pass a double degree (which implies good things about your intelligence). I... don't actually know. Throughout my entire school career, I was the guy for whom maths came easily. I don't know what's normal there. Actually, it may be possible to narrow down what you're missing in mathematics. (If we do find it, it won't solve all your math problems immediately, but it'll be a good first step) Let's start here: Define "division".

No one is smart enough. But if you mean, specifically, smart enough to then I think the question is kinda backwards. "Am I too stupid to try to improve my thinking?" -- it's like "am I too sick to try to improve my health?" or "am I too weak to try to improve my strength?" or "am I too poor to try to get more money?". Now, no doubt all those things are possible. If you really can't reason at all, maybe you'd be wasting your time trying to reason better. And there are such things as hospices, and maybe some people are so far in debt that nothing they do will get them out of poverty. But those are unusual situations, and someone who is headed for a good result in a challenging subject at a good university is absolutely not in that sort of situation, and if the stuff on Less Wrong is too hard for you to understand the fault is probably in the material, not in you. A fine example of the kind of "easy" task human brains (even good ones) are shockingly bad at. I just attempted a randomly-chosen 2-digit multiplication in my head. I got the wrong answer. Am I just not very intelligent? Well, I represented the UK at two International Mathematical Olympiads, have a PhD in mathematics from the University of Cambridge, and have been gainfully employed as a mathematician in academia and industry for most of my career. So far as I can tell from online testing, my IQ is distinctly higher than the (already quite impressive) Less Wrong average. It is OK not to be very good at mental arithmetic. (Having said which: If there were something important riding on it, I'd be more careful and I'm pretty sure I could do it reliably. I did a few more to check this and it looks like it's true. So I may well in fact be better at multiplying 2-digit numbers than you are. But the point is: this is not something you should expect to be easy, even if it seems like it should be. And the other point is: Even if you are, in some possibly-useful sense, less intelligent than you would like to be,

For folks who post here morale and akrasia are usually much bigger problems than brain hardware.

... You are not too stupid. You are really, really, seriously, not too stupid. That's something that you might want to work on, but it's not a general intelligence failure. There are some tricks that can be learned (or discovered) and employed to multiply by specific numbers more quickly; alternatively, practice will help to speed up your mental multiplication.