When I read your answer, I was thinking, (seriously no offense because I know you are really smart) I don't know for sure that this definition works for complex numbers.

It does; complex numbers are just another type of number. We'll get to them shortly.

And then I was thinking that mathematics relies on definitions and deductive reasoning and intuition cannot give the certainty of deductive reasoning, thus it might be a fallacy to think that something simple and intuitive is an accurate model of mathematical reality... then I remembered that it was taught in kindergartens even...

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.

Define "multiplication"

X*Y : I have Y sets of X objects, how many objects do I have?

Exactly.

"addition"

X+Y : I have X objects. I am given Y objects. How many objects do I have?

Perfect.

It's easy to visualize imaginary numbers as another type of object 'x', and I am given y objects. So I have x + y imaginary objects and X + Y real objects.

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).

"subtraction"

X-Y : I have X objects. Y objects are taken away from me.

Right.

Then it makes me wonder what other exceptions to manipulation there is

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".

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.

It seems that my mind lights up with too many questions when I learn math, many of which are difficult to answer. (My professor does not have much time to meet students for consultations and I don't think I want to waste his time).

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:

In Math, I just barely by the skin of my teeth scraped together a pass in Calculus with a C-. [...]“Scott Alexander, who by making a herculean effort managed to pass Calculus I, even though they kept throwing random things after the little curly S sign and pretending it made sense.”[...]I don’t want to have to accept the blame for being a lazy person who just didn’t try hard enough in Math.

Things such as why dividing by zero doesn't work confuses me and I often wonder at things such as the Fundamental Theorem of Calculus.

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.

I am concerned that I am not smart enough

No one is smart enough.

But if you mean, specifically, smart enough to

aspire for rationality

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.

I find it difficult to multiply two 2-digit numbers in my head

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, that is not reason not to aspire to rationality. And the other other point is: It's clear that your intelligence is, at the very least, perfectly respectable.)

Well, About 3-5 percent of the best students in a cohort can expect to get First Class Honours. It basically means 97th percentile, or 95th percentile, depending on the quality of the students. The 75th to 95th percentile can expect to get Second Class Honours.

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 must admit that this question stunned me. I don't actually know.

...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".

I noticed that many articles in the sequences confuse me at times because I can think of multiple interpretations of a particular paragraph but have no idea which was intended. Also, many actions/thoughts of Harry in HPMOR confuse me. I might have interpretations of the events but I don't think those interpretations are likely to be correct. Is this normal?

This seems normal to me. What is intended is very often not an easy question to answer.

I have edited the post though, I think that saying that I am on track to receive First Class Honours in both is too optimistic.

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).

I just really don't get why I don't do well in math, which I assume would be the best measure of one's fluid intelligence. Things such as why dividing by zero doesn't work confuses me and I often wonder at things such as the Fundamental Theorem of Calculus. It seems that my mind lights up with too many questions when I learn math, many of which are difficult to answer.

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).

Is this normal too?

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:

Things such as why dividing by zero doesn't work confuses me

Define "division".

Should I aspire for rationality or am I too stupid?

...

...consistently do relatively well in Physics, Chemistry, Engineering and Programming modules.

You are not too stupid.

I'm in a double degree in Chemical Engineering and Business and on track to receive First Class Honours in both.

You are really, really, seriously, not too stupid.

Yet, I find it difficult to multiply two 2-digit numbers in my head.

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.

Hi! Lesswrong first came to my attention when I read HPMOR. I took a 2-year course in Knowledge and Inquiry - which includes critical thinking and epistemology (also includes philosophy of science). I was a Christian but reading some articles on Lesswrong and reading counter-arguments to Christianity convinced me otherwise (trying to reduce confirmation bias and trying to falsify the belief of Christianity).

Pardon me for taking this opportunity to express one concern I've had for more than a year. I'm a college student and I am concerned that I am not smart enough expect a net gain in utility by aspiring to rationality (added in edit).

I don't do well in Math (about 60th percentile for multivariable calculus), but consistently do relatively well in Physics, Chemistry, Engineering and Programming modules. (consistently in top 8 percent of students in top university in Asia). I'm in a double degree in Chemical Engineering and Business and on track to receive at least Second Upper in both (First Class in one and Second Upper in the other seems to be the most likely outcome, though of course I am striving for First Class in both. I am usually too pessimistic when it comes to grades and honours).

Yet I find it difficult to multiply two 2-digit numbers in my head. I always forget what numbers I was trying to multiply and the progress of the multiplication so far. I tried Dual N-Back and had to work for half an hour to pass n=2. I can't remember numbers and always make tons of errors in my mathematics work (not switching signs for one or more terms when factoring out a negative number, for example, or just plain getting stuck).

I'm worried that my fluid intelligence just isn't enough. I'm also quite sad at the expectation that my fluid intelligence will decrease throughout my adulthood. I can't find any convincing evidence (maybe a study or something) that fluid intelligence cannot be fully described by mathematical ability (if effort exists). Should I aspire for rationality or am I too stupid?

Thanks!

Edit: I'm also in another predicament - I am no longer a Christian, but I still go to church every week. I treasure the friendship and companionship of my Christian friends. They are really nice and caring people. I cannot predict reliably what their reactions would be to me revealing the current status of my beliefs. I meet them only once a week in church and if I were to stop going to church, our friendship would most likely perish.

There are other benefits to going to church as well: Here, church is a marketplace for contacts and relationships. I believe going to church would help me in the future if I were to go into business.

However, my parents and all my relatives who are descended from my paternal and maternal grandparents are Christians, and most of my extended family beyond that are Christians. My parents are devoted Christians and it would break their hearts to find out that I am no longer a Christian. My relatives would judge me and proclaim me a failure. Most of our Asian community would do the same (Where I'm from, it is considered odd, or mad not to have a mainstream religious belief. We are categorized by religion as much as we are categorized by race). Even if I were to succeed financially, they would say that I am not someone to be trusted because I am somehow immoral for rejecting Christianity.

Would anyone care to offer my some advice?