Question for Mathematicians: How do you think?

Question

I'm curious about people who are very good at math. How do you think? When you see a complex equation how do you think of it as? Does an image of how it looks on a graph automatically form in your mind? Can you easily distinguish where the different parts of it came from? Does it "talk" to you and tell you what it is, or is it a big mystery until you actually run some numbers through it?

How do you perceive things?

Answer 1

Hmm. By nature, I'm probably a pretty good mathematician (got a Harvard PhD in the subject when I was 19). I think a big part is dealing comfortably with the levels of abstraction. I'll add polynomials as fluently as I add numbers. I'll deal with vector spaces made up of functions just like I'll deal with one made up of 3-tuples. When I first saw a Rubik's Cube, I took out pencil and paper and tried to view it as a permutation group (I didn't successfully solve it that way, but that was my early instinct).

There's a theory that there are a lot of different kinds of "intelligences", and that they amount to being good at recognizing different kind of patterns. If that's valid, I'd say that in math the patterns center on logical relationships (and also chains of inference).

Certainly, for me it is NOT the case that I understand geometry particularly well. Heck, I'm probably below average in that kind of recognition. Rather, it's in the LOGICAL structures where everything seems to be clearer to me than to most other people.

Answer 2

99% of people who are good at math are so because they use it so much. And it begins to be just like anything you do frequently. You get to the point where some of it you can do subconciously (people with more experience can do more without thinking). You asked about imagining a graph. If I saw a y= function, my natural instinct would be to automatically visualize a graph. For complex functions, it would take a second to figure out the graph, and for very complex functions I wouldn't be able to figure out the graph without a computer. But for instance, if I saw "y=x^2" I would automatically see a big "U" in my head. I guess because that's what it means. Like when someone says "dog", you see a picture of a dog in your head, it works the same way.

For complex equations, I would generally just solve it algebraically. This is just a step-by-step process, although sometimes you may do several steps at once in your mind. With practice, you're able to solve more and more complex equations without a pen and paper.

Now you specified, "good at math". But I think maybe you're asking about geniuses. The .01% of people who have some unexplainable mathematical abilities. The people who can do long math operations very quickly in their head via means unknown even to them (although, in japan (or is it china?) a lot of students are so good with an abacus that they can imagine it in their head and actually solve very large multiplication/power problems extremely quickly just by seeing the abacus in their head... it's weird)

But for most people, it's just a matter of practice. And I also think that most people experience numbers differently. I think the majority of people see the numbers in the head as they work a problem, like they were working it on paper (although much faster). Other people perceive numbers differently, in ways I can't describe, but I think that the important thing isn't so much being able to manipulate the numbers (we have computers for that) but being able to manipulate equations, to solve problems, and to understand physical problems.

Answer 3

I usually read the questions MORE than three times and highlighting ALL the given information in the questions or graphs. Then identify EXACTLY what the questions ask me to solve. After all that, trying out DIFFERENT METHODS to solve the problem is the next step. It all comes down to how motivated you are at trying to solve a particular problem, I guess…

Besides, everybody is a Mathematician. But not everybody chooses to be one.

Answer 4

I get drunk and then I ask the question what it thinks about itself. It generally answers with something like, well x wants to equal 4 because otherwise you will feel really bad in the morning. Then I look at 4, and realize that it is the magical answer to the equation, (3 tries to be the answer, but if 3 were the answer, zombies would invade the world, and nobody would want that). So clearly 4 is the answer.

Answer 5

Complex function!
It's not as simple as u can think of,
You can imagine, one simple example of complex function,
Integer number = 2
Integer number = 3
Davide it , 2/3 = 0.666666666666>>>>
So, value of this division, is indefinite still,
it's a complex quantity. it will have One pole, means, it's division will become, Infinite at one point.
so, it's nothing but single order complex function.
Similarly, another types of complex function, Will have Pole 2 times, 3 times etc.
So, That functions value is becoming indefinite at that time, either zero, or infinite or indefinite.
So, How can we solve, the Indefinite part of complex function?

Simply, using the full function as sum of two quotients.
i.e. perfect part + complex part
For above example 2/3 = 0.666666666666......
where 0 is perfect part
0.66666666666........, is imperfect part.
Imperfect part can be solved by introducing one Calculated key constant value = i
so, 2/3 = 0 + (0.666666,,,,,) i
So, we are going to multiply such an key constant that will suppress our complex recursion of solution.
This is all about complex function.

Answer 6

there is nothing like a patteren , whenever we get ne question we just think wht it is and then think with some simple methods and we can only say tht we have same mind which u have we are not doing neting special accept using it.ya one thing is clear we are not fearing from ne problem just be confident and try to solve it.thats it

Answer 7

first of all we think with our brains as you do.second do not be afraid of the problems.third write down the things you are given

Answer 8

practice, practice, practice
then it will just click for you
keep trying