Ask a Mathematician!

[IdiotsOpposite]

I got bored, and I didn't see any mathematician threads since october. So in a fit of boredom, I made this one! Joy!

Got any questions about the way math works? Just need help on a homework problem? Just ask me, here! What could go wrong?

Note: There's only one question I'll ban, and that is questions like this:

"What is 48/2(9+3)?"

Why? Because this question starts flame wars.

 

[_random_]

Does .(9)=1 and why or why not?

 

[IdiotsOpposite]

Does .(9)=1 and why or why not?

Ooh! I've actually got the answer to this in a post I made on a different forum.

Short answer, yes.

Long answer, yes, and here's why: Why .999... = 1, A Treatment With Infinite Series

 

[Crezth]

What is infinity?

 

[Integral]

What is your mathematical background?

 

[mech654]

Why is the Square root of 2 is considered a irrational number?

 

[IdiotsOpposite]

What is infinity?

Infinity is really rather strange. It's not a number, but in some cases we can treat it as one. (Note that this "some cases" does not extend to arithmetic!) The best way to think about infinity isn't as a number, but as a limit, in my opinion. Of course, every math person thinks different things about infinity!

Personally, I use the Projective Real Line, which is like the line of real numbers that's so familiar, but with an "endcap" put at each end of the line, such that the line forms a circle. This "real circle" has one end zero and the other end "infinity". You still can't really do arithmetic with it, though.

What is your mathematical background?

Registered math major in college. My latest completed courses are Vector Calculus and Linear Algebra, and I'm working on Differential Equations right now. After that, I'm off to Real Analysis!

Why is the Square root of 2 is considered a irrational number?

Because it was proven to fit the definition of an irrational number. The proof is simple, and so I can reproduce it here. First note, however, the definition of an irrational number: a number that cannot be expressed as a ratio of two coprime numbers a and b. So let's use this!

First, assume that sqrt(2) is rational. From this assumption we'll derive a contradiction, showing that the assumption is untrue.

Then, if sqrt(2) is rational, it can be expressed as a ratio of a and b, i.e. sqrt(2)=a/b. This can be rearranged to give b sqrt(2) = a. Square both sides and we get 2 b² = a² . What this tells me is that a² is even. Since a² is even, a must be even (this is easy to prove, try it!) and a = 2c for some number c. So we can rewrite this as 2 b² = (2c)² = 4 c² . Divide left and right side by two to get b² = 2 c² . This tells us that b² , and thus b is even. But wait! We said that a was even. a and b cannot BOTH be even, because two even numbers are not coprime. Thus, contradiction! Therefore, sqrt(2) is irrational.

A definition: For those of you who do not know what "coprime" means, i.e. the non-math crowd, two numbers are coprime if they do not share any factors. For example, 3 and 14 are coprime, because the only factor of 3 is 3, and the factors of 14 are 2 and 7. 2 and 6 are NOT coprime, as the factor of 2 is 2, and the factors of 6 are 3 and... 2

Hope that long-winded explanation helped!

 

[Zelig]

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

 

[IdiotsOpposite]

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

I simply remind myself that what I'm doing is the purest of all possible professions.

 

[Azash]

How did you grasp mathematical thinking? Do you have any tips on how to do so?

 

[Monsterzuma]

whats your favorite polynomial? show us a pretty graph.

 

[_random_]

I simply remind myself that what I'm doing is the purest of all possible professions.

And what do you mean by that? Also, what are your thoughts on the following comic?

 

[Crezth]

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

Hey man, nothing is harder than signing a piece of paper or firing someone. Those CEOs work harder than the rest of us combined!

 

[mech654]

Thank you, for the explanation. I am definitely Subscribing to this.

 

[Integral]

Registered math major in college. My latest completed courses are Vector Calculus and Linear Algebra, and I'm working on Differential Equations right now. After that, I'm off to Real Analysis!

It's a fun ride. I majored in math in college (along with economics).

How did you grasp mathematical thinking? Do you have any tips on how to do so?

I can second this question, even now. I "think" topologically and do fine with analysis, but for the life of my I could not do abstract algebra. I think it was a mental block or something.

My answer: it took me a long time to learn how to think like an analyst. I did lots of practice and wrote lots of proofs, many of them poorly. Eventually you build a repository of examples, counterexamples, and "standard proofs" (both strategies and results) and get into a rhythm so that things become intuitive. Again: intuition is relative and grows with you, and for me it took a lot of practice.

You learn tricks along the way. A good chunk of real analysis is judiciously adding 0 or multiplying by 1 in a clever way (at the undergrad level).

I survived topology by having a long list of examples and counterexamples in my head, which also came with practice.

I never learned to think like an algebraist.

 

[IdiotsOpposite]

How did you grasp mathematical thinking? Do you have any tips on how to do so?

Gah. I had a nice long paragraph here, but then my internet froze and I lost it all. Oh well, something for the Rants thread, I guess.

In short, my main strategy throughout most of my learning was threefold: I visualized what I was trying to find. (This often doesn't work!) I memorized handy little rules to help me through. And finally, I tried to go above and beyond whatever the professors were teaching, all the time. Talk about finding limits? I looked up the epsilon-delta method. Talk about the product rule? I proved it, which went a long way toward remembering it.

Basically, I guess my best piece of advice could be to try to go above and beyond whatever you're learning. Once you know the underlying theory, the formula suddenly becomes cake (or perhaps pi ) to remember!

whats your favorite polynomial? show us a pretty graph.

Hmm. I haven't gotten asked that question before! My favorite polynomial would probably be the simple x^2, always the first thing we differentiate, the first thing we integrate. But that's not a very pretty picture. So let's look at y=x⁴-5x²+5, instead. It has a cool W-shape!

As for non-polynomial things, my favorite is the Riemann Zeta function of 1/2 + i x. Why? Well, it's extremely important (particularly in finding the zeros), and it just plain gives a cool graph! For example, here's a graph of the Riemann Zeta function in polar coordinates:

And what do you mean by that? Also, what are your thoughts on the following comic?

Well, I consider mathematics to be pure. You don't need to work with reality or experimentation at all with mathematics. And moreover, mathematics is the only scientific discipline where you can, with only a sheet of paper, prove that something is irrefutably true! That's what I mean by the purest possible profession.

As for my thoughts on the comic, it certainly does paint mathematicians in a better light, doesn't it? But while it's true that not many people will know what you do, I wouldn't say that none of the people who DO know are able to fire you. I should hope the head of the math department wherever you work knows what you're doing!

 

[IdiotsOpposite]

Thank you, for the explanation. I am definitely Subscribing to this.

Thanks for taking the time to read it! Man, this whole thread is an ego-boost.

It's a fun ride. I majored in math in college (along with economics).



I can second this question, even now. I "think" topologically and do fine with analysis, but for the life of my I could not do abstract algebra. I think it was a mental block or something.

My answer: it took me a long time to learn how to think like an analyst. I did lots of practice and wrote lots of proofs, many of them poorly. Eventually you build a repository of examples, counterexamples, and "standard proofs" (both strategies and results) and get into a rhythm so that things become intuitive. Again: intuition is relative and grows with you, and for me it took a lot of practice.

You learn tricks along the way. A good chunk of real analysis is judiciously adding 0 or multiplying by 1 in a clever way (at the undergrad level).

I survived topology by having a long list of examples and counterexamples in my head, which also came with practice.

I never learned to think like an algebraist.

Ah, yes, the infamous multiplying by 1. That's how I learned the integral of sec(x), and I've never forgotten that! (Although I wish there was a better method of doing it!)

 

[pi-r8]

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

 

[IdiotsOpposite]

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

Oh fine. Math major then.

 

[Crezth]

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

Whence come this arrant pedantry?