How smart do you have to be to study Mathematics as a Major ?

[S2000magician]

If you decide now on a math major, can you change it later?

Much of upper division mathematics – especially pure mathematics (algebra, topology, number theory, analysis) – involves constructing proofs. This requires an absolutely solid understanding of the theory, skill in logic, and a soupçon of creativity. Where do you see yourself in those areas?

For the first, let’s use your calc 1 for a quick test: without peeking now:

• What, exactly, do we mean when we say that the limit of f(x) as x approaches c is k?
• What, exactly, is the derivative of f(x) with respect to x?
• What, exactly, do we mean when we say that f(x) is continuous at x = c?
• What’s the quotient rule?
• What’s Rolle’s Theorem?

That should be enough.

[SamCryBaby]

1. As we approach x from both sides of c , y seems to get closer to k ? So basically without ever touching the actual point of C, (C-g,C+g) all get closer to k while g is as tiny of a number as you can imagine and even smaller than that.
2. The derivative of f(x) would be the slope of the graph at that particular point.The whole formula for computing it would be the same for whatever slope we desire to calculate between two points .However the best estimate is to imagine these two points getting very very very close.I think the formula would be an approximation since we are never close enough but I guess the whole notion is the slope at a given point.The rate of change is a byproduct of computing the slope.
3. I am not sure on this one,one would be where the limits on both sides would be equal to the actual value of the function, or that we could draw the graph without talking our pencil of the paper at that specific point and its not so smaller and not so larger horizantal neighbors.
4. I know the formula but not th underlying principle.But my phone is not friendly for writing down the formula in the correct format.However I think I should look at the underlying principle.
5. IDK honestly .I googled it though

Sorry for the bad layout.I am on my phone.

[S2000magician]

“Exactly” means, well, “exactly.”

1. Given any ε > 0, there is a δ > 0 such that . . . .

[Greenman72]

@S2000magician

• What’s Rolle’s Theorem?

Named after Myron Rolle, an NFL player and Rhodes Scholar.

(He wasn’t a very good NFL player, and he attended Florida State, but by mere virtue of the fact that he’s a Rhodes Scholar and played football professionally, I think this qualifies as BSD.)

[S2000magician]

@Greenman72

@S2000magician

• What’s Rolle’s Theorem?

Named after Myron Rolle, an NFL player and Rhodes Scholar.

(He wasn’t a very good NFL player, and he attended Florida State, but by mere virtue of the fact that he’s a Rhodes Scholar and played football professionally, I think this qualifies as BSD.)

Ding, ding, ding, ding, ding!

Give that man a cheroot!

[bchad]

I figured Rolle’s theorem was: “If you can, get a Royce.” It then became known as Rolle’s Royce.

As for the limit and derivative definitions, I seem to recall it gets worded in the following way: K is the limit of f(x) at X if for every arbitrarily small e, there is an e’ such that f(X+e)=K+e’ . Hmmm something doesn’t seem quite right here because as e shrinks to 0, e’ is supposed to as well.

Continuity is a bit easier once limits have been defined. I believe that continuous functions are functions where for all X in the domain, the limit for positive e is the same as the limit for negative e, AND the function has a legitimate real (or complex, if you’re doing complex numbers) value at X. If a function has a limited domain, there’s presumably some way to word it so that the points at the extremes of the domain only have to be approached from one side.

Im pulling this out of my head from several decades ago, and didn’t look at Wikipedia. Am I close??

[system]

My cousin studied applied math at Harvard. Is he a baws?

[S2000magician]

The point is that mathematics involves very precise definitions; if you don’t understand those definitions, you cannot master mathematics.

The answers, for whatever they’re worth, are:

1. Given any ε > 0, there is a δ > 0 such that |f(x) − k| < ε whenever 0 < |x − c| < δ.
2. It’s the limit as h approaches zero of (f(x + h) − f(x)) / h.
3. The limit as x approaches c of f(x) = f(c).
4. d/dx (f(x) / g(x)) = (df(x)/dx × g(x) – f(x) × dg(x)/dx) / (g(x))²
5. Given a function f(x) continuous on [a, b] and differentiable on (a, b), there exists a point c, a < c < b, such that d/dx(f(c)) = (f(b) – f(a)) / (b – a).

[Palantir]

Sam, do you have any facility with writing code? Does it come to you fairly naturally or do you struggle with it?

[SamCryBaby]

@Palantir

Sam, do you have any facility with writing code? Does it come to you fairly naturally or do you struggle with it?

I have been OK with Python,once I did practise enough.Trouble is I don’t have any yardstick to gauge myself with.I have been pretty good at discrete math, if that counts.The trouble is using my brain was not my top faculty before coming to the U.S.Only after really experiencing the opportunities available here, I really fell in love with science and intellect.The university I attended in the home country was more like a brothel, you only have to come from a disadvantaged background like mine (educationally speaking) to appreciate the learning.

[IsThereAny]

@SamCryBaby

@Palantir

Sam, do you have any facility with writing code? Does it come to you fairly naturally or do you struggle with it?

The university I attended in the home country was more like a brothel

Haha wtf

[S2000magician]

@StallionDis

My cousin studied applied math at Harvard. Is he a baws?

No.

I’d be more impressed with a mathe degree from Harvey Mudd than Harvard.

[Mobius_Strip]

@S2000magician

The point is that mathematics involves very precise definitions; if you don’t understand those definitions, you cannot master mathematics.

The answers, for whatever they’re worth, are:

1. Given any ε > 0, there is a δ > 0 such that |f(x) − k| < ε whenever 0 < |x − c| < δ.
2. It’s the limit as h approaches zero of (f(x + h) − f(x)) / h.
3. The limit as x approaches c of f(x) = f(c).
4. d/dx (f(x) / g(x)) = (df(x)/dx × g(x) – f(x) × dg(x)/dx) / (g(x))²
5. Given a function f(x) continuous on [a, b] and differentiable on (a, b), there exists a point c, a < c < b, such that d/dx(f(c)) = (f(b) – f(a)) / (b – a).

I think the emphasis on math definitions and proofs is absolutely terrible, destroys intuition, emphasizes memorization, kills the enthusiasm and pleasure from doing math (yes, solving math problems is highly enjoyable), and for the most part is completely worthless to most students who don’t intend to pursue a PhD.

so basically,

1. f(x) can be made arbitrarily close to k by picking x which is close to c.

2. the instantaneous rate of change of f(x) with respect to x

3. f© exists and f(x) can be made arbitrarily close to f© by picking x which is close to c.

4. Looking for an expression that quantifies the impact of an infinitesmall change dx to the quotient of two functions h(x)=f(x)/g(x)? Well the product rule is easy to remember and understand intuitively: turn the expression around so that f(x)=h(x)g(x), and let’s say f(x) represents the area of a rectangle while h(x) and g(x) are its sides. How much does the area change with respect to x if we increase both sides a little? h(x) increases to h(x+dx) and g(x) increases to g(x+dx). The “length” increase is then h(x+dx)-h(x)~h’(dx) and correspondingly the width increase is ~g’(dx). So the “extra area” added is f(x+dx)-f(x)=h’(dx)*g+g’(dx)*h+h’(dx)*g’(dx). Clearly then the change in the area f(x) with respect to a small change dx is [f(x+dx)-f(x)]/(dx)=h’g+hg’+h’g’(dx), and the last term drops out since it approaches zero as (dx) approaches zero. So now that I know the product rule f’=h’g+hg’, I can immediately come up with the quotient rule by substituting h=f/g and solving for h’: h’=(f’g-fg’)/g^2.

5. Right, suppose a car leaves town at a certain time and travels at constant speed until it gets to another town some time later. Another car leaves the same initial town at the same time as the first car, and drives at a continuous variable speed until it gets to the other town at the same time as the first car. Clearly somewhere along the way the second car’s speed was the same as the first car’s constant speed - it can’t be higher at all times if they arrive at the same time, and it can’t be lower at all times for the same reason. It must have been lower at certain times and higher at other times, so if it is continuous and there are no jumps in the speed clearly it was equal to the constant speed of the first car at some point along the road.

[S2000magician]

@Mobius_Strip

I think the emphasis on math definitions and proofs is absolutely terrible, destroys intuition, emphasizes memorization, kills the enthusiasm and pleasure from doing math (yes, solving math problems is highly enjoyable), and for the most part is completely worthless to most students who don’t intend to pursue a PhD.

Out of curiosity, have you ever taught university mathematics?

(Serious question; not snarky.)

[Palantir]

Math is a set of definitions, claims, and proofs, how can you have too much emphasis on them?

[bchad]

It’s useful to know how to prove things and it’s interesting and sometimes useful to see and learn what is required to prove something that seems intuitively true. However, the theorem-proof, theorem-proof style of teaching does alienate a lot of people who probably would be good applied mathematicians, and that is sad. I get that the theoreticians and abstractioners need to do the proofs.

on the other hand, there is a view of math as the analysis of logical systems of thinking: i.e. what conclusions can be drawn from what premises. I wish that there were more classes that took this approach with respect to natural language, because there is value there, too.

[comp_sci_kid]

Math is modern day magic. Master it and you can master anything

[S2000magician]

@bchad

However, the theorem-proof, theorem-proof style of teaching does alienate a lot of people who probably would be good applied mathematicians, and that is sad.

High school geometry is nightmarish.

On the other hand, one of the most enjoyable classes I took had the innocuous title of “Introduction to Mathematical Methods”. It was a class in learning how to prove things. We covered some set theory, some point-set topology, some abstract algebra (semigroups and groups, mostly), and so on. It was a really fun class, taught by the gentleman who became my favorite mathematics professor. (He was a bit confused, however, about why an upper division math class had an accounting major enrolled in it.)

[Turd_Fergeson]

@bchad

It’s useful to know how to prove things and it’s interesting and sometimes useful to see and learn what is required to prove something that seems intuitively true. However, the theorem-proof, theorem-proof style of teaching does alienate a lot of people who probably would be good applied mathematicians, and that is sad. I get that the theoreticians and abstractioners need to do the proofs.

on the other hand, there is a view of math as the analysis of logical systems of thinking: i.e. what conclusions can be drawn from what premises. I wish that there were more classes that took this approach with respect to natural language, because there is value there, too.

i think your second paragraph naturally follows and requires the first. The dirty work is training your brain to think precisely and logically (if I know this to be true, then x must also be true) and I don’t think there’s better training than the theorem-proof training. It’s been over a decade since I wrote a proof, but I use the thought process it has developed every day and it’s invaluable. My old bosses, MBAs and one engineer, used to comment that they never met a person who could break down and analyze a problem the way I could. I think they just never worked with someone who went through the training of a pure math major.

[Greenman72]

@S2000magician

(He was a bit confused, however, about why an upper division math class had an accounting major enrolled in it.)

TBH, I’m kinda confused about that, too. I have rarely, if ever, met an accountant who was interested in higher-level math. If you can do high school algebra, you have more than enough math skills to be an accountant.

And it seems that if you enjoy higher-level math, you wouldn’t be interested in accounting, since it’s all about mind-numbing, relatively useless, arbitrary government rules. Sure, some of the ration analysis has a little mathy-ness to it, but not exactly Myron Rolle theorem.