In context of his post, I thought it was clear he was making a joke about crazy people who visit math professors (and the like) with their pseudo math written on the side of a grocery bag explaining the theory of everything and the meaning of life, only to force the professor to tell them they are wrong.
If anybody is on the fence about going to grad school for mathematics, you should listen to this. It is the truth.
Not going to take me up on the hug on behalf of economists everywhere? lol j/k
BTW, nice video 
BUMP
So I’m chilling on Facebook and notice this little ditty on there…
"If you choose an answer to this question at random, what is the chance you will be correct?
a) 25%
b) 60%
c) 50%
d) 25%"
I believe there is a correct, non-paradoxical answer to this question that is not part of the choices. Do I have to relax the assumption that I’m constrained to choosing one of a-d? Am I just stupid?
As defined, the question itself is too vague and poorly defined…If you believe there is one non-paradoxical answer, that is your belief . Whether you choose to restrict your assumption is your choice given it is your belief. As for asking this question, “Am I stupid,” you do realize you give permission to somebody to say that you are stupid when you ask this question, right?
There are several undefined variables that make it impossible to come up with an answer more specific than "it depends. " e.g., what defines correct here? Can I pick more than 1 choice? Is the correct answer listed here?
It’s one of those stupid facebook chain questions. It’s a satirical paradox poking fun at probability.
Anyways, I’m taking Intro to Probability next semester (MA 421), my advisor told me it was the 2nd hardest course in the department. Any experience guys? Also I haven’t taken Linear Algebra yet. So plan on taking both of those courses at the same time.
Why are you taking a 400 level math course before you’ve taken Linear Algebra?
A 400 level math course will involve heavy proof writing. Also, Linear Algebra is typically a gateway course where they focus more on proof writing than in elementary 100 level courses. If you aren’t intimately familiar with the following, you are in for a world of hurt:
- Combinatorics: (ok, maybe a 400 level course is not necessary, but VERY VERY helpful for probability because you need to COUNT things. Generating functions are also needed later on though.
- Discrete Mathematics: Absolutely essential. If you didn’t have 1), you’ll need to know things about sets, elementary counting rules, elementary probability stuff like Baye’s Formula, conditional probability, basically everything before you start studying discrete (and eventually continuous) distributions (this course(s) should be where you have had previous experience with probability),
- Set Theory: Arguably even more essential than 2, since a lot of the elementary results you will derive rely heavily on 3. Actually, any serious math course will rely on 3.
- Calc III: In the event you actually survive your 400 level probability course, you will need this to derive results for multivariate distributions. Also essential background for things like (auto)covariance.
I don’t know your mathematical background, but it sounds very odd that you are taking the courses you are taking in the order you are taking them.
I’ve already taken Calc 1-3. Currently taking Diff. Equations (341), and 225. 225 is the pretty much the intro course for writing proofs. So propositions, set theory, and relations I know a little about.
Though I’ve been told to take Linear Algebra before probability.
So far, it sounds like you’ve taken the “mechanical” mathematics courses. I say mechanical because that is what it is: Here is problem type A. Here is the formula/method for solving A. These courses are in all honesty, distilled versions of Real Analysis. You’re told what something is and the emphasis is on how to solve it, but you are never required to question why it solves it.
But that’s not even your biggest problem. You seem to only have a continuous mathematical background. If you don’t know how to work with discrete objects, you are already at a disadvantage compared with the other students that are taking that class. Not only do you have to catch up on learning that stuff, you also have to learn the actual material in your probability course. For instance:
Can you prove DeMorgan’s law’s if there are only two sets A & B? What about if A and B represent finite intersections of sets?
Do you see how to use of DeMorgan’s laws to prove the other?
Can you prove/use the Binomial Theorem?
Given 3 sets and their intersections/unions of elements, can you find a the number of elements that are in a specific combination of unions/intersections of those sets? What about if there are more than 3 sets?
How many divisors does 1030 have that are not a multiple of 5, 7, and 33? How many relatively prime divisors does 1030 have?
Finally, probability problems are typically word problems. For instance: From an ordinary deck of 52 cards, what is the probability that the 6th card is a 10 given than three 10’s have already been drawn?
You should definitely listen to whomever else told you to take Linear Algebra before Probability. You actually have to rely much more on abstract reasoning to get through it instead of using Maple/Wolfram/nameyourCAShere, so use it to build your proof writing and abstract reasoning skills. You should also take Discrete Math as well, just so that you have a foundation to build Probability on. Otherwise you are setting yourself up for failure.
It’s also entirely possible that that 400-category Intro to Probability course is in fact one of those probability courses designed for non-math majors to fulfill upper-division/stat requirements (and from which real Stats/Math majors cannot use for upper division Math/Stat ).
If that’s the case, then lacking some of the Math background won’t be too big of a problem.
Yeah, I’m required to take 421 to fulfill my statistics sequence for my major. But I’m probably just going to hold off until fall to take 421 I guess.
Hi!
@Azn_Boy: I’m taking Econ 310: Probability and Statistics at UW-Madison. In terms of getting a feel for probability courses, I already took a Finite Math course (which is essentially intro to probability) a couple of semesters earlier. You’ll learn some Axioms of probability, try to prove them with basic knowledge, and also use Permutations to solve equations.)
Here’sa link to my current course. I’d link my Finite Math course, but my teacher only re-wrote the book on the blackboard for lecture and never assigned homework; he just had two exams the whole semester based off random problems he pulled out of the book.
I’m a Biochemisty Major and only had to take up to multi-vector calculus. I’m in a quantum mechanics class now and finally have a use for all that calculus.
For anybody taking Abstract Algebra and doesn’t know this about dihedral groups, there is actually a way to calculate what a reflectionreflection, reflectionrotation and rotationreflection is. (rotationrotation is easy).
Scroll down to “Group Structure” and you will find a set of functions that calculate this. The tricky part is converting the table you might be using for say, D_3 or D_4 to and from the R’s and the S’s. You must relabel the reflections in counterclockwise order like they do with the triangle shown. Once you do the calculation though, it’s just a matter of finding the labels that correspond to the D_n you’re working with.
Another useful identity for dihedral groups is reflectionrotationreflection = rotation^(-1)
Any one you know of any good introductory math books? I’m not talking about textbooks or anything like that. Basically, as I’m in my second year of college my appreciation for math is growing. Since I was pretty much raised into math through school (like most of us), a lot of what I know was just spoon-fed to me and I never really questioned anything as I took it all in. So now I have this strange insecurity and I kinda feel clueless. It seems like there are so many questions I should be asking, but I don’t really know where to start. So yeah. I’m pretty much looking for something that will help start me on my journey, I guess I should say, on math.
Hey everyone. I’m about to graduate with a master’s in statistics. I definitely agree that Discrete Math and Linear Algebra are great precursors to a probability & stats course.
Quick question… I’ve been thinking about trying to get an MBA. How hard would it be to get one come from a math/stat background with no econ/business experience?
so this has been bothering me for a very long time
e^(i(pi)) +1 = 0
so we supposedly proved that statement in Calculus the other day, but I’m not satisfied with the proof, I asked the teacher to elaborate on it, but he just said,
“there’s the proof, math doesn’t lie”
But im not content with just staring at the numbers, I understand the why, but it feels to bland and not complete. Is there any books articles that talk about it? I want something like verbal reasoning.
Ask questions. I will answer them. Then I can better suggest things.
I’ll give you a couple to ask yourself:
- Why does algebra work?
- What is calculus all about?
- How does geometry relate to calculus?
No clue. Ask your advisor.
Is the proof the equation itself? Or did you use Taylor series maybe? It makes sense if you had complex variables since e^(i(pi)) literally evaluates to -1. What was the proof they gave?
it was taylor series.
We first started of with doing MacLawren series for cos, then sin.
and then when he told us to do a MacLawren series for e^(ax), where i = a. We all came to the conclusion that the domain for this function should be + numbers only.
we proceed and then chose pi as our x variable, and came to the conclusion that
e^(ipi) = cos(pi) + isin(pi), and when solved, cos(pi) +isin(pi) = -1, therfore e^(ipi) = -1
but its so mind boggling, that the math isn’t enough. The question, “what does it all mean”, isn’t helping me neither. The more I look into it, the more im bothered by it.
It doesn’t look like you’ve taken complex variables.
Much like the integers are a subset of the rational numbers, and the rational numbers are a subset of the real numbers, so too are the real numbers a subset of the complex numbers. These are numbers in the form
a+ bi, where a and b are real numbers.
The truth is, you’ve been working with complex numbers your whole life. Except you have only been working with complex numbers in the form a + 0*i, where a is a real number.
The whole a+bi additive notation is a convenience. When you see a+bi, you are essentially talking about a coordinate (a,b). Since you’ve been only working with real numbers, you’ve been working with coordinates in the form (a,0) where again, a is a real number.
Think of the xy-plane. You have an x-axis and a y-axis. Yet you cannot solve an equation as simple as x^2 = -1. There is no real number that satisfies said equation. If however, we accept sqrt(-1) = i as a solution, then this opens up the possibility of solving other equations with numbers that have a negative number inside a square root.
I won’t derive the complex number system here, but it can be shown that the sum of two complex numbers and the product of two complex numbers (where multiplication is defined as (a+bi) (c+di) = (ac-bd) + (bc+ad)i) ) always spits out a complex number. You should plug in b = d = 0 to see that the usual multiplication works with this definition. You can also go here for more info
Anyways, with this system in mind, we can now think about the complex plane. Think of the same xy-plane you’ve always used, except now, ONLY the x-axis is composed of every single real number you’ve ever used. The y-axis does NOT have any real number (although again, points ON this y-axis are points whose coordinates are themselves real numbers) (and I suppose since the x- and y-axes intersect at 0, I supopse they have a common point there)Much like in the real xy-plane, when you are on the x-axis, your y-coordinate is 0 (that is, you deal with points of the form (a,0), where a is a real number ) so too, when you are on the complex plane, and you are on the x-axis, you are dealing with points in the form (a,0). Every single point on this line is a real number. Notice your y-coordinate has always been zero. This is because again, you only had need of real numbers. But now you have discovered complex numbers and so you need to consider what something like (1,1) means in the complex plane. Well, it means: go right one, and up one. This is the complex number 1+1*i, or more colloquially, 1+ i.
the e^i(theta) notation is a representation of a coordinate in polar form. It’s like taking two different routes to get to the same point (-1,0). In rectangular coordinates, you start at the origin, walk one unit to the left, and you’re at (-1,0). But with polar form, you declare points by rotation of a radius. So your point on a graph is contingent on a radius r, and an angle of rotation theta. So, rather than thinking in terms of rectangles, you look at points in terms of where they are on a circle or radius r. Since we’re merely looking at coordinates of thr form (a,b) = (rsin(theta),r cos(theta)) , we use the additive notation and write this point, which lies on said circle of radius 1, as cos(theta) + isin(theta). It can be shown that
e^i(theta) = cos(theta) + isin(theta), so this can be taken as truth.
So basically, this whole notation is a fancy way of saying this:
You can take the polar route to the point (-1,0) as follows: Form a circle of radius 1 at the origin (ie, the point (0,0) ), go out the length of the radius (which is one) and stay right there, then rotate said radius eminating from the origin 180 degrees counterclockwise (i.e. Pi). The point I end up at as I hang onto the very radius that is being rotated is the point (-1,0) in the complex plane, which, informally speaking, can be likened to the point (-1,0) in the real plane.
or to put it another way:
e^(i*pi) = cos(pi) + isin(pi) = -1 + 0i = -1.
From algebra, we then get the well-known identity
e^(i*pi) + 1 = 0.