John

Hm. I think [math]a_n = \frac{\sin{n} + 1}{n}[/math] works. WolframAlpha seems to think it does, anyway.

I'd say O(n). The innermost loop only iterates through four values, since k starts at N - 2.

Well, I wouldn't necessarily say he's a crank, but he is an ultrafinitist, so while his videos may be decent overall (and I've heard they are), bear in mind that he's philosophically opposed to infinite sets, irrational numbers, and even finite numbers large enough that they cannot be physically represented. This goes against the mainstream foundations of modern mathematics. He's also attempted to reformulate trigonometry into rational trigonometry, which is very much fringe mathematics. As an example of his position, a few articles expressing his views can be found at his page here: http://web.maths.unsw.edu.au/~norman/views.htm I don't know how much all this will factor into any videos he's done on geometry, but it's something to look out for.

What you're asking about is the inverse, not the converse, and the inverse is not necessarily true. For instance, consider [math]\sum_{n = 1}^{\infty} (-1)^n a_n[/math] where [math]a_n = \begin{cases} \frac{1}{n^2} & n \textnormal{ is odd} \\ \frac{1}{n^3} & n \textnormal{ is even}. \end{cases}[/math] Then our series is [math]\sum_{n = 1}^{\infty} (-1)^n a_n = -1 + \frac{1}{8} - \frac{1}{9} + \frac{1}{64} - \frac{1}{25} + \frac{1}{216} - \frac{1}{49} + \cdots[/math] which converges even though [math]a_n[/math] is not monotonically decreasing. The converse of the alternating series test would be that if the series is convergent, then the absolute values of the terms are monotonically decreasing and the limit is zero. Since the converse is the contrapositive of the inverse, we see that the converse is not necessarily true either.

But it's clear what the relationship must be. We must have [math]\varepsilon < \min(r, b - a)[/math]. If you want an equality, use the fact that [math]\min(x, y) = \frac{x + y - |x - y|}{2}[/math].

In the U.S., there are loads of different available scholarships and grants, though of course there's no guarantee of getting them since they're mostly competitive or tied to certain qualities not every student will have. The government offers Stafford loans, which are granted regardless of financial need, though whether interest accumulates during your degree does depend on whether you have financial need, there is a maximum amount you can take out each year and overall, and how much you can actually take out each year is tied to the cost of attendance for your school. There are also Pell grants available from the government, though they're based on financial need. Individual states may also offer some assistance. For instance, I'm from Louisiana, and there's a program there called TOPS, which pays full tuition and fees for any public school in Louisiana for four years as long as you take certain classes in high school and achieve a certain GPA and ACT score (and assuming you attend college shortly after high school). Other than that, it's similar to what hyper mentioned above. People go to school while working. Given the odd hours that taking classes sometimes imposes on availability for work, some people wind up working multiple jobs. It's difficult, especially in a degree like physics, but it's doable. Of course, there are also online degrees available in some subjects, or at least individual online courses available. The perception of online classes/degrees varies (as an example, the medical school in the city I come from won't accept online classes for any of its specific prerequisites, even the non-science ones). Another option is to prioritize some job and take classes during non-business hours, though this may entail taking far longer than four years to complete a degree. Depending on the job and possibly the degree, an employer may even offer some financial assistance.

C has a very simple syntax, and those seven statements constitute pretty much all there is in terms of controlling the execution of code. Other than that, it's a matter of becoming familiar with the various data types, operations we can perform on those data types, knowledge of the functions built into the standard library, and construction of new functions. There are also sometimes shortcuts for certain statements, e.g. in C, we have the ternary operator ?:, used like this: condition ? statement1 : statement2; which can take the place of if(condition) { statement1; } else { statement2; } Other languages (like Python) provide more shortcuts to performing various actions, more built-in functions, etc. These collectively are referred to as "syntactic sugar." For instance, the Hello World program in C is #include <stdio.h> int main(void) { printf("Hello world.\n"); return 0; } while in Python it's print("Hello world.") So the number of words required to write a program depends on the language. Still, I'd say the seven control statements I listed in my previous post are enough to at least express decent programs in pseudocode.

I'm not sure exactly what you're asking, but in terms of program control, if we look at C, we have goto (as noted) to immediately execute some labeled statement, if/else to make a decision based on some condition, switch to enumerate a variety of outcomes based on the value of some variable (similar to an "if/else if/else if/.../else" statement), for, while, do/while to loop through the same block of code as long as some condition is met, break to immediately exit a loop and move on to the first statement after the loop, continue to immediately force another iteration of a loop without executing further code within the loop, and return to allow a function to pass some result back to whichever function called it.

Just noticed this should say "counterclockwise" rather than "clockwise." We could go clockwise too, I suppose, but then we'd need [math](|z_1|, \theta_1) < (|z_2|, \theta_2) \iff \left(|z_1| < |z_2|\right) \textnormal{ or } \left(|z_1| = |z_2| \textnormal{ and } \theta_1 > \theta_2\right)[/math], since angles increase in the counterclockwise direction. Well, in this case, the inequalities are just between magnitudes and angles, both of which are scalar values.

I think it's more accurate to say that certain physical systems can be modeled using the mathematics of complex numbers. Whether complex numbers (or indeed, real numbers) exist in their own right is a matter of philosophy. Edit: One representative discussion can be found here: http://philosophy.stackexchange.com/questions/451/do-numbers-exist-independently-from-observers.

There is an imaginary solution mathematically, but what I was referring to was the physical interpretation. That is to say, physically, there is no time at which the ball is at a height of four meters so long as it's following the parabolic trajectory defined by -t2 + 3 = h. We mustn't confuse the mathematical model of a system with the system itself. As for applications of complex numbers in physics, I'll have to leave that for those who know more about physics and applied mathematics. My current knowledge there doesn't extend much beyond knowing that there are applications.

Alright, I see what you're saying. The situation then is that you're standing at a height of three meters, dropping a bowling ball, and then trying to determine at what time the bowling ball will be at a height of four meters. But a height of four meters doesn't lie on this parabola, so there's no solution. Of course, in practice, you could drop the bowling ball onto a machine that will catch the ball and launch it with enough force to send the ball four meters or higher into the air, but then, mathematically, you're dealing with a different parabola. Edit: Just randomly decided to search for "imaginary time" and this came up: http://en.wikipedia.org/wiki/Imaginary_time. But it's something you'd probably want to discuss in the Physics forum.

Well, we can always just go by the magnitude to get a sense of relative sizes. We just have to accept that in that context, certain apparently unequal numbers are equal in magnitude, e.g. 1 has the same magnitude as i and -1. The difficulty arises when we try to put the complex numbers into some order such that two unequal numbers have unequal positions in the order, e.g. even though 1 and i have the same magnitude, if we want to order the complex numbers, we'd like either i < 1 or 1 < i (but not both), and fulfilling either requirement can be tricky, especially once we start adding operations into the mix. Hm, I'm not sure what you're asking here. The parabola has real solutions for all t, but in practice the Earth will keep the bowling ball from following the parabola past 3 meters, so in practice (disregarding wind and other such adjustments we'd need to make), the bowling ball will follow the parabola until it bounces, then follow another until it bounces again, and so on until it comes to rest. The parabola only describes the ball's path from the moment you throw/drop it until the moment it collides with some other object, so at the risk of upsetting any philosophers of science lurking around, the fact that the graph of the parabola continues down past h0 - 3 meters doesn't have a particularly profound meaning. Edit: To answer your question, the solution is real not imaginary, and it lies one meter under the ground.

I'll try. Just let me know which part(s) need clarification beyond what Wikipedia can provide. Yes. We can induce a total order on the complex numbers by moving up along the imaginary axis and then right along the real axis, i.e. [math]a + bi < c + di \iff [(a < c) \textnormal{ or } (a = c \textnormal{ and } b < d)][/math]. So for example, i < 100i < 1 < (2 - 100i). It's an odd sort of ordering if we're looking at the numbers in terms of their conceptual magnitude, but as far as I can tell it does satisfy the axioms. We can also induce a partial order on the complex numbers using the concentric circles of magnitude mentioned earlier, counting clockwise around each circle before moving to the "next." It's probably easier to think about this in terms of polar coordinates, in which we can say that [math](|z_1|, \theta_1) < (|z_2|, \theta_2) \iff \left(|z_1| < |z_2|\right) \textnormal{ or } \left(|z_1| = |z_2| \textnormal{ and } \theta_1 < \theta_2\right)[/math]. However, this is a bit unsatisfying since it means, among other things, that 1 < -1.

We need to be careful with our terminology here. The standard order we use with the real numbers is not a well-order, since a well-order requires that any non-empty subset has a least element, which is not true of the reals (consider, for example, the interval (0, 1)). It is, however, a total order. If we assume the axiom of choice, then we have the (logically equivalent) well-ordering theorem, which states that any set can be well-ordered. So while the standard order on the reals is not a well-order, if we assume the axiom of choice, then there does exist a well-order for the reals, as well as a well-order for the complex numbers. Whether the order is meaningful, much less useful, is another story. However, what we can't achieve with the complex numbers is an ordered field structure, since the square of any non-zero element of an ordered field must be positive, but we have i2 = -1. Edit: If we try to order the complex numbers simply according to their magnitude, then we find that infinitely many complex numbers share any given magnitude. Looking at the complex plane, what we see is each magnitude |z| represented by a circle of radius |z|. This is what studiot was saying. So for instance, looking strictly at magnitude, -1 = i = 1 = -i. What we can do in that case is define the equivalence relation ~ such that z1 ~ z2 if |z1| = |z2|. But all we accomplish with that is mapping each equivalence class to a real number, thus it's not exactly a mathematical breakthrough and doesn't really accomplish what we want.