Ask me any pre-Grad Studies Math question and I’ll try to answer and walk you through any difficulty in “getting it.”
This is the inspiration for this thread.
It may take five days for me to be able to start answering, though. Things will probably have quieted down enough by then. No doubt others will beat me to the punch with answers if I am not able to parcel out some time before Thursday, April 9th.
- Jack…
is BACK!
Does geometry count? If so:
In proposition 31 of Lobachevsky’s Theory of Parallels we’re introduced to the ‘boundary line’ or oricycle, which is the curve where all perpendiculars drawn from the midpoint of all possible chords are parallel to each other.
If I understand it properly (which I’m pretty sure I don’t), that means you can in theory extend the oricycle 360 degrees, forming a circle - and again, no matter what chord you draw, the perpendicular will be parallel to the others. Wouldn’t this result in, say, a semi-circle where the diameter is composed of two parallel lines? If so, doesn’t the oricycle need to be, by definition, infinitely large, because all the diameters are parallel, and so the center of the circle is…uh, something which my background in Euclidean geometry has no explanation for?
In the Euclidean plane, yes, the oricycle would be infinitely large. But in the hyperbolic plane, it’s not. See Wikipedia for a picture of one.
That may have come across as unfair to questioners and a bit silly. While others are of course free to chime in, I feel I have to assume SOME responsibility for “my” thread.
But it’s moot at this point. While I was unable to connect yesterday, I will be able to participate this afternoon and tonight, and the same for Tuesday (both) and Wednesday morning. These may be token participations, but I will be present (and unaccountable for. :)) By late afternoon Wednesday, I should be able to take on several questions. 
- Jack…
is BACK!
How can mathematicians claim to be so smart when they keep trying to convince the world that pie are squared?
I’m gonna throw a PI in your face, inspector, if you don’t come up with something more “SERIOUS”,*** dog*** –
AND in Planck time!
- Jack…
is BACK!
What are matrices used for?
I know I was there for that day of algebra, but I think I must have been sitting next to a pretty girl or something, because there’s just nothing on that 3x5 card in my brain.
From Wkipedia on Linear Algebra:
“One of the applications of linear algebra is the solution of simultaneous linear equations. The simplest case is when the number of unknowns is equal to the number of equations. Therefore, one could begin with the problem of solving n simultaneous linear equations in n unknowns.[1]” *
Matrices in the most basic form contain numbers. They can also be entirely variables, or a mix.
Something I found out more recently was that a certain kind of matrix could take data from a 3-D space and make a “visual space” for computer graphic simulation. IOW, you can program something like a space vehicle moving in a simulation, then use such a matrix to plot the actual graphics across a PC screen, making them look real. You could also project a real-world 3-D situation. For instance, from actual space/motion data from a tape of an auto in motion.
- Jack
The rules of matrix manipulation formailize such techniques for any n equations of n variables.
OK, but how does it happen in the real world, that you have a system of simultaneous linear equations that need solving? I mean, where do they come from in the first place?
Explain Euler’s Identity.
I’m actually in the middle of writing a numerical solver for ordinary differential equations, which is one of the most important pieces of software in any engineering discipline. Over the course of solving a differential equation, my software will solve hundreds of thousands of systems of simultaneous linear equations.
Solving simultaneous linear equations is only one application of linear algebra, though. It’s like calculus in that coming up with activities that don’t use linear algebra in some way is extremely difficult. In particular, modern statistics makes extremely heavy use of matrix methods, so any discipline that uses statistics–and that’s pretty much all of them–uses linear algebra.
(Underlining mine.)
It seems to me that ultrafilter’s “numerical solver for ordinary differential equations” leads to the same questions: “OK, but how does it happen in the real world, that you have a system of ordinary differential equations that need solving? I mean, where do they come from in the first place?”
I, at least, think of an application “in the real world” as something like an engineer calculating stresses on a bridge, or that statistician above finding the probability of an event. Has anyone got an example or two at that level?
I just tried this in this thread. I’ll repeat the basic outline here, but the linked posts will really contain the explanations.
A) Realize that complex numbers represent combinations of rotation and scaling (which we can also think of as ratios between two-dimensional positions). In particular, i means “rotate 90 degrees”. (More detailed explanation of this). This is the most important thing to understand; everything else follows almost immediately from this.
B) Take the definition of e^z as “The ratio between the final and starting value of a quantity which grows for one unit of time and whose rate of growth is always z * its current value”. In particular, using our previous observation about i being 90 degree rotation, we have that e^(ix) is “The total effect after one unit of time if I keep moving in such a way as that my velocity at any moment is x times as large as my displacement from the origin, but rotated 90 degrees from this.” I.e., “The total effect after one unit of time if I keep moving in such a way as that, while facing the origin, my velocity is x times as large as the distance between us but pointing to my side”.
C) Since I am constantly moving to the side, neither forward towards the origin or backwards away from it, we see that my distance from the origin is neither decreasing nor increasing; thus, I must be tracing out a circle around the origin. Since my velocity is x times as large as the radius of this circle, we see that I am in fact moving at a speed of x radians per time unit [this is just the definition of “radian”]. Thus, after one unit of time, I will have rotated x radians. And so we can conclude that e^(ix) = rotation by x radians. [More detailed explanations of parts B) and C) on this page]. In particular, e^(ipi) is rotation by pi radians, which is to say, exactly half of a full revolution; since this is the same as multiplying all co-ordinates by -1, we have that e^(ipi) = -1.
D) Finally, optionally, one can easily further establish that rotation by x radians is also the same as cos(x) + i*sin(x), since this observation is essentially just rephrasing the trigonometric definitions of cosine and sine. See this post. It’s not really necessary for understanding the gist of Euler’s theorem, though.
Explain calculus in a sentence or two and how it might be useful in everyday life for the everyman.
I’d say that calculus provides tools to evaluate changes in related things. IF you can establish a mathmatical relationship you can predict the impact of changes one place on other things. A real world example would be figuring out whether increasing the price of an item will result in a net increase (get more per unit) or a net decrease (lower volume of sales) in revenue.
That is my summary without using much math at all.
What’s up with Fourier analysis? Applications actually seem to make sense to me, but I don`t understand the concept and never had a clue what one actually ‘does’. For instance, if you are learning about Fourier transforms, what is your homework going to be like that night?
How does calculus do that more than algebraic equations and statistical correlations?
There are certain relationships that simply can’t be expressed without calculus. For instance, Newton’s law of cooling states that the rate at which an object’s temperature approaches that of its surroundings is proportional to the difference between the two temperatures. This is not an algebraic equation, and any attempt to express it as an algebraic equation will be deficient in some way.
Interesting. I have no idea what that really means, but thanks for the example 
I guess I’ll have to add physics and math to the ever-growing “list of things to learn” for if I ever return to college.
Thanks! That actually makes sense!