The mathematics/physics/engineering study group thread

[Qhartb]

So what kind of stuff does college algebra cover?

Throw out some topics!

What "algebra" really is can be boiled down to the statement:

if x = y then f(x) = f

For introductory algebra, the idea is usually to apply this repeatedly to solve for an unknown. x and y are almost always reals and f is usually a basic function such as addition or multiplication by a constant or principal square root. Most of these functions are invertible and defined everywhere and when they're not, that's taught explicitly (eg, don't divide by zero, sqrt(x^2) = abs(x), multiplying both sides by zero doesn't help ).

For higher-level algebra, x and y are rarely reals. They're more often matrices, vectors, sets, or members of a generic field, ring, group or other algebraic construct. Instead of caring about x and y, you care about the properties of the set they belong to combined with a set of functions defined on that set.

 

[Saturos]

If 3x>9, then which of the following must be true?

A. x>3

B. x>3

C. x<3

D. x<3

The answer is B because dividing by threes gives x>3

My question is, what exactly is meant by saying ,"dividing by threes" in this scenario?
Stupid simple question, I know, but the logical reasoning just seems to elude me for some reason. I guess I'm just over complicating things way too much. I tend to do that with maths.

Also, what does < and > mean?

 

[dfc05]

If 3x>9, then which of the following must be true?

A. x>3

B. x>3

C. x<3

D. x<3

The answer is B because dividing by threes gives x>3

My question is, what exactly is meant by saying ,"dividing by threes" in this scenario?
Stupid simple question, I know, but the logical reasoning just seems to elude me for some reason. I guess I'm just over complicating things way too much. I tend to do that with maths.

Also, what does < and > mean?

Dividing by three means that you divide both sides of the equation by three. As in:
3x/3 > 9/3
x > 3

What you do to one side you have to do to the other!

The only tricky thing about the < > math is that when you mult/divide by a negative number, you have to flip the < or > sign. But if you just work it out and test a number or think about it, you can figure that out.

< means "less than or equal to."



As for advanced/college algebra, I took a class that did a little bit of abstract algebra. I don't remember much about it except that we used letters in place of numbers, and symbols in place of things like "x / + -". Proving things, going into the basics of algebra like what exactly a "function" is. It was kinda fun. It was in high school though so we only did a couple of months of it. In college you mostly take calculus, and differential equations if you're in math/engineering. There's also discrete math, which is proofs, logic, and set theory, but it's mostly just math majors, comp sci, and electrical engineers. I took it because I was sick of chemistry and discrete math was easier than advanced chem. Then there are a whole lot of other math courses where I look at the course description and I have no idea what the heck they're about.

 

[Saturos]

Dividing by three means that you divide both sides of the equation by three. As in:
3x/3 > 9/3
x > 3

What you do to one side you have to do to the other!

The only tricky thing about the < > math is that when you mult/divide by a negative number, you have to flip the < or > sign. But if you just work it out and test a number or think about it, you can figure that out.

< means "less than or equal to."

Oh ok, so just like "equations", meaning algebraic problems where you find equalities: (i.e. 3x + 4 = 5x + 2 x=1) equations involving only greater than, less than (i.e. like 3x>9) are treated just like any other equation?

 

[mindless_moder]

are math jokes game ? .

All the maths functions were at a party last night and everyone was enjoying themselves except e^x who was on his own in a corner. So then lnx came over to him and asked "why aren't you mixing with everyone else?" so then e^x replyed "I tried to integrate, but nothing happened!"

Why should you never drink and do calculus at the same time?...it's wrong to drink and derive.

bunch of functions sitting in a bar, minding their own business, when a differential operator bursts in; everyone dives for cover, except "e to the x" who sits unfazed at the bar; an air of "you can't harm me" about him..

"aha" says the differential operator, "but I differentaite with respect to y!!"

ZAPP!!

yeah im going to shut up now

 

[ríomhaire]

I had a maths exam today and there were 3 questions I couldn't do. Helplife2.net, you know what to do:

Show that if a and b are non-zero real numbers, then the value of
[U]a[/U] + [U]b[/U]
b   a
can never lie between -2 and 2.

Hint: consider the case where a and b have the same sign separately from the case where a and b have opposite sign.

Let

y=sin^-1 (x/(1+x^2)^1/2)

[SIZE="1"](that's y equals arc-sin of x over square root of 1 plus x squared)[/SIZE]

Find dy/dx and express it in the form

a/(a+x^b)

[SIZE="1"](a over a plus x to the power of b)[/SIZE]

where a and b are natural numbers

Integrate this:
(2x^3)/(x^2-1) dx

[SIZE="1"](two by x cubed over x squared minus 1)[/SIZE]

The limits are 4 and 2

 

[mindless_moder]

Integrate this:
(2x^3)/(x^2-1) dx

[SIZE="1"](two by x cubed over x squared minus 1)[/SIZE]

The limits are 4 and 2

did you try partial fraction decomposition ? , the degree of the numerator is higher than the degree of the denominator so you can do long division , decompose the fraction and then integrate the sum. I THINK.

 

[99.vikram]

Riomhaire:

The first answer is simple. Differentiate the function with respect to a and b and equate them to 0.
You get a = {+b, -b}. Hence proved that 2 and -2 are extreme values.

Second answer can be obtained by replacing x by tan(k). Instant result.
y = k. But dx = sec(k)^2 * dk. Thus dx = (1+tan(k)^2) * dy.
Hence we get dy/dx = 1/(1+x^2).

Third answer can be obtained using product rule for integration. Take the function as [2x^2] * [x / (x^2-1)]
:cheers:

 

[Cavalry]

This may seem stupid in comparison to the maths your all talking about, but I don't get how to find the area of a square that is inscribed into a circle.

If the apothem is 16, and the circumference is 34, how do you determine the squares area?
I know I have to cut the square into four equilateral triangles, but is the side of each triangle the same as the apothem?

 

[ríomhaire]

did you try partial fraction decomposition ?

Don't know what that is. I'm doing secondary school maths here, not third level.

The first answer is simple. Differentiate the function with respect to a and b and equate them to 0.
You get a = {+b, -b}. Hence proved that 2 and -2 are extreme values.

But it wasn't a calculus question D=
I also don't understand why that proves they're the extreme values. Sorry.

Third answer can be obtained using product rule for integration.

Not on our course. In fact our maths teacher once told us there was no product rule for integration, only for differenciation.

I'll look at your other answer later. Thanks for the help all the same.

 

[99.vikram]

But it wasn't a calculus question D=
I also don't understand why that proves they're the extreme values. Sorry.

I assumed you had already finished calculus.
(This particular result is due to the properties of Maxima & Minima.)

Not on our course. In fact our maths teacher once told us there was no product rule for integration, only for differenciation.

Check the Wikipedia entry here.
You'll probably learn this stuff later.

 

[mindless_moder]

sorry you dont really need partial fraction , just do the long division and integrate .
you should get 2x + 2x/(x^2-1) after dividing , which is pretty easy to integrate.

 

[ríomhaire]

Second answer can be obtained by replacing x by tan(k).

Oh god that was so easy. How did you know to replace x with tan(k)?

I assumed you had already finished calculus.
(This particular result is due to the properties of Maxima & Minima.)

You see there were 8 questions on our paper, 3 of which are calculus. This was not in one of these three questions so I didn't even think of using calculus. I was trying to do it with normal algebra. My friend proved it using limits. I tried it again myself today and I accidentally came up with the result that it has to lie inside the range of 2 and -2, instead of of outside it. Annoyed me to say the least.

sorry you dont really need partial fraction , just do the long division and integrate .
you should get 2x + 2x/(x^2-1) after dividing , which is pretty easy to integrate.

You know. It's kinda funny how after doing so many integration questions I now think about them along certain lines and it didn't even occur to me that you could divide it

 

[99.vikram]

Oh god that was so easy. How did you know to replace x with tan(k)?

Whenever you see:
(1 + x^2)^0.5 --> try a cot() or tan() substitution
(1 - x^2)^0.5 --> use sin() or cos()
(x^2 - 1)^0.5 --> use sec() or cosec()
Then the term in parantheses gets simplified and cancels out with dx.

Example:

x^3 * dx / (1 - x^2)^0.5

Let sin(k)=x.
=> sin(k)^3 * (cos(k)dk) / cos(k)
=> sin(k)^3 * dk

 

[mindless_moder]

You know. It's kinda funny how after doing so many integration questions I now think about them along certain lines and it didn't even occur to me that you could divide it

i know EXACTLY what you mean , it happens alot during tests because you tend to overestimate the problems.

 

[xcellerate]

...nevermind

 

[xcellerate]

Last year's equity book value per share was $21.70 . This year's is $25.77 . The number of shares increased from 38,800 last year to 41,490 this year (but maybe there was a stock split; maybe not). This year the company has New retained earnings of $185,200 . Find the value of Net equity issues for the company this year.

A. Net sales equal $55.7 thousand
B. Net repurchases equal $42.1 thousand
C. Net sales equal $42.1 thousand
D. Net repurchases equal $48.4 thousand
E. Net repurchases equal $55.7 thousand

 

[Raziaar]

Can somebody please elaborate to me what exactly is heat?

Like... I figure it has to be on a sub atomic level, but what is it exactly composed of, for example when you have heat radiating off a hot iron or anything else. What sub atomic particles comprises the heat that radiates off molecules or atoms?

 

[AKIRA]

this is the most ****ed up assignment i've ever received(sorry for the length..if tl;dr then that's OK ):

Spoiler

COSC1P03 Assignment 5: Sorting

Due: July 8, 2008 @ 7pm EST, submitted online and coversheet in drop box.

You are working for a PC repair shop called Fixem Computer Systems. Every day, many computers are brought in for service. Normally, the computers are put on a shelf, and any technician who wants, takes the computer off the shelf if: they feel like fixing it, they think it?s an easy fix, or they feel guilty that nobody has fixed the system yet. You see that this must change, so you have a meeting with the service manager.

The service manager doesn?t want to step on any toes, so in order to simulate the system accurately, the manager decides to estimate how much time each job will take and how desirable the job is so that your program will take into consideration that clients with little problems expect fast service and workers prefer fixing certain system-problems over others. (Don?t forget to consider how long the system has been waiting to be fixed).

Create a class that defines a job. A job should have a number of minutes it has been sitting on the shelf, an amount of time that it takes to fix, and a rating out of 10 (1 being awful work and 10 being fun) for how fun it is to fix the problem. In addition, make jobs searchable and comparable.

Based on the definition of a job, define how you believe the system should be sorted so that workers are forced to pick the right machine to fix next. The only caveat is that you must take at least 2 of the 3 fields into consideration. (This makes the jobs comparable.)

Read in one action at a time from the file. Each time you read a job, you will have to place the new job at the right location in the queue based on your personal definition. (Your ?enter? method will not actually work the same way as a normal queue). This is a ?modified? queue so you will have insertion in the middle or even at the front at times.

If you encounter a ?Leave? then you must call the leave method of your queue which simulates a worker taking the next job off the queue to work on it.

Your queue ADT should have all the methods defined in the book and will be implemented using arrays as described in class.

(time to fix, time on shelf, job funness rating)
IE
60 37 5
120 12 8
Leave
40 12 10
Leave
Leave

Please make sure exceptions are taken care of and that in your comments you explain why you sorted the jobs the way you did.

Just output the state of your queue at each state change to an ASCII file.


Marking Scheme:

Assignment 5, sorting & queues:[50]

[Commenting][13]
[2]Variable Dictionary
[5]Description of the purpose of the program & reasoning behind sorting
[2]@param and other @statements used correctly
[4]Every method is commented

[Main Class][28]
[10]Program works and gives perfect output for valid input
[-15]Program doesn?t work for ?normal? cases or doesn?t compile
[6]Program deals with exceptions appropriately
[6]All queue methods exist and work properly **Counts as Exercise 5**
[6]Queue implements searchable and comparable

General Style: [9]
[2]Minimal global variables
[1]Variables and classes have names that make sense
[1]Spacing and tabbing are easy on the eyes
[2]Appropriate information hiding techniques used
[1]Only one main method (unless including test harness)
[2]Output is nicely formatted/spaced

Workaholic Bonus ? You may only choose 1 of the 2: [25]

[5] Used an interface in case this implementation could be improved.
[14] Provided an additional implementation of this queue using linked lists (superior to arrays in this case).
[6] Provided a test harness usable for either queue implementation.

OR

[5] Used an interface in case this implementation could be improved.
[15] Created an alternative way of reading the data ? all data is read into a linked list one at a time regardless of priority and sorting. It is then sorted using bubblesort.
[5] Provided a test harness for the new implementation.

 

[Vegeta897]

Can somebody please elaborate to me what exactly is heat?

What I said in AIM kekekeke.

 

[dfc05]

Can somebody please elaborate to me what exactly is heat?

Like... I figure it has to be on a sub atomic level, but what is it exactly composed of, for example when you have heat radiating off a hot iron or anything else. What sub atomic particles comprises the heat that radiates off molecules or atoms?

Heat ain't a particle. It's energy... as in, the energy from molecules vibrating and bouncing around and stuff.

 

[theotherguy]

Can somebody please elaborate to me what exactly is heat?

Like... I figure it has to be on a sub atomic level, but what is it exactly composed of, for example when you have heat radiating off a hot iron or anything else. What sub atomic particles comprises the heat that radiates off molecules or atoms?

Heat is the kinetic energy of atoms in a medium vibrating. When you place your hand on something hot, it hurts because the atoms of it are literally smashing into the molecules of your hand and breaking them apart. Imagine that a skillet is made of billions of billiard balls connected by springs. When you place it on the stove, the energy of the fire causes the atoms to get knocked out of place, and gain energy. Imagine now that the fire is a cue stick, smacking a billiard ball. The energy of the ball will be transferred to the rest of the collection as it strikes the balls around it, and eventually it will cause the entire collection to vibrate and shutter. If you keep striking the billiard balls on the bottom with cue sticks, the vibrations will become larger and larger until the entire thing breaks apart.

When we say something is "radiating" heat, we mean to say that it is emitting infrared radiation, which is energy released in the form of photons from very high energy atoms. Sometimes as is the case with very hot metals, the photons that are radiated are in the visible range, and we can see a red glow.

Heat can be transferred in a few ways:
1. Convection: This is where a fluid (like air or water) is touching a heat source. The atoms of the fluid are given energy by this source, and the density of the fluid changes, causing hot atoms of the fluid to rise and cooler ones to take their place, slowly transferring heat from one end of the fluid to the other.

2. Conduction: This is where two solids are touching. Heat can move between the solids because high-energy atoms smash into lower-energy atoms, which, by momentum alone, transfers through the entire cool body until both bodies are of equal temperature.

3. Radiation: High-energy photons, electrons, protons or neutrons are emitted by certain chemical and nuclear reactions, as well as by very hot solids. These particles travel through space, and if they strike something, will generate heat by transferring energy to it.

 

[soulslicer]

QUESTION

Nuclear Fusion Reactor.

1. I know how the stuff works, but I am curious to know..Nuclear Fusion is a countinous unstoppable reaction from what I know. Elements would keep getting formed till Iron or Lead if I am not wrong, since it has the most stable Nucleus and lowest mass excess. So why doesn't this happen in the Nuclear Fusion Reactor, what's that stopping it from going beyond helium?

2. Also, it's a ****ing 1 Million degrees in there. Okay, so the magnetic field holds the plasma, but it definmitely can't hold the 1 million degree radiation right? What epic material is that being able to handle that kind of heat without vaporising.

-soulslicer

 

[Raziaar]

Thanks guys! I guess I was just confused when I read the line on wikipedia that said "In the history of science, the history of heat traces its origins from the first hominids to make fire and to speculate on its operation and meaning to modern day particle physicists who study the sub-atomic nature of heat."

I had already known about the vibration thing for years after learning how a microwave works to cook food by exciting it and causing the molecules to jitter around... I was just confused and thinking there was something else involved due to that line on wikipedia.

 

[Saturos]

Heat is the kinetic energy of atoms in a medium vibrating. When you place your hand on something hot, it hurts because the atoms of it are literally smashing into the molecules of your hand and breaking them apart. Imagine that a skillet is made of billions of billiard balls connected by springs. When you place it on the stove, the energy of the fire causes the atoms to get knocked out of place, and gain energy. Imagine now that the fire is a cue stick, smacking a billiard ball. The energy of the ball will be transferred to the rest of the collection as it strikes the balls around it, and eventually it will cause the entire collection to vibrate and shutter. If you keep striking the billiard balls on the bottom with cue sticks, the vibrations will become larger and larger until the entire thing breaks apart.

When we say something is "radiating" heat, we mean to say that it is emitting infrared radiation, which is energy released in the form of photons from very high energy atoms. Sometimes as is the case with very hot metals, the photons that are radiated are in the visible range, and we can see a red glow.

Heat can be transferred in a few ways:
1. Convection: This is where a fluid (like air or water) is touching a heat source. The atoms of the fluid are given energy by this source, and the density of the fluid changes, causing hot atoms of the fluid to rise and cooler ones to take their place, slowly transferring heat from one end of the fluid to the other.

2. Conduction: This is where two solids are touching. Heat can move between the solids because high-energy atoms smash into lower-energy atoms, which, by momentum alone, transfers through the entire cool body until both bodies are of equal temperature.

3. Radiation: High-energy photons, electrons, protons or neutrons are emitted by certain chemical and nuclear reactions, as well as by very hot solids. These particles travel through space, and if they strike something, will generate heat by transferring energy to it.

A very facinating explaination theotherguy! I've been studying heat transfer methods/principles myself recently, and I must say, I've never seen a better explanation 'in layman's terms' before! Seriously, I'm not being facetious. You said your a Carnegie Mellon student right? Maybe you become a professor. That was a good explaination.

 

[Mr Stabby]

QUESTION

Nuclear Fusion Reactor.

1. I know how the stuff works, but I am curious to know..Nuclear Fusion is a countinous unstoppable reaction from what I know. Elements would keep getting formed till Iron or Lead if I am not wrong, since it has the most stable Nucleus and lowest mass excess. So why doesn't this happen in the Nuclear Fusion Reactor, what's that stopping it from going beyond helium?

It will int theory be stoppable and startable. I think the helium or triterium when formed is extractable, It's possible that to fuse helium requires even more energy, so the system isn't capable of it.

2. Also, it's a ****ing 1 Million degrees in there. Okay, so the magnetic field holds the plasma, but it definmitely can't hold the 1 million degree radiation right? What epic material is that being able to handle that kind of heat without vaporising.

-soulslicer

It will in theory all be held by the magnetic field, however that hasn't actually been achieved yet.

 

[Hazar]

x = (Pi+3)/2
2x = Pi+3
2x(Pi-3) = (Pi+3)(Pi-3)
2Pix-6x = Pi^2-9
9-6x = Pi^2-2Pix
9-6x+x^2 = Pi^2-2Pix+x^2
(3-x)^2 = (Pi-x)^2
3-x = Pi-x
Pi = 3


-1 = -1
-1/1 = -1/1
-1/1 = 1/-1
sqrt(-1/1) = sqrt(1/-1)
i/1 = 1/i
i = 1/i
i * i = 1
-1 = 1
​

am i doin it wrong?

 

[Vegeta897]

x = (Pi+3)/2​

2x = Pi+3
2x(Pi-3) = (Pi+3)(Pi-3)
2Pix-6x = Pi^2-9
9-6x = Pi^2-2Pix
9-6x+x^2 = Pi^2-2Pix+x^2
(3-x)^2 = (Pi-x)^2
3-x = Pi-x
Pi = 3​

-1 = -1
-1/1 = -1/1
-1/1 = 1/-1
sqrt(-1/1) = sqrt(1/-1)
i/1 = 1/i
i = 1/i
i * i = 1
-1 = 1​

am i doin it wrong?

From my own guess, I think the fallacy in those proofs is forgetting about giving a value the "positive or negative" notation when taking the square root of something.