## And That Is The Rest Of The Story

Ask me anything   throw idea's my way!   Hey everyone, I'm just a 16 year old guy trying to find his place in this crazy messed up world we've built for ourselves. My interests include math, physics and anything funny! You'll find that I blog a lot of math, but I like to keep things personal as well, so I'll often post things about my life. I like to think that I try my hardest to make sure that everyone I know thinks for themselves, and I will NEVER take something as fact unless you can prove it; and may I say, good luck proving it to me!

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Electromagnetism →

Alright guys, I have reached a question that I can not answer. I’ll give you some background, and then present the question. I really need an answer, my sanity depends on it.

So I was in physics the other day talking about right hand rules. If you don’t know what these are,…

There we go! the last part was what I was looking for. Thanks a bunch!

— 2 days ago with 9 notes
#physics  #Special Relativity  #electrons  #electromagnetism  #fields
Electromagnetic Fields →

The first one I believe is an arbitrary rule designed purely to fit in with the remainder of our mathematical construct of reality; that our coordinate system and algebraic convention are designed so that i x j = k. Hence why it’s a right-handed coordinate system, it’s the idea that we must make…

I think the thing that I was trying to get at was the whole spin thing. I get the whole special relativity causing a magnetic field thing, that makes sense. What I don’t understand is why the direction is the way it is. I think you got onto it a bit there with the electron spin, but I have no way of tying it back in with the direction of flow in the magnetic field.

My physics teacher and I thought that it had something to do with the spin of the electrons involved, as that is also a quantity that receives a polarity (plus or minus), but we can’t seem to take the spin idea and turn it into the way the the magnetic field lines align due to the fact that spin has no physical interpretation to us.

Any ideas there?

— 2 days ago with 4 notes
Anonymous asked: You are a very smart person. That's very attractive. Just thought I would share this with you.

you… you can stay ;)

— 2 days ago with 5 notes
Electromagnetism

Alright guys, I have reached a question that I can not answer. I’ll give you some background, and then present the question. I really need an answer, my sanity depends on it.

So I was in physics the other day talking about right hand rules. If you don’t know what these are, they are conventions that tell you how a magnetic field will be generated from an electric current, and vice versa.

There are the three we are presented with in grade 11 physics, and I had my concerns with all of them. Here is the first law:

If a current travels down a wire, use your thumb on your right hand to represent the direction of the current (positive to negative), the way that your fingers curl is the direction of the magnetic field, which will always be perpendicular to the direction of the current

This right hand rule is actually the physical representation of the cross product of the vector I, and the vector r that connects the center of the wire to a point on the magnetic field B, where r is perpendicular to both I and B. Now, I hope you can see where this is going. What physical process is causing the vector field B to be perpendicular to both I and r, as well as point in the direction that it does; the direction the cross product would give?

I have a problem similar to this with the second right hand rule, which states that if I curl my fingers in the direction of the current in a coiled wire, my thumb will point to the N pole of the coil. This of course implies that the magnetic field points in a certain direction, which just seems weird.

Now, the last right hand rule has to do with the direction of a force. If a wire has a current flowing in one direction perpendicular to a magnetic field, then placing your thumb in the direction the current is going, and pointing your fingers in the direction the external magnetic field is flowing (N poles to S poles), then the direction your palm faces will give the direction of the force acting on the wire.

This one has an easy component, and a tougher one. The reason the cross product works for the force in this case is because of the way the magnetic field moves around the wire, and the direction the external magnetic field is pointing. This also raises the same issue: Why does the magnetic field around the wire point the way it does?

Alright, now that I have that off my chest, I think I should also add in something a little cool that I discovered when we brought up Maxwell’s Equations today. Here is their form, where E is the electric field, and B is the magnetic field

$\triangledown \cdot \mathbf{E}=\frac{\rho }{\epsilon _0}$

$\triangledown \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$

$\triangledown \cdot \mathbf{B}=0$

$c^2\triangledown \times \mathbf{B}=\frac{\partial \mathbf{E}}{\partial t}+ \frac{\amthbf{j}}{\epsilon_0}$

where

$\epsilon_0c^2=\frac{10^7}{4 \pi}$

$\frac{1}{4\pi\epsilon_0}\approx 9\times 10^9$

p is the field density and j is the current density.

If you think about it, the laws that link the two together depend on time derivatives, so assuming B and E do not change with time (fixed current through a wire, and a non-changing magnetic field), we get the new equations:

$\triangledown \cdot \mathbf{E}=\frac{\rho }{\epsilon _0}$

$\triangledown \times \mathbf{E}=0$

$\triangledown \cdot \mathbf{B}=0$

$\triangledown \times \mathbf{B}=\frac{\mathbf{j}}{\epsilon_0 c^2}$

under this interpretation, the two forces are not related, which is why you can only create an electric current when a magnetic field is moving (shoving a magnet through the hole in a copper coil), and why only a moving electric current will create a magnetic field around a wire (Still don’t know why the direction works out the way it does).

Pretty cool right? PLEASE MESSAGE ME THE ANSWER TO THIS QUESTION OR SEND ME A SUBMIT THINGY PLEASE PLEASE PLEASE!! You don’t want to see me cry do you?

DISCLAIMER! this experiment is repeatable in a vacuum with no external rotation, magnetic field (other than ones specified), and no gravity.?

— 2 days ago with 9 notes
#math  #mathematics  #maths  #physics  #mathema

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Wow! This is actually really great, and most of these classes seem to have sets of videotaped lectures for the classes!

HOLY CRAP

Just gonna leave this here…

(via imagessimulated)

— 4 days ago with 109097 notes
How to argue correctly

Alright, today I want to talk about *rummages through books I stole from my schools unused textbook room* ah yes, logic should do nicely. I think what I am going to try to do is come up with a short guide line to proving that you are right the correct way. I know quite a few people who like to yell at me for my use of logic, but don’t actually understand it themselves. Sometimes I’ll go in for a proof by contrapositive in my Law class and get nothing but stares and chants of “you can’t do that” or “I don’t get it” which apparently makes me wrong. I hope this helps you to understand the logic you are using, and finally begin to understand what it means to be correct in a fundamental way. At the end I’ll put down some guidelines for arguing your position well, and all that jazz.

The idea of logic is given from its definition in the American College Dictionary: “The science which investigates the principals governing correct or reliable inference.” Which really just means that logic is the thing we use to try to understand when things are true or false.

When using logic, we have to be careful. Ordinary languages such as English just aren’t good enough at removing ambiguity from what is being said; to avoid this, specific words are given singular meanings, which can then be used to construct sound logical arguments. This is called syntax or grammar. What is meant when we say something is called semantics, and we have to be careful that we properly define the semantics when constructing a logical sentence.

I used to have a whole lot of fun with the idea of grammar and semantics, as we are not really given the tools to understand it very well in school. This has lead to many arguments where someone would be trying to prove something to me, but their semantics were terribly defined to the point where their argument actually was too ambiguous to draw any conclusions. Apparently people don’t like it when you point this out though, because then you are being “too picky with their language choices” and are “misusing logic” while in actuality you are using a logic more fundamental than the idea of implication or inductive statements. Yes I am bitter, and yes that was a bit of a rant… moving on…

The first thing you need to know about logic is that instead of whole sentences, logicians like to abbreviate things to single variables. I could let the sentence “It is cold outside” be represented by the letter A, and then be able to use it in this form.

Not only are sentences given abbreviations, but so are propositions, such as the word implies and the phrase “if and only if”. Here are a list of a few of them

A^B (A and B) Conjunction
AvB (A and/or B) Distinction
A->B(A implies B OR if A then B) Conditional
A<->B (A if and only if B OR A iff B for short) Biconditional

There are two kinds of or statements. The one above means and/or and the other means either a is true of b is true; also denoted A OR B and A XOR B respectively. There are a few other symbols that kind of fit in here, parentheses; which tell you which parts of the argument should come first or tell you that a specific proposition is only applicable to certain sentences, and ~ which is the negation of a statement (We’ll talk more about that in a second).

You can then have quantifiers, which come in varieties as well

$\exists _x$ Which is read: “There exists some x”

$\forall _y$ Which id read “For all y”

These statements are negations of each other, that is the opposite of all is not none, it is some (E.g. not all) , and the opposite of some is not some again, but all (E.g. Not some). It’s also fun to get people on things like this as well.

So, now that we have this down, we should introduce truth tables. I am not going to explain myself, but feel this picture sums up nicely.

Armed with this, we can now tackle a tough question: How do we know when something is true? Well, we can first talk about Tautologies.

A tautology is a statement where all outcomes on a truth table (the last column) are all true.

Take the statement

$A\rightarrow (A \wedge A)$

There are two options, A is true, or A is false. If A is true, then A implies A and A is true. If A is false, then ~A implies ~A and ~A which is still true (try to prove this to yourself using the truth table above and the truth table for  A implies B). If we have a tautology, we can change the right arrow to a double right arrow, making the statement

$A\Rightarrow (A \wedge A)$

This is just syntax though, and you may see it differently elsewhere. I personally have another book on my shelf which doesn’t mention tautologies, so has no use for the single arrow. Instead it uses the double arrow, which serves its purpose.

Now, in English class, you were probably taught about tautologies implicitly. On of the most fundamental theorems about tautologies is that if the statements P1, P2, P3,…Pn as well as Q is made into a conditional statement in the following way:

$P_1\wedge P_2 \wedge P_3\wedge ... \wedge P_n\rightarrow Q$

then a tautology is formed when Q is true for all cases that P1 P2 P3… are all true. In English class, this would mean that to prove something is true using deduction, you would have to construct a tautology where every factor you can come up with implies some statement which is true if and only if all of the other factors are correct. It would be your job in any essay to prove that all of the Pn are correct, which would complete your argument. This implies that your argument is only valid when all of your major premises tautologically imply your conclusion.

We can actually prove this, but before I do that, I’ll get into proof methods.

Say you want to show that something implies something else, E.g. P implies Q. From what we learned about tautologies, this proof becomes a simple one, all we have to do is assume that P is true, and show that it implies that Q is true. This has shaped the scientific method, and has been given the name, modus ponens. It’s formal statement is:

$P\implies Q, P\vdash Q$

(read P implies Q, P is true, therefore Q is true).

Notice the double arrows there to signify the tautology.

We then go on to if and only if statements, which actually are two tautologies stuck together.

$P \iff Q=(P\implies Q)\wedge (Q \implies P)$

(again notice the double arrows) This can be proven using a truth table to show that they share the same outcomes for all truth values of P and Q. All you have to do is then assume P is true and prove that Q is true, assume Q is true and prove that P is true. For those of you who are slightly more mathematically inclined, a good proof to try when trying to understand this law is proving that

$S\cap T=S\cup T \iff S=T$

where S and T are sets.

Sometimes though, you don’t have information about P and you want to show that it implies Q. The two proof methods outlined above are direct proof methods, but others may be used. My favorite is the proof by contrapositive. Using truth tables, it is easy to see that

$(P\implies Q)=( \rightharpoondown Q\implies ~\rightharpoondown P)$

(sorry for changing “not” symbols on you, LaTex ad my keyboard don’t exactly match up) This is just another way of proving that P implies Q, and is actually really useful when you are given a statement for P such as x^2>1 and a conditional for Q like x>1. I used this in law to prove that the right to conscience is more fundamental than our right to freedom of thought.

Another one is proof by contradiction. This proof method is used to show that P is true in the first place when you want to use it in a conditional statement. You assume that P is false and derive a contradiction. Fun trial proof for this one:

Being gay is completely natural.

Proof.

take the negation to be: Being gay is not natural.

Then this would imply that nothing in nature should be gay.

Because other species exhibit homosexuality, we arrive at a contradiction, so our original assumption must have been false, and homosexuality is natural.

$\square$

Now, there are other times when you have three or more statements involved in a conditional or biconditional, but those are treated under the tautology portion of this post. You can also have statements like:

$P\implies (Q \wedge R)$

$P\implies (Q \vee R)$

$(P \vee Q)\implies R$

but breaking them up like you would any statement makes their proof possible.

So I guess it’s time for the guidelines. I should put in a thing that lets people know the math is done…

MATH HAS CEASED LET THE GUIDELINE COMMENCE

Alright, I’ll do this in point form

- Syntax and semantics are important when trying to prove your argument, so if your proof is shrouded by bad grammar and has many meanings, people are allowed to call you out.

- You have to consider everything when getting into a tautological proof. Proving that one thing implies something else is great, but if a third thing implies it as well, you could run into a contradiction if that third thing is false, which is the opposite of a tautology and probably the opposite of what you want to do.

-If you have proved that something implies something else, that is good, but please don’t forget to prove the first thing is true.

- Understand how to prove conditional statements.

-Know the difference between “if” and “if and only if”

-Be open to different proof types

- Know that the negation of “For all x” is “there exists some x” and that the negation of “There exists some x” is “For all x” (useful in proof by contradiction and contrapositive.

-Know which OR you intend to use, the inclusive and/or or XOR the exclusive.

-Understand that not all conditional statements are tautologies, and it is up to you to prove that by showing that when each statement  in the premise is true all at once, the outcome is true. DO NOT ASSUME IT IS A TAUTOLOGY.

I’ll continue things along this line when I finish reading the book.

I guess I could also tell you how to be good at being a person while I’m at it.

-When you are wrong admit it. If there is a flaw in your proof which someone has pointed out, you can try to patch it, but if someone has proved you wrong by using a tautology involving all necessary premises, you are wrong. Stop and accept it.

Finally I would love it if tumblr would stop pumping out poof logicians. Ethos an Pathos are not proof methods, though they are rhetoric devices. Try to use logic or logos more often to make your point than using a sad story or your position within the social hierarchy.

Good night, and try as always to think for yourself. Logic is a tool, kind of like a screw driver. You wouldn’t use a hammer to screw in a screw, so don’t use fallacy, ethos and pathos in place of sound logic.

— 5 days ago with 20 notes
#math  #mathematics  #maths  #mathema  #logic  #english  #fun  #tumblr
Anonymous asked: You are extremely intelligent.

Why thank you, but there are others who are more intelligent than I am. I am just a little good at one thing XD

— 1 week ago with 2 notes
Riemann hypothesis

I found a potential proof for the Riemann hypothesis in the European journal of mathematics. Take a read!

Anyone want to fact check it for me?

— 1 week ago with 15 notes
#math  #mathematics  #maths  #mathema  #physics
Math Contest

So I did the Canadian Senior mathematics contest today and I am feeling so great.

You see, when I go to do math contests, something horrible always happens. In grade 9, I went and tried to do one without a calculator (The Pascal), which of course went horribly for someone who hadn’t studied counting arguments yet; I think I got a 60%. In grade 10 (The Cayley), I had a civics project to do the night before and was running on about three hours of sleep(you can read about them here and try past contests); again, about 60%. This is the first time that I have actually tried one and not only had all of my brain functioning, but also knew what to do inside and out. It actually felt really good.

The test was proof oriented, and a lot of the questions were centered around when a solution to an equation exists and proving specific results for sequences and series.

I know I didn’t get all of the questions right, but that wasn’t the point anyway, I showed all of my work and learned a lot about what I can and can’t do. Actually, I did get the hardest question on the test, which wasn’t actually that bad in all honesty. I’ll see what I remember about it…

A sequence of A’s and B’s is called an (m,n)-sequence if when beginning with some letter, the “m”th letter and the “n”th letter afterwards are both different. For instance, the (1,1)-sequence can be written, ABABABA… OR BABABA…

a) how many different outcomes for a (2,2)-sequence are there?
b) Prove that the (1,2)-sequence can not exist.
c) Show that if an (m,n)-sequence exists, so does a related sequence (mr,nr) where r is any positive integer.
d) For what values of m and n do (m,n)-sequences exist?

Wow, that was pretty good for a 12 hour gap XD

The question before that was not as hard, but was also given a level B rank (Hardest on the test). I think it looked like this:

a) fully expand the expression (a-1)(6a^2-a-1)
b) for what values of x does

$6Cos^3(x)-7Cos^2(x)+1=0$

c) for what values of x is

$6Cos^3(x)-7Cos^2(x)+1<0$

That is all I remember for the hard ones XD Have fun, test your Canadian Senior level math skills!!

— 2 weeks ago with 12 notes
#math  #maths  #mathematics  #physics

So I found this in the paper this morning (The Beacon Herald). I know how a lot of the time there seems to be nothing going right in the world’s news, especially with the recent hurricane in the Philippines and Rob Ford’s … well, whatever that was. Here is something uplifting; a sign that the times are changing for LGBTQA people.

In one light, this is a terrible thing to appear in a news paper; it means that there is still prejudice in the general populous and that LGBT(Q and A probably should have been mentioned as well, but we are taking baby steps) people are still having to put up with religious persecution. On the other hand, Newspapers are having none of it, and fighting back one case at a time.

Here is the article in full, it actually the most amazing thing I’ve read in a paper in a while.

"Dear Amy: I recently discovered that my son who is 17, is a homosexual.
We are part of a church group and I fear that if the people find out they will make fun of me for having a gay child.
He won’t listen to reason, and he will not stop being gay. I feel as if he is doing this just to get back at me for forgetting his birthday for the past three years - I have a busy work schedule.
Please help him make the right choice in life by not being gay. He won’t listen to me, so maybe he will listen to you.” - FEELING BETRAYED

"Dear  Betrayed: You could teach your son an important lesson by changing your own sexuality to show him how easy it is. Try it for the next year or so: Stop being a heterosexual to demonstrate to your son that a person’s sexuality is a matter of choice - to be dictated by ones parents, the parents’ church and social pressure.
I assume that my suggestion will evoke a reaction that your sexuality is at the core of who you are. The same is true for your son. He has a right to be accepted by his parents for being exactly who he is.
When you “forget” a child’s birthday, you are basically negating him as a person. It is as if you are saying that you have forgotten his presence in the world. How very sad for him.
Pressuring your son to change his sexuality is wrong. If you cannot learn to accept him as he is, it might be safest for him to live elsewhere.
A group that could help you and your family figure out how to navigate this is Pflag.org ( In Canada, see pflagcanada.ca) This organization is founded for the parents, families, friends and allies of LGBT people, and has helped countless families through this challenge.  Please research and connect with a local chapter. ” -Amy Dickinson

— 2 weeks ago with 28 notes
#LGBTQA  #LGBTQ  #LGBT  #News  #Funny
Anonymous asked: Do you know how to take the determinant of a 4x4 matrix? I think my TA does it a kind of weird way…

I had to learn 8 chapters of linear algebra to do this, I hope you are happy ;)

Alright, there are a few ways to do it, so I thought I would start with the permutations way, and then show the decomposition way.

With determinants, you have to realize that it is all based on permutations, more specifically, they all follow the form:

$\sum_{\sigma \in S_n}sgn(\sigma)a_{1\sigma(1)}...a_{n \sigma(n)}$

Where sigma is any permutation of n numbers. The only thing that doesn’t make total sense is the Sign of sigma which is defined as:

$sgn(\sigma)=\left\{\begin{matrix} 1 \, if \, \sigma \, is\, even\\ -1 \, if \, \sigma\, is\, odd \end{matrix}\right.$

You can tell if the permutation is even or odd by counting the number of pares of numbers in which the leading number is greater than the following number. If there are an odd number of such pares, then the permutation is odd, and if there are an even number, then the permutation is even.

for instance, the permutation 231 is odd, and the identity permutation 123 is even

To get a better idea, we can use a bigger example. Is 35142 even or odd?

now, we have to notice that 3 does not only precede 5, but also 1,4, and 2. So, the pares that satisfy the condition involving 3 are: (3,1), and (3,2) with five now, we can see that there are three pairs that satisfy the condition: (5,1) (5,4) and (5,2) finally, moving to four we see that there is one last pair: (4,2). Because there are 6 pares, which is an even number, we know that the sign of this permutation is +1.

That being said, there are 4! or 24 different ways of permuting 4 objects, meaning the determinant will be a 24 term term expression if we were to do it this way. Ew.

We can try something else however, we can break it up into smaller determinants.

when we want to take the determinant of a 2x2 matrix, we use  all of our permutations; 12 and 21, which gives us;

$\begin{vmatrix} a_{11} &a_{12} \\ a_{21}&a_{22} \end{vmatrix}=\sum_{\sigma \in S_n}sgn(\sigma)a_{1\sigma (1)}a_{2 \sigma(2)}=a_{11}a_{22}-a_{12}a_{21}$

for a 3x3 matrix, we have a bit more work to do. We get our summation expression:

$\begin{vmatrix} a_{11} &a_{12} &a_{13} \\ a_{21} &a_{22} &a_{23} \\ a_{31}& a_{32} &a_{33} \end{vmatrix}=\sum_{\sigma}sgn(\sigma)a_{1\sigma(1)}a_{2 \sigma(2)}a_{3 \sigma(3)}$

Which then gives us:

$a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{23}$

interestingly enough, we can group the terms by their leading a_{nm} value (corresponding to the top row of the matrix), and then take the determinant in reverse to get:

$\begin{vmatrix} a_{11} &a_{12} &a_{13} \\ a_{21}&a_{22} &a_{23} \\ a_{31} &a_{32} &a_{33} \end{vmatrix}=a_{11}\begin{vmatrix} a_{22} &a_{23} \\ a_{32}& a_{33} \end{vmatrix}-a_{12}\begin{vmatrix} a_{21} &a_{23} \\ a_{31}& a_{13} \end{vmatrix}+a_{13}\begin{vmatrix} a_{21}&a_{22} \\ a_{31}&a_{32} \end{vmatrix}$

There is a similar operation in four dimensions, which I guess I should try out. Here is the list of all 24 permutations, so we then begin

1234(even), 1243(odd), 1423(even), 4123(odd), 1324(odd), 1342(even), 1432(odd), 4132(even), 3124(even), 3142(odd), 3412(even), 4312(odd), 2134(odd), 2143(even), 2413(odd), 4213(even), 2314(even), 2341(odd), 2431(even), 4231(odd), 3214(odd), 3241(even), 3421(odd), 4321(even).

To save you the time and effort (I wish I could say that for myself) I’ll just give you the end result.

$\tiny \begin{vmatrix} a_{11} &a_{12} &a_{13} &a_{14} \\ a_{21} &a_{22} &a_{23} &a_{24} \\ a_{31} &a_{32} &a_{33} &a_{34} \\ a_{41} &a_{42} &a_{43} &a_{44} \end{vmatrix}=a_{11}\begin{vmatrix} a_{22} &a_{23} &a_{24} \\ a_{32} &a_{33} &a_{34} \\ a_{42} &a_{43} &a_{44} \end{vmatrix}-a_{12} \begin{vmatrix} a_{21} &a_{22} &a_{24} \\ a_{31} &a_{32} &a_{34} \\ a_{41}& a_42 &a_{44} \end{vmatrix}+a_{13} \begin{vmatrix} a_{21} &a_{22} &a_{24} \\ a_{31}& a_{32} &a_{34} \\ a_{41} &a_{42} &a_{44} \end{vmatrix} -a_{14}\begin{vmatrix} a_{21} &a_{22} &a_{23} \\ a_{31}&a_{32} &a_{33} \\ a_{41}&a_{42} &a_{44} \end{vmatrix}$

Now, you could always start here, and it would give you the right answer, but you could also try using some theorems.

If a matrix A has a row or column of zero’s, the determinant is zero.

if the matrix has two identical rows or columns, the determinant is zero.

if A is triangular, the determinant is the product of the diagonal elements.

another cool thing is that if you interchange two rows or columns, the determinant will only change sign of the determinant, and you are allowed to add and subtract rows and columns without affecting the determinant. (as long as it is a square matrix)

I hope this helps :)

— 3 weeks ago with 6 notes

Do you ever feel like you’ve let yourself down to the point where you feel there is no use in picking yourself back up?

— 1 month ago with 30 notes

First picture is of all the pure math books I own, and the second is of my full bookshelf after I went to the library sale in town, and raided my schools math department for old textbooks :) What do our bookshelves look like? Reblog with pictures! :D

EDIT!
Authors names on all Pure Math books because I want you guys to get them too :)
Calculus 6E and 7E were written by James Stewart
Projectue Geometry and College Mathematics were both written by Frank Ayres
Linear Algebra was written by Seymour Lipschutz
Mathematical Thinking was by Gilbert Vanstone
First Concepts of Topology was written by Chinn Steenrod
Statistics Without Tears (Actually a christmas present for my math teacher later in the year!) was by Derek Rowntree
Natures Numbers and Visions of Infinity were written by Ian Stewart
Dictionary of Mathematics was written by Sybil P.Parker
And finally, Dr.Eulers Fabulous Formula was written by Paul J. Nahin

— 1 month ago with 21 notes
#math  #mathematics  #maths  #mathema  #physics
School Systems

So I have a bit of a poll to do. Do you guys prefer the way that School systems across Canada and the United Stated (I have no idea how Britain runs its schools, but I am assuming it is similar) segregate children based on ability, or do you find it cruel?

I have some very strong opinions about this in one of these directions obviously; I would like to hear what you guys think, and hear the reasons you have for both sides. One opinion does not make an informed statement, especially when it is just my own; but I will give my opinion after I get a few people telling me theirs though :)

— 1 month ago with 19 notes
#education  #math  #mathematic  #school  #ability
Refutation

First off thanks for viewing the site.

Up until now I have been extremely patient, solicitous, understanding and have tried to take my viewers by the hand and gingerly walk them through the math, even though the math is mind numbingly simple. Herein lies the rub.  You as with most of the viewers equate simple with simplistic.  Yours, as has their, refutations been rather weak and facile.  Some pathetically so.

You are 16 years old.  You haven’t even begun life.  Still living with mommy and yet you presume to lecture a grizzled veteran of the sciences on how one should prioritize one’s life.  “ I enjoy it to the end, and will never take it seriously enough to let it stress me out;”. Spoken like a child with all the experience of a child. I can’t even call it arrogance. It’s look, Mommy, I’m a big boy!!  You are now about to be schooled in how the big boys do it.

A.    Your summary of the proof is spot on so you get the math. Great.

B.     Your refutation is completely fallacious and irrelevant.  BS only works in corporate America.  It finds no traction in math.

“To understand the mistake, we have to ask, is 2+2 the same as 1+3?

Well sure they equal to the same thing, but are they actually the same? Is having two things in your right hand and two things in your left hand the same has having one thing in your left hand and three things in your right hand? If you think so, I bet your equilibrium would beg to differ.”

Do you actually think this has any mathematical relevancy? Stick to the math.  Leave these inchoate ramblings to lesser minds. Point out the mathematical error (like I will do to your refutation).  Anything else is pure gossip, and as we know,  in gossip anything is possible.

C.       “At this point in the proof, he has abandoned the Pythagorean theorem in favor of the relationship between sine and cosine. “

This is absolutely incorrect.  I never abandoned the Pythagorean theorem.  This is another sticking point that lesser minds not versed in flowing with the algebra just can’t seem to get their flaccid minds to grasp.  The Fermat equation, no matter the value of n, a,b, and c, can always be expressed in terms of the Pythagorean theorem.  I’ve explained this ad nauseum.  I’ve proven it at least three different ways. Besides it is impossible to divorce the relationship of the sine and cosine from the Pythagorean theorem and vice a versa. I can’t be too harsh on you because you are too young to know any better but you had best tighten up your game if you want to challenge me or anyone knowledgeable in the field.

D.    At this point in the proof, he has abandoned the Pythagorean theorem in favor of the relationship between sine and cosine. It could just have easily of been written that:

$Cosh^2(\theta)-Sinh^2(\theta)=1$

so

$c^nCosh^2(\theta)-c^nSinh^2(\theta)=c^n$

Again your refutation, your logic, your math is spectacularly incorrect. It’s utter nonsense. You are correct in saying I could have just as easily expressed it in terms of the cosh and sinh but your algebra is the algebra of a 16 year old.

Right triangles (This is from wikipedia hyperbolic triangles)

If C is a right angle then:

The sine of angle A is the ratio of the hyperbolic sine of the side opposite the angle to the hyperbolic sine of the hypotenuse.

The cosine of angle A is the ratio of the hyperbolic tangent of the adjacent leg to the hyperbolic tangent of the hypotenuse.

The tangent of angle A is the ratio of the hyperbolic tangent of the opposite leg to the hyperbolic sine of the adjacent leg.

The hyperbolic sine, cosine, and tangent are hyperbolic functions which are analogous to the standard trigonometric functions.

So had you really grasped the algebra you would know that it would be

(not sure if this will show so the equation should be c^n((tanhb/tanhc)^2+
(sinha/sinhc)^2=1)

But again why even raise the argument?  It does nothing to disprove the proof. It’s just noise. If anything it supports my position. It’s lazy thinking and a lazy attempt at doing real math and the ultimate sin is it completely obliterates your arguments.

E.     The proof is definitively proven via my latest steradian proof and the square-cube law.  Until you and like-minded viewers can somehow develop enough mathematical insight to grasp the fundamental concept that the unit circle, right triangle, and the Pythagorean theorem lie at the heart of any, I repeat, any, Fermat equation where ,a^n+b^n=c^n then you are just doomed to a blissful ignorance and vapid chants of “you can’t do that”.

F.      Lastly, I hope I wasn’t too harsh but your arrogance needs to be tempered and your ignorance diminished.  This is a margin proof of Fermat’s last Theorem.  It’s obvious you have not the slight idea of the implication of this.  It is one of the greatest theorems in mathematics.  Yes Wiles has proved it but it’s not what Fermat would have created.  What I have written is something he could certainly have discovered.  That why this is such a big deal and is worth of a bit of stress.  If I may use a musical analogy, right now you are a decent piano tuner.  You have a long way to go before becoming a composer.

YarYou are more than welcome to yell back at me at me via my email listed in the video.

-Willie Johnson (completely unedited)

Alright, I am going to do this paragraph by paragraph. I am truly sorry that your equations did not come out, but I assume they were all from Wikipedia anyway.

Up until now I have been extremely patient, solicitous, understanding and have tried to take my viewers by the hand and gingerly walk them through the math, even though the math is mind numbingly simple. Herein lies the rub.  You as with most of the viewers equate simple with simplistic.  Yours, as has their, refutations been rather weak and facile.  Some pathetically so.

You are 16 years old.  You haven’t even begun life.  Still living with mommy and yet you presume to lecture a grizzled veteran of the sciences on how one should prioritize one’s life.  “ I enjoy it to the end, and will never take it seriously enough to let it stress me out;”. Spoken like a child with all the experience of a child. I can’t even call it arrogance. It’s look, Mommy, I’m a big boy!!  You are now about to be schooled in how the big boys do it.

Alright, the age fallacy… here we go again. If anyone on here ever tells me I am too young to understand something (math related at least), you will no longer be on good terms with me. I actually found this paragraph extremely rude and degrading, I feel almost obligated to begin another age related battle, but I will not. I instead want you to know that Ethos does not work on me; that is I do not bow to “Higher Powers”; not that being older should give you any power over me. That fact that you try to tell me that my understanding of how math should be done (in a manner which removes negative emotion and attachment to results) is a fallacy in itself. Letting your emotions guide you to what you think is right is definitely a problem, and can lead you to believe that you are right in cases when you are not.

On top of that, I actually am proud of myself when I finish a proof or find a flaw in one (whether they are proofs done by others, or by me). I do not however, seek validation from the world, meaning I will admit if I am wrong, and try desperately to understand what happened if I am. If I have a “Look mommy, I’m a big boy!” attitude, then you have something along the lines of a “Look world, I deserve to be famous for anything I do!” attitude, which to me just says existential dread. You do not need to be the one who proves Fermat’s Last Theorem, and you are not entitled to to the proof.

The thing is, unlike you, I am not arrogant. Arrogance is for those who are afraid to admit they are wrong; you are arrogant. I am self-aware; I know that I am not perfect, and I can tell when someone has made a mistake that I have made in the past. I would like for you to see what your error was, but your ego is getting in the way. You are never going to get better as a Mathematician this way. Telling me that this is how the big boys do it was also a very stupid thing to say. You want to talk about being high on yourself with no support; there is an example. This is not actually how the big boys do it, the big boys engage in intelligent conversation (not pissing matches with others), and use theorems intelligently with the intent to make their arguments as sound as possible. The big boys use math and ideas that you have regretfully not learned yet.

A.    Your summary of the proof is spot on so you get the math. Great.

B.     Your refutation is completely fallacious and irrelevant.  BS only works in corporate America.  It finds no traction in math.

“To understand the mistake, we have to ask, is 2+2 the same as 1+3? Well sure they equal to the same thing, but are they actually the same? Is having two things in your right hand and two things in your left hand the same has having one thing in your left hand and three things in your right hand? If you think so, I bet your equilibrium would beg to differ.”

Do you actually think this has any mathematical relevancy? Stick to the math.  Leave these inchoate ramblings to lesser minds. Point out the mathematical error (like I will do to your refutation).  Anything else is pure gossip, and as we know,  in gossip anything is possible.

So let me get this straight: I understand the proof you did, but somehow the contradictions that you innately arrive at are made irrelevant by either my age or the fact that you don’t want to hear it. Great.

Moving on, the example I gave was the simplest example that I could have possibly given you for the idea that two numbers (or functions in this case) can be broken up in many different ways, and a proof for one way does not imply a proof for all. These questions have actually baffled greater minds than you or I for centuries, and were brought into full light by ideas made in the previous century in set theory. The fact that I was hinting at with my “ramblings to lesser minds” was that there is no unique way to break up 1 into functions of theta uniquely, and that it does not make sense that a proof using only one way should be valid for all ways of breaking up one, as seen in the contradiction I derived. You also have to understand that this is a blog and I am trying to make my ideas appealing to the largest demographic possible, where as you are just trying to appeal to yourself.

.       “At this point in the proof, he has abandoned the Pythagorean theorem in favor of the relationship between sine and cosine. “

This is absolutely incorrect.  I never abandoned the Pythagorean theorem.  This is another sticking point that lesser minds not versed in flowing with the algebra just can’t seem to get their flaccid minds to grasp.  The Fermat equation, no matter the value of n, a,b, and c, can always be expressed in terms of the Pythagorean theorem.  I’ve explained this ad nauseum.  I’ve proven it at least three different ways. Besides it is impossible to divorce the relationship of the sine and cosine from the Pythagorean theorem and vice a versa. I can’t be too harsh on you because you are too young to know any better but you had best tighten up your game if you want to challenge me or anyone knowledgeable in the field.

I am going to have a field day with this one.

For starters, yes you actually did remove yourself from the Pythagorean Theorem a little ways into the proof. You began the proof with stating the Pythagorean Theorem, which is: The sum of the squares of the two legs of a right angled triangle will equal the square of the hypotenuse. You then deduced the relationship between Sine square and Cosine squared, which looks like:

$\forall_\theta\, Sin^2(\theta)+Cos^2(\theta)=1$

That was all well and good, until you realise that this theorem can be proved in non-geometric ways using Euler’s Formula expansion for sine:

$Sin(\theta)=\frac{e^{i\theta}-e^{-i\theta}}{2i}$

And cosine:

$Cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2}$

Which does not rely on the Pythagorean Theorem, as Euler’s Formula:

$e^{i\theta}=Cos(\theta)+iSin(\theta)$

Can be proved in many ways which do not make use of the Pythagorean Theorem, such as the proof by infinite summation. This means that the formula that you derived is not intrinsically related to the Pythagorean Theorem, and any combination of n functions which yield one may be used instead. You have to remember that the Sine and Cosine functions are not just a relationship with angles, but are also functions in their own right, and should be recognised as such.

Again, this has nothing to do with my age, but with my mathematical ability, and seeing as the equation you derived is no longer unique to the problem, any solution you get will be true only for that expansion (how is that for “big boy math”).

“At this point in the proof, he has abandoned the Pythagorean theorem in favor of the relationship between sine and cosine. It could just have easily of been written that:

$Cosh^2(\theta)-Sinh^2(\theta)=1$

so

$c^nCosh^2(\theta)-c^nSinh^2(\theta)=c^n$

Again your refutation, your logic, your math is spectacularly incorrect. It’s utter nonsense. You are correct in saying I could have just as easily expressed it in terms of the cosh and sinh but your algebra is the algebra of a 16 year old.

Alright then, I would really like to know how algebra for someone my age and algebra for someone your age are different. If I am correct in expressing the equation in the way that I did, why is it that I am suddenly wrong in multiplying by c^n just as you did in your “proof”. Honestly, I would like you to meet a boy names Jacob Barnett; a 15 year old boy working at the Perimeter Institute for Theoretical Physics, and then try telling him that algebra is different for people younger than you. Age is not equivalent to knowledge, nor is It directly proportional… arrogance seems to be pretty common in the general populous however.

Right triangles (This is from wikipedia hyperbolic triangles)

So had you really grasped the algebra you would know that it would be…

But again why even raise the argument?  It does nothing to disprove the proof. It’s just noise. If anything it supports my position. It’s lazy thinking and a lazy attempt at doing real math and the ultimate sin is it completely obliterates your arguments.

I cannot see your math in this part, but will respond to the parts that need responding to.  For one, whoever taught you that Wikipedia was a reliable source for math needs to seriously rethink their life. Sure it has the formulas, but there is no way they are teaching you how to use them, what their restrictions are, and if any of the information is 100% correct.  Moving on…

Your next sentence bugs me. You seem to be asserting that if I simply understood the algebra involved, I would be so much more enlightened and would see your point of view. I am going to assume that above you give a substitution for sine of theta in terms of sinh of theta and cosh of theta, as well as for cosine. This was not what I was getting at in the slightest with the error I pointed out. The formula you derived is equivalent to the one you gave in terms of sine and cosine; the point of the error was to separate the variables out in a different way, using the relationship between sinh squared and cosh squared. This expansion gave a contradiction, as it is not based on the unit circle, and therefore is not equivalent to the original expansion.

I am thinking that you took my substitution for

$Cosh^2(\theta)-Sinh^2(\theta)=1$

To be a fallacy in this regard, but it is actually far from it. It is different from the definition by the unit circle, which has been previously proven to not be related to Fermat’s Last Theorem in more than one way. What I did was real math (slightly too complicated for you even), and the fact that you cannot see that makes me wonder what you think real math is. It is big boy math though, so I get it if you are a little uneasy with it.

.     The proof is definitively proven via my latest steradian proof and the square-cube law.  Until you and like-minded viewers can somehow develop enough mathematical insight to grasp the fundamental concept that the unit circle, right triangle, and the Pythagorean theorem lie at the heart of any, I repeat, any, Fermat equation where ,a^n+b^n=c^n then you are just doomed to a blissful ignorance and vapid chants of “you can’t do that”.

Are you referring to this: http://970279249279632413.weebly.com/ because if you go to page 4 line 11, you will see that you have written:

$Sin^3(\theta)+Cos^3(\theta)=1$

I think a graphing calculator would disagree with you on this one.  Besides that, you do not write formally enough for the proof to be understood fully. Even if it was correct, the way that you write math makes it very difficult for any logic (and I do mean any) to gleam though. I can see misuses of the equal sign, uses of the “is congruent to” sign which should really be the “is related to” sigh R. Because you obviously do not have a grasp on the use of these symbols, you do not use the logic correctly when you prove that the Pythagorean Theorem is related to the relationship between sine of theta squared and cosine of theta squared. Relations produce disjoint subsets called equivalence classes, and the Pythagorean Theorem is just one of these equivalence classes (I gave another one above using Euler’s Theorem). Again, big boy math; you probably don’t understand.

.      Lastly, I hope I wasn’t too harsh but your arrogance needs to be tempered and your ignorance diminished.  This is a margin proof of Fermat’s last Theorem.  It’s obvious you have not the slight idea of the implication of this.  It is one of the greatest theorems in mathematics.  Yes Wiles has proved it but it’s not what Fermat would have created.  What I have written is something he could certainly have discovered.  That why this is such a big deal and is worth of a bit of stress.  If I may use a musical analogy, right now you are a decent piano tuner.  You have a long way to go before becoming a composer.

It is not that I do not have any idea of the implication of a marginal proof of Fermat’s Last Theorem at all. The problem is that I am always going to refuse to accept that something is correct when there are fundamental errors that are obvious to even an “ignorant 16 year old”.

Frankly, a marginal proof of Fermat’s Last Theorem would be cool, but it wouldn’t be ground breaking. Wiles got a lot of fame around the world because he had solved Fermat’s Last Theorem, but gained notoriety in the mathematics community because he had proved the Taniyama-Shimoura  conjecture; a conjecture which was thought to not be subject to proof with modern day mathematics.  You have not done any work on the subject, nor have you shown anything that is remotely close to a breakthrough, heck, your proof that the Pythagorean Theorem was related to Fermat’s Last Theorem wasn’t even correct on a fundamental (I am talking logic here) level. Sorry to be rude, but if you can’t take the heat, stop giving me stupid arguments based on my age and give me a real mathematical argument.

Finally, about your composer rant. To you I may seem a Piano Tuner, but in all actuality, I do proofs harder than this on a daily basis (and I do them correctly as well). If I was to go through conservatory levels, I would probably give myself a level 1 (the lowest level of composition if you don’t know).  I understand that I am not a level 5(or whatever lies beyond) though, and will work towards it every day of my life. You however seem to be stuck in the role of the atonal tenor who gets put in the back of a children’s choir. Until you accept that you cannot sing and make an effort to learn how, you will forever be dissonant and out of touch with real composition.

Sorry, but you actually do have a thing or two to learn from this sixteen year old who has “barely lived”, and I think that says a lot about who you are as a person. Math is not a contest, have fun, admit you are wrong, and grow as a person.

-Math-a-Magic

— 1 month ago with 11 notes
#math  #maths  #mathematics  #mathema  #physics  #FLT  #submission

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