/sci/ - Science & Math

Where is the lie?

I don't know about lie, but looking back my math teachers basically taught math as magical formulas rather than showing it with simple logic

Your picture isn't a lie necessarily. The lie is that that's a proof that 0.9999...=1. Teacher probably didn't prove 0.1111...=1/9 or 1/3=0.333... simply because kids will take those as fact.

Abstract things are beautiful because they can be almost anything. That's how I see advanced math that you weren't lied to but just didn't quite know about at the time outside of simple logic.

>>5217778

Also.

You cant really divide 1/9, since the number will be infinite, therefor 1/9 isnt the same as 0.111...., and there for you cant say that 9 x 1/9 = 9 x 0.1111.....

>>5217725
This. So much fucking this. Even now, I get in deep shit working simple math because I've forgotten some simple formula and have to look it up.
At my old high school, a math class consisted of this:
1. Walk into class, find a few problems from yesterday's lesson on the board, do them.
2.The teacher gives us a formula and tells us what each term means.
3.He tells us to read the section out of the book and gives us a homework assignment.
If we were lucky, he'd notice when we were all stuck on a problem and do it for us, give us a minute's explanation, and then leave us to the rest.
>tfw modern high school math textbooks actually tell kids 'use your TI-83 graphing calculator for this part
>Tfw I know for a fact that the textbook I read never explained how to find the inverse of a 3x3 matrix. Finding the determinate was done using a long-winded formula, but the book never told anyone how to modify the matrix before multiplying, only to punch it into your calculator and hit x^-1

>.999...=1
Oh, damn; thanks for the reminder, anon.

Not sure about mathematics but I was working with tautologies in logic and thought I found something neat.

>A=A
This is the law of identity.
>(A->A)&(A->A) by equivalence of the first
>A->A by simplification of the second
>Av~A by implication of the the third
This is the law of excluded middle.
Starting a new chain:
>A=(AvAvA...) by addition from the first.
>A=(A&A&A...) not sure how but it is a tautology.
This the law of idempotence.

I was looking at these and somehow arrived at these:
>A=((A=A)->A)
>A->(A=A)
Not sure what the applications are, but I got excited.

>>5217788
IIRC, there's a trollscience proof somewhere that pi=3. Something about inverting the corners of a square infinitely until you achieve a circle, and measuring the perimeter.

>>5217719
Your proof in the picture is wrong, here's a video that explains why in an easy to understand way with doodles. Enjoy,
https://www.youtube.com/watch?v=wsOXvQn3JuE&feature=relmfu

>>5217798
its pi=4
take a unit square and turn it into a unit "circle" by making infinite right-angle cuts that don't change the perimeter

>>5217804
>Just put 2 on both sides.
>She literally puts a 2 before .222... and 2 before the X.
My sides are logically equivalent to my sides.

>>5217814
I think the reason bad math is so funny is because you can show it to some people and actually confuse them.

>>5217719
The only reason that proof is so odd for some of us to accept is because we have to reconcile the fraction system and the decimal system.

the entire family of anti derivatives of 1/x is ln(x)+c

>>5217798
>>5217808

>>5217719
>So, what else have my math teachers lied to me about. Pleb math btw.

I can think of a two lies:

(1) Calculus teachers first say that "dy/dx" is just a single piece of notation,
and should not be interpreted as division. Then, a few chapters later, they
contradict themselves and start interpreting it as actual division -- for example,
they take dy=f'(x)dx and then divide both sides of the equation by dx.
They need to just stop with all this artificial complexity of "first it's not division
and then it is division". Just keep it simple and explain it as division from the
beginning -- it will help us understand the concept a lot better.

(cont...)

>>5218157

(cont...)

(2) Teachers (and textbooks too) will draw an arrow on the x,y plane and call
it a "vector". No, it's not a vector, goddamit -- stop lying to us about that.
A vector is a list of numbers and/or variables -- nothing more. When you draw
an arrow, it actually represents a concept called "translation"
(or "displacement" in physics) which requires TWO vectors that are equal to
each other: (Δx,Δy)=(2,-3) for example. Teachers and textbooks need to
stop using the word "vector" as a lazy-ass shortcut for "point" and
"translation" -- it's confusing the hell out of students. Get it right, people: a
"vector" is something like "(5,9)". We describe a "point" in analytic geometry
by using TWO EQUAL vectors like "(x,y)=(3,8)". And we describe "translation"
in analytic geometry (or "displacement" in physics) by using TWO EQUAL
vectors like "(Δx,Δy)=(2,-3)". Don't draw a point and call it a "vector" -- that
confuses the shit out of students. And don't draw an arrow and call it a "vector"
goddamit -- an arrow actually has a grand total of SIX vectors associated with it:
two equal vectors to describe the base point, two equal vectors to describe the
tip point, and two equal vectors to describe the translation for which the arrow
is symbolically showing.

>>5218165
Can you explain what you mean by "equal vectors"?

>>5218165
That is just stupid and false though. A vector is not a list of numbers or variables. A vector is any element of a vector space. For example, functions are vectors since you can add functions and get new functions. And a point being "TWO EQUAL vectors"? You clearly don't know what you are talking about. It is maybe confusing since we kind of alternate between viewing R^n as a vector space and just a topological space, but if one considers more general spaces which are not vectorspaces it is very clear that we cannot describe points in terms of vectors. We cannot in general add points, for instance (if you have a sphere, can you add two points on it and get another point on the sphere?).

>>5218157
becasue its not division, dx/dy is something different from dx and dy. any good calc course shows you the theorems showing that dx/dy, as in dividing dx by dy, can be set equal to dx/dy (the one symbol).

which is why we call it dx/dy and not D.

>>5218165
>No, it's not a vector, goddamit

You dont' seem to understand what a vector is. A vector is any element of a vector space, that is one that is closed, under linear combination, obeys the associative and distributive laws appropriately and contains a zero element. Arrows in a plane, based and the origin, and added by joining them end to end completely meet this requirement.

>>5218165

>an arrow actually has a grand total of SIX vectors associated with it:
two equal vectors to describe the base point, two equal vectors to describe the
tip point, and two equal vectors to describe the translation for which the arrow
is symbolically showing.

You have absolutely no idea what you're talking about, and are filling the space with a kind of mathematical word salad

Welcome to /sci/!

>>5218203
>A vector is not a list of numbers or variables. A vector is any element of a vector space.

Learn to wikipedia, please:

en.wikipedia.org/wiki/Vector_%28mathematics_and_physics%29

On that page, you will find about 20 uses of the word "vector", for all kinds of different applications. You decided to pick one of those applications and give me lecture about how that particular application uses the word "vector".

What they all have in common is the one pure, base meaning of "vector": a list of numbers, or more generally a list of expressions that each evaluate to a number.

>>5218203
>And a point being "TWO EQUAL vectors"? You clearly don't know what you are talking about.

If you give me the vector (2,3) and claim that it represents a "point", then you are not giving me enough information. I am free to interpret that vector as the point (r,theta)=(2,3) if I wish. Did you mean for me to interpret it as polar coordinates or rectangular coordinates? You didn't say, so your usage of (2,3) as the description of a point is provably incomplete.

To complete the description, you must indicate what those vector components stand for. The way you do that is to set the coordinate value vector (2,3) equal to a coordinate variable vector such as (x,y). Once you do that, you get (x,y)=(2.3). Anything less than that is ambiguous.

>>5218228
Holy shit, get a load of this guy.

>>5218228
please tell me the list of numbers representing the following vector:
the infinite straight line going trough the origin and the point 1,1,1 in the vector space of all the the straight lines going through the origin

>>5218240
wat? so you dont even know what polar coordinates are? the vector (2,3) is the vector with 2 on the first axes and 3 on the second. you are talking about a transformation from one such vector space to another where each point r,theta is mapped to an x and y.

>>5218197
>Can you explain what you mean by "equal vectors"?

Certainly.

If you have a point described by the following coordinates:

x=4
y=9

Then you can rewrite that using vector notation, like so:

(x,y)=(4,9)

Now, you have two vectors (x,y) and (4,9) that are set equal to each other. Hence, "two equal vectors".

Notice that you need the (x,y) side of the equation to fully describe how you are specifying the location of the point. If you simply have the vector (4,9) then you haven't specified how to interpret those coordinates, which makes it ambiguous (is it polar? is it rectangular?).

>>5218258
hahahahahahha.

lelz, this is gold. im saving all your post and making a new "millionaires is just a game of luck" post. "you need 6 vectors to describe an arrow"

>>5218252
>wat? so you dont even know what polar coordinates are?

I'm fully aware of what polar coordinates are.

A point in polar space is specified as (for example): (r,theta)=(3,pi/2)

A point in rectangulare space is specified as (for example): (x,y)=(5,8)

In both cases, the point is specified by setting two vectors equal to each other.

The first vector specifies the coordinate variables. The second vector specifies the coordinate values.

Remember that a "vector" (no adjective in front of it) is nothing but a list of numbers (or variables or expressions that represent numbers) separated by commas and enclosed by parenthesis. That is all. Vectors are used in dozens and dozens of applications, where they take on all kinds of application-specific meanings. Some of the other posters in this thread are taking application-specific uses of vectors and (incorrectly) assuming that the pure-math concept of "vector" is the same as their application-specific meaning.

>>5218258
I understand what you're saying, but this seems like a very strange notation. In the (a,b) notation of describing a point on a graph, specifying "a" is on the x axis, and "b" is on the y axis is implicit in the notation; there is no reason to redefine it.

And besides, the location of that point is dependent on your choice of an origin, so the "arrow" to the point is implicit in the notation too.
An arrow between two points is just subtracting two vectors, a process which makes another vector. In other words, you can say the "arrow" between two points is a single vector, because it is the result of addition of vectors.

>>5218278
> "you need 6 vectors to describe an arrow"

Read my post again. I did not say that you "need" 6 vectors to describe an arrow.

If you draw an arrow, there are 6 vectors involved in that arrow's full description:

The arrow has a base point -- for example: (x,y)=(2,3)

The arrow has a tip point -- for example: (x,y)=(5,7)

The arrow symbolically represents the translation from the base to the tip: (Δx,Δy)=(3,4).

A "vector" is a list of things enclosed in parentheses. If you count above, you'll see 6 things enclosed in parentheses. Thus, there are 6 vectors involved.

Many people focus on that last vector above (3,4) as what they mean when they draw that arrow. That's fine. But it's also very helpful to understand that there are other vectors involved in the formally full and complete description.

>>5218287
no, the vector (x,y) is taken as the point when you travel x from the y axis and y from the x access. you are talking about a transformation between to vector spaces.

>>5218312
saved.

>>5218312
Excellent summary.

However, you've missed the essential detail that each of your vectors must be represented with an arrow, which each take 6 vectors of their own, and therefore an arrow takes 36 vectors.

>>5218310
>you are talking about a transformation between to vector spaces.

Nope. I'm talking about something much simpler.

In the simple case, a point can be described like so:

x=3
y=8

When you convert that to vector notation, you get:

(x,y) = (3,8)

- - - - - -

Here's another way to look at it. Let's say that I draw the x and y axes and I plot a point somewhere.

Next to that point I write "(x,y)", meaning the generic coordinates of the point.

Then I decide that I want to go ahead and assign values to x and y for that point: 3 and 8 respectively.

So then I also write "(3,8)" next to the point.

Since they represent the same point, I go ahead and set them equal to each other:

(x,y) = (3,8)

As a result, I have two vectors that are equal to each other. One vector defines the coordinate variables of my space (x,y) and the other one defines the coordinate values (3,8).

Hence, it takes two equal vectors to formally and completely describe a point. The (3,8) by itself isn't enough because it doesn't say how to interpret those numbers. It would be kind of like me saying that my weight is 35. You would then ask me: 35 what? 35 pounds? 35 kilograms? So then you realize that you actually need two things to understand a person's weight: a number and a unit of measurement. That's basically the same reason why you need two vectors to describe a point (from a rigorous formal perspective).

>>5218346
Perhaps you should also explain to them how you need two equal functions to describe a curve.

>>5218343
>However, you've missed the essential detail that each of your vectors must be represented with an arrow

You've gotten the idea of a "point" confused with the idea of "translation".

We use arrows to symbolize translation.

Points are just symbolized with a dot.

When you draw the arrow, you have two points (to dots, that is): one at the base and one at the tip.

The arrow itself suggest the translation from the base point to the tip point, which is determined by taking the tip and subtracting the base, like so:

base: (x1,y1)=(3,4)
tip: (x2,y2)=(7,9)
translation: (x2-x1,y2-y1)=(4,5)

So when you're done, you just have the 6 vectors total. (Remembering, of course, that a "vector" in pure math is simply a list of numeric-valued variables or expressions enclosed in parentheses.)

>>5218372
Hmm... I see. So there is in fact an infinite recursion of arrows needed to represent any vector, making them computationally intractable and best left to the number theorists.

>>5218372
Do you need an (x,y) =(a,b) argument for each point? You compared it to defining the unit system, and so can't you just define it once for an arbitrary number of points?

>Vectors are points in space
Hello, I'm <span class="math">f(x)=2+x^2[/spoiler], I want to be a vector too!

>>5218346
Clearly, you need three equal vectors to achieve this goal.
(the horizontal position on the graph, the vertical position on the graph) = (x,y) = (3,8)

>>5218353
>Perhaps you should also explain to them how you need two equal functions to describe a curve.

In this thread, I'm assuming that most of you have gotten badly confused about "vectors" vs. "points" vs. "translation" (or "displacement" or whatever your textbook like to call it). So I thought I would illuminate the way it works from a rigorous, formal perspective -- because I think that can be very helpful.

A few of you guys are trolls. That's ok, I understand. I probably would have been a troll here myself a few years ago, before I started getting into the formal stuff.

>>5218399
>formal
explain to me please what is a co-vector.

>>5218390
Your the (2,0,1) point in the space with the basis {1, x, x^2}.

>>5218386
>Do you need an (x,y) =(a,b) argument for each point?

Excellent question.

As a shortcut, you're certainly free to omit the "(x,y)=" from the beginning of every point in a diagram, as long as somewhere it's clear what the coordinate variables are. (For example, if you just label your axes "x" and "y", then that's good enough.)

But just because you drop the "(x,y)=" from the description of each point, don't forget that from a formal perspective, it's still there.

What I'm giving you guys is the formal view of this from pure math. Just because I say that a point is fully and formally shown as "(x,y)=(2,3)" doesn't mean that you can't take some shortcuts when it can save your hand from getting writer's cramp!!!

>>5218421 Silly argument missing the point
Hello, I'm <span class="math">f: \mathbb R \rightarrow \mathbb R\qquad x\mapsto f(x) = x\exp(-x^2)[/spoiler], can I be a vector too?

>>5218165
>>5218203
>>5218197
>>5218197
>>5218214
>>5218214
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>>5218258
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>>5218343
>>5218372
>>5218372
>>5218399
..........

Holy hell /sci/, I post a troll thread and all I receive in return is butthurt. Man you people are no fun.

>>5218393
>Clearly, you need three equal vectors to achieve this goal.
>(the horizontal position on the graph, the vertical position on the graph) = (x,y) = (3,8)


Nope. The elements in a vector cannot be geometric objects.

Your first so-called "vector" consists of geometric objects, so they're disqualified.

The elements of a vector need to be number-valued. They can be numbers, variables, and expressions. They can even be complex numbers too.

The only requirement from a pure math perspective is that a "vector" (with no adjective in front of that word) is a comma-separated list of number-valued expressions or variables that are surrounded by parentheses. That's all.

The biggest problem I usually see is when a point gets confused with a translation (or a "displacement" in physics).
Both a point and a translation are represented using vectors. (A point is (x,y)=vector, and a translation is (Δx,Δy)=vector.)
Then the teacher starts using the word "vector" interchangeably for both "point" and "translation".

Holy shit do those students get confused by that. They see those arrows and they get all confused about whether the arrow represents a point or a vector or a translation or whatnot.

>>5218441
>Holy hell /sci/, I post a troll thread and all I receive in return is butthurt.


Just as there's a fine line between "trolling" and "stupid", there is also a fine line between "butthurt" and "trying to help people who might sincerely not understand".

>>5218474
Excuse me?
Don't go round spouting bullshit like that.
Your definition of a vector might suffice for a physicist, but it's not mathematically correct.

>>5218390
Someday people will understand that <span class="math">f(x)=2+x^2[/spoiler] isn't a function but an identity, that <span class="math">f(x)[/spoiler] is not the function <span class="math">f[/spoiler] but the value of the function <span class="math">f[/spoiler]evaluated at <span class="math">x[/spoiler], and that all of this matters and is not just your math teacher being annoying on small notation problems.

>>5218441
oh god, my sides and my jimmies

>>5218372
>Remembering, of course, that a "vector" in pure math is simply a list of numeric-valued variables or expressions enclosed in parentheses.
>numeric-valued variables or expressions

What the hell? They just have to belong to a field. Not all fields are fields of "numeric-valued variables" or whatever you're talking about. They can be functions. They can be vectors. They can be matrices. They can be sets. They can be fields. They can basically be any mathematical object you want.

The fact that you're saying that the coordinates of a vector are numbers makes me think you have never studied linear algebra or heard about vector spaces.

>>5218493
>Your definition of a vector might suffice for a physicist, but it's not mathematically correct.

Sorry, I've just read too many textbooks that start off with, simply:

"a vector is a list of 1 or more number-valued expressions"

or words to that effect. I've just read it too goddam many times. I'm as skeptical as the next guy, but there comes a point when I've got to start taking it seriously when I see the same thing over and over again from a broad variety of authors.

I'm well aware that various mathematical applications have their own shades of meaning on vectors. When you look up "vector" on wikipedia, you'll see dozens of shades of meanings, each putting their own adjective in front of the word "vector".

The general problem in this thread is that you guys studied "X vectors", where X is some adjective or some branch of math. Then you extrapolate from that that the general word "vector" (without any adjective in front of it) means the same thing as the "X vectors" that you're familiar with. Then you start wwhharrggarrbling all kinds of stuff that pertains to the X branch of math.

>>5218208
10/10 I'm outta here

>>5218535
>"a vector is a list of 1 or more number-valued expressions"

These textbooks were either not math textbooks, or highschool textbooks. If you really think so high of yourself that you don't admit your maths knowledge is at most of highschool level, here are pointers for you:

>http://en.wikipedia.org/wiki/Vector_space
>A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field.
>or generally any field.

Then you can click on the little blue "field" link and end up on:
>http://en.wikipedia.org/wiki/Field_%28mathematics%29
and read, within the example of fields:
>If X is an algebraic variety over F, then the rational functions X → F, i.e., functions defined almost everywhere, form a field, the function field of X.

Functions on algebraic varieties aren't exactly "number-valued expressions", are they?

Uneducated brat.

>>5218542
wat

>>5218165
Given any two elements x and y in <span class="math">\mathbb{R}^2[\math], x, y, x+y, x-y, and y-x are all vectors given standard definitions of vector addition and scalar multiplication.[/spoiler]