Mathematics - 2017-01-06 (page 6 of 6)

arctic tern

23:00

all finite extensions of finite fields are normal... if F is finite and F(a)/F is finite and a has multiplicative order m, then F(a) is the splitting field for x^m-1.

Animesh Ashish

Can anyone help me out as to how to see all the expression in understandable form?

Mahmoud

An explanation for why can't $\sin(a+b)=\sin(a)+\sin(b)$ ?

arctic tern

@AnimeshAshish the number of expressions that are understandable is probably in the millions or tens of millions...

mercio

$\sin(a+b)$ can be $\sin(a)+\sin(b)$, for example if $a=0$

Mahmoud

It looks obvious from the diagram, but I don't get why do we multply by $\cos$

Simple Art

23:03

@Mahmoud Trivial. If $a=b=0$, the statement is false.

arctic tern

if a=b=0 then the statement sin(a+b)=sin(a)+sin(b) is true not false

Mahmoud

I know right, but I want to have a geometric intuition behind the fuzzy formula.

mercio

well you have a diagram

with something of size $\sin(a+b)$ (the dotted arrows)

Mahmoud

And a weird text,

The full height of the blue triangle (sin(a)sin⁡(a)) can’t be used, since the red triangle doesn’t extend as far. (Why? When we add angle bb, we’re moving at a steeper angle with the same hypotenuse. We gained vertical distance and lost horizontal distance.) We’re effectively “sliding back” sin(a)sin⁡(a), reducing it by a factor of cos(b)cos⁡(b).
The full height of the red triangle (sin(b)sin⁡(b)) can’t be used either, since it’s at an angle. We’re “turning” sin(b)sin⁡(b), reducing it by a factor of cos(a)cos⁡(a).

mercio

something of size $\sin(a)$ and something of size $\sin(b)$

and they are not positioned in a way that they match up

arctic tern

23:06

@Mahmoud let's start with the bottom. the blue triangle. the right side has height sin(a). do you understand why the dotted line has height sin(a)cos(b)?

Mahmoud

@arctictern No, I don't

mercio

do you understand why it's shorter than the gray arrow

arctic tern

@Mahmoud look at the hypotenuse of the blue triangle. the whole hypotenuse has length 1 by assumption. how long is it from the origin to the top of the dotted line segment?

Animesh Ashish

@arctictern : can you tell me why does the "dollar sign" or rather "$" at the start and end of the expression is used? AND THE FORWARD SLAshes/backslashes in the mathematical expression?

Mahmoud

cos(b) ?

mercio

23:08

besides, if $\sin(a+b)$ were equal to $\sin(a)+\sin(b)$ you would get $0 = \sin \pi = 2 \sin (\pi/2) = 2$

arctic tern

@AnimeshAshish it's called LaTeX, it's how we typeset mathematical equations. see "LaTeX in chat" in the room description, upper right corner.

@Mahmoud correct. so the smaller blue triangle within the whole blue triangle is scaled down by a factor of cos(b). in particular that means the height sin(a) is scaled down to cos(b)sin(a). make sense?

Mahmoud

$\LaTeX$ Is kinda cool, and powerful @AnimeshAshish

Animesh Ashish

@Mahmoud : DO I have to learn it for the chat? Or is there a software to convert al the LaTeX written programs into understandable format?

arctic tern

@AnimeshAshish See the link I told you to click. You don't need to understand LaTeX to read it (your browser will turn it into math automatically once you follow the directions), but you'd need to know LaTeX in order to type some yourself.

Mahmoud

@arctictern Yes, $\sin(a)=\frac{\sin_2(a)}{\cos(a)}$

arctic tern

23:12

$\sin_2(a)$??

Mahmoud

The smaller height we're interested in.

arctic tern

I guess you mean the height of the dotted blue line segment.

okay. now work on figuring out the sin(b) to sin(b)cos(a) part.

Mahmoud

@AnimeshAshish It won't take a lot of time, en.wikibooks.org/wiki/LaTeX/Mathematics

Animesh Ashish

@Mahmoud : Gonna start asap!

Mary Star

@arctictern Why are all finite extensions of finite fields normal?

Mahmoud

23:17

@arctictern How ?

arctic tern

@Mahmoud the right side of the red triangle and the red dotted line form a smaller triangle. the angle at the bottom they make is a (do some basic geometry to figure this out). since the hypotenuse of this small triangle is sin(b) and its angle is a, its base (the dotted red line) is cos(a)sin(b).

Simple Art

@AnimeshAshish You learn something new everyday

arctic tern

unless you don't

Simple Art

um\

if I may interject

if $\sin(x)>0$ and $\cos(x)>0$, then

Mahmoud

@arctictern Thanks $\displaystyle{\huge{:)}}$

Simple Art

23:22

$\sin(x+y)=\sin(x)\cos(y)+\sin(y)\cos(x)<\sin(x)+\sin(y)$

@Mahmoud $\displaystyle\huge\ddot\smile$

Mahmoud

Oh good one.

arctic tern

@Mahmoud you mean you didn't already know the algebraic proof?

Mahmoud

@SimpleArt $\displaystyle\huge ت$

Simple Art

lool

Mahmoud

@arctictern I did, but It doesn't help memorize it

Adeek

23:26

@KajHansen thanks a lot for that link. I didn't know about his videos. They are really something.

I will be going through them.

Mahmoud

@Adeek What videos ?

Lucas Henrique

Hey there

Mahmoud

Hey.

Null

$\displaystyle\huge\heartsuit$

SteamyRoot

Cute.

Null

23:33

"And here we have a classic example of the Butt-Curve, can someone calculate its B-value?"

Mahmoud

Bye.

Animesh Ashish

Can anyone tell me how I render latex on my smartphone? I am using Chrome Browser.

Ted Shifrin

Hi @Mahmoud and tern ....

BTW, @Mahmoud, if you're interested, I have a different proof of the addition rule for $\sin$ based on areas. If you email me, I can send you that.

hi @Steamy, @Null, et al.

SteamyRoot

Ohi

frostedcake

is a non-stationary turning point, defined by f'(x) != 0 and f''(x) = 0 a point of inflection that is not an extrema?

as opposed to a stationary turning point defined by f'(x) = 0 and f''(x) = 0

Ted Shifrin

23:43

But even the "stationary turning point" you just gave needn't be a turning point.

Try $f(x)=x^3$ or $f(x)=x^5$.

A point of inflection is only guaranteed by $f''(x)=0$ IF the second derivative actually changes sign at the point.

Simple Art

$$\huge\color{red}{\heartsuit}$$

3

frostedcake

for some reason dollar signs are showing up in your answers

Ted Shifrin

Yes, that's how we type math in here.

See "LaTeX in chat" over there >>>> ^^^^

Simple Art

@frostedcake Please check the LATEX in chat on the top right

Ted Shifrin

@SimpleArt: Is that a general declaration?

Simple Art

23:45

what is?

Ted Shifrin

the red heart

Simple Art

Oh, yeah

frostedcake

aha

Animesh Ashish

Can latex be rendered in smartphone browsers?

Ted Shifrin

Yes, but getting that to bookmark is trickier (at least it was for me) ...

Simple Art

23:46

despair.com/collections/demotivators/products/love

@TedShifrin Agreed

frostedcake

but according to my textbook a non-stationary point of inflection can be defined as above

Animesh Ashish

Can anybody teach me how?

There is no 'latex in chat' thing coming from my smartphone.

Simple Art

@AnimeshAshish Sorry, I actually never figured out how to do it on phone

Can you edit bookmarks?

Ted Shifrin

@AnimeshAshish: I did figure it out and I think robjohn gave instructions on the webpage. This is the link: math.ucla.edu/~robjohn/math/mathjax.html

Simple Art

Like, open up options in whatever browser and find the bookmark editting option?

Animesh Ashish

23:48

Yes, I do

Simple Art

Then copy and paste the block of text at the bottom of TedShifrin's link into the URL of a bookmark

frostedcake

@TedShifrin Sorry I meant why wouldn't my rule necessarily be a point of inflection

Simple Art

then, whenever you come visit us, you can click that bookmark to get the math looking fancy

and you'll see my pretty heart.

Ted Shifrin

@frostedcake: I was warning you about needing to change from concave up to concave down to have an inflection point. What about $f(x)=x+x^4$? I don't think 0 is an inflection point. Look at it.

Simple Art

:O

Ted Shifrin

23:51

Never mind. I was right. Look at $f''(x)$, @frostedcake. It's 0 at 0, but positive everywhere else. So no change of concavity.

Simple Art

Its the end of the world!!!!! $\color{white}{\text{dies dramatically in the background}}$

frostedcake

sorry for which graph?

Simple Art

Btw, hit that "select all text" to see my invisible messages./

Ted Shifrin

For $f(x)=x+x^4$.

Animesh Ashish

@SimpleArt : I am sorry, are you telling me?

Simple Art

23:53

@AnimeshAshish Please see comments above.

Ted Shifrin

Heya, @MikeM. I escaped icemageddon (it cost me $175 to change my Delta ticket — those shysters were letting people traveling today and tomorrow change for free, but not with my Monday ticket date).

robjohn

@TedShifrin why concave up instead of convex?

Ted Shifrin

@robjohn: I prefer convex and concave, but 99% of today's calculus books say concave up and down.

Hi, btw. :)

robjohn

@TedShifrin hey there

Mike Miller

Congrats.

Ted Shifrin

23:56

Yeah, I'm too old for all the stress, @MikeM. How did your lecture for all the dignitaries go? :)

Mary Star

Let $p$ be a prime, $n\in \mathbb{N}$ and $f=x^{p^n}-x-1\in \mathbb{F}_p[x]$ irreducible. We have that $a\in \overline{\mathbb{F}_p}$ is a root of $f$.
We have that $\mathbb{F}_p(a)$ is a finite extension of $\mathbb{F}_p$.
How can we compute $|\mathbb{F}_p(a)|$ ?

Mike Miller

It was ok. 10 people came. I was nervous and not satisfied with what I did.

Ted Shifrin

Well, another time or two and you won't get nervous.

arctic tern

@MaryStar If a is a root of a polynomial f(x) irreducible over F, then F(a)/F has degree deg(f)

frostedcake

@TedShifrin that's really weird

Ted Shifrin

23:58

Teachers/books make mistakes, @frostedcake ... or sometimes students misinterpret.

Simple Art

@robjohn Haha, a funny pain of my life. When I tried to first learn calculus...

Mary Star

@arctictern Yes, we have that $[\mathbb{F}_p(a):\mathbb{F}_p]=\deg f$, but can we calculate also $|\mathbb{F}_p(a)|$ ?

frostedcake

@TedShifrin is it because maybe they work for a certain degree of functions? as in up to x^3 ?

Simple Art

despair.com/collections/demotivators/products/love

arctic tern

@MaryStar If K/F has degree n then K is isomorphic to F^n as a vector space over F.

Ted Shifrin

23:59

That's not a good excuse, @frostedcake. There are zillions of functions out there (including trig functions and exponentials, etc.).

Hi again, tern :)

arctic tern

hello ted

Transcript for

$Mathematics$ Mathematics