The h Bar

ACuriousMind

16:00

Well, you will need the addition theorems, no?

heather

okay, almost done, quick question: what's sin(x)*sin(y)?

i can't remember for the life of me

DanielSank

Just leave it as that.

Don't simplify anything.

heather

alright

DanielSank

Seriously, don't simply anything at all yet.

heather

one moment, let me type it out

DanielSank

16:02

@ACuriousMind heather's result will be whatever it is. If we use the angle addition fomulae, then we'll see it's equal to what I wrote, proving $R(\theta)R(\phi) = R(\theta + \phi)$, which is nice. As a bonus, and to me the more interesting thing, this provides a lightning quick way to derive the addition formulae if you forget them.

AccidentalFourierTransform

@ACuriousMind IIRC, it is possible to define $\sin,\cos$ through the addition "theorems" (that is, they are not theorems anymore). It is fun: there are thousands of different ways to define trig functions, from ODE's to series to integrals, etc.

DanielSank

^

ACuriousMind

@DanielSank I remember the addition formulae by looking at $\exp(\mathrm{i}\phi)\cdot \exp(\mathrm{i}\theta)$, there I don't even have to do matrix multiplication ;)

DanielSank

@ACuriousMind That's not any faster, tbh.

It's almost the same thing.

ACuriousMind

Yeah, I agree

DanielSank

16:04

Real/imaginary parts = x/y components.

so whatever, they're similar.

AccidentalFourierTransform

@ACuriousMind oh no, you're one of those too

DanielSank

@AccidentalFourierTransform What are "those"?

I am fascinated by the fact that ACM used a Roman "i"...

...why...?

AccidentalFourierTransform

@DanielSank well, you use $|v\rangle$ to denote vectors.

ACuriousMind

To be honest, I can't recall the last time I needed an addition formula, though :D

AccidentalFourierTransform

even when doing abstract algebra

and ACM writes $\mathrm i$ instead of $i$

heather

16:05

oh, darn

DanielSank

@AccidentalFourierTransform Meh, it works.

@AccidentalFourierTransform yeah that one is weird to me.

ACuriousMind

@DanielSank Because it's not a variable. $i$ is a variable, $\mathrm{i}$ is the imaginary unit (although it is of course good style to avoid using the variable $i$ when having an imaginary unit around)

AccidentalFourierTransform

you two disgust me

with your weird notation :-)

DanielSank

@ACuriousMind Meh. Ok.

I buy that.

AccidentalFourierTransform

I dont

it looks awful

ACuriousMind

16:07

@AccidentalFourierTransform I also write $\mathrm{e}$ not $e$ for Euler's number :)

DanielSank

I can see the logic in that.

I will never argue against clear notation.

For any reason.

AccidentalFourierTransform

@ACuriousMind yeah I do that too. But its not the same

heather

ack

what have I done

AccidentalFourierTransform

@heather what are you doing haha

heather

@AccidentalFourierTransform, I have no idea =P

I thought I'd typed it right

DanielSank

16:07

Does bmatrix work in mathjax?

ACuriousMind

@AccidentalFourierTransform Now, if I had an upright $\pi$...

heather

I used it on all the earlier ones @DanielSank

ACuriousMind

@DanielSank Yes

DanielSank

@ACuriousMind $$\text{p}$$

AccidentalFourierTransform

@ACuriousMind lol

heather

16:08

$\begin{bmatrix}
(\cos(\phi))(\cos(\theta))+(\sin(\theta))(-\sin(\theta))&(-\sin(\phi))(\cos(\theta))+(\cos(\phi))(-\sin(\theta))\\(\cos(\phi))(\sin(\theta))+(\sin(\phi))(\cos(\theta))&(-\sin(\phi))(\sin(\theta))+(\cos(\phi))(\cos(\theta))\end{bmatrix}$

where've I gone wrong...

I have the & in there

DanielSank

Not sure but that's a lot of unnecessary parentheses.

heather

begin and end

AccidentalFourierTransform

yeah the code it looks fine

heather

@DanielSank yeah

AccidentalFourierTransform

idk

ACuriousMind

16:09

You might need spaces around the &

heather

$\begin{bmatrix}(\cos(\phi))(\cos(\theta))+(\sin(\theta))(-\sin(\theta)) & (-\sin(\phi))(\cos(\theta))+(\cos(\phi))(-\sin(\theta)) \\ (\cos(\phi))(\sin(\theta))+(\sin(\phi))(\cos(\theta)) & (-\sin(\phi))(\sin(\theta))+(\cos(\phi))(\cos(\theta))\end{bmatrix}$

DanielSank

oh, lol

ACuriousMind

Also around the `\\`

DanielSank

Put spaces around the `\\`

heather

goodness, there

okay

DanielSank

16:10

\(^.^)/

AccidentalFourierTransform

now delete all the parentheses

heather

i was wondering what crime I'd committed

ACuriousMind

@DanielSank scheme might have tainted her ;)

heather

@AccidentalFourierTransform will do

DanielSank

lolololol

heather

16:10

@ACuriousMind no, it's just a bad habit I've got =P

AccidentalFourierTransform

ok, where were we?

I think it has something to do with $\mathrm e^{i\pi}+1=0$

right?

DanielSank

Well, @heather take the top left element of your matrix.

heather

$\begin{bmatrix}\cos(\phi)\cos(\theta)+\sin(\theta)-\sin(\theta) & -\sin(\phi)\cos(\theta)+-\cos(\phi)\sin(\theta) \\ \cos(\phi)\sin(\theta)+\sin(\phi)\cos(\theta) & -\sin(\phi)\sin(\theta)+\cos(\phi)\cos(\theta)\end{bmatrix}$

DanielSank

It should be equal to the top left element of my matrix.

So, $$\cos \phi \cos \theta - \sin \theta \sin \phi = \cos(\theta + \phi) \, .$$

AccidentalFourierTransform

@heather ah, almost. If you delete the parentheses you have to move the minus sign to the front: $(a)(-b)=-ab$

heather

16:12

@AccidentalFourierTransform, gotcha

DanielSank

...which is exactly the angle addition formula for cosine.

So, we just proved something neat: if you compose two rotation matrices, it's the same as a single rotation matrix for the sum of the angles.

heather

whoa

well, okay, now I'm questioning: how'd you get your matrix?

DanielSank

Oh, I just used the rotation matrix you got and stuck $\theta + \phi$ in for the angle.

You derived the form of an arbitrary rotation matrix and I applied to the case of an angle $\theta + \phi$.

AccidentalFourierTransform

@DanielSank but you didn't prove that if $\theta,\phi$ are a pair of angles, then so is $\theta+\phi$

DanielSank

@AccidentalFourierTransform Huh?

AccidentalFourierTransform

16:15

you might get problems with the domain of definition and global obstructions

@DanielSank Im just messing with you lol

DanielSank

I have no idea what you mean.

:|

heather

but why is it $\theta +\phi$ instead of something else whacky?

i mean, i guess it makes sense intuitively

ACuriousMind

Like all cats, it seems the Fourier transformed version is also a bit of a troll ;)

heather

@ACuriousMind, lol

DanielSank

If I rotate by 10 degrees, and then by 4 degrees, I better get a rotation by 14 degrees.

I gotta go.

heather

16:16

@DanielSank, well, thanks, that was awesome!

have a good day

DanielSank

you too

heather

or morning or whatever

DanielSank

morning

heather

@DanielSank thanks =)

@DanielSank right, because you're only two hours behind me

DanielSank

Take some time to digest what we did.

heather

16:17

will do

DanielSank

@ACuriousMind and @AccidentalFourierTransform can help.

obviously

@heather Nah, I'm on east coast at the moment.

AccidentalFourierTransform

well, there is something else that is fun

DanielSank

ciao

AccidentalFourierTransform

@heather what is the rotation matrix for $\theta=\pi/2$?

@DanielSank bye

heather

well, you just sub in pi/2 for theta

and then simplify

AccidentalFourierTransform

16:18

@heather yep

Let's call that matrix $I$

heather

so $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$

AccidentalFourierTransform

$$I=\begin{pmatrix}0&-1\\+1&0\end{pmatrix}$$

yes, thats it

heather

right

AccidentalFourierTransform

well, what is fun is that the matrix $I$ is essentially the same thing as the complex number $i$

heather

?

AccidentalFourierTransform

16:20

for example, can you calculate $I^2=II$?

(just matrix multiply $I$ with itself)

heather

sure, one moment

okay, it'd be

$\begin{bmatrix}-1&0\\0&-1\end{bmatrix}$

i believe

AccidentalFourierTransform

exactly, $I^2$ is minus the identity

heather

oh, huh, yeah

AccidentalFourierTransform

$I^2=-1$, just like the complex number $i$

heather

huh

that's interesting

but are there any other similarities?

AccidentalFourierTransform

16:23

yes

for one thing, the rotation matrix $R(\theta)$ from before

can be written as

$$\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}\cos\theta+\begin{pmatrix}0 & -1 \\ +1 & 0\end{pmatrix}\sin\theta$$

right?

what's going on? why isnt it working?

ACuriousMind

The $I$ would be called a linear complex structure in mathematics, because you can essentially define multiplying a vector by $a+b\mathrm{i}$ by saying it's the same as applying $a E+ b I$ to it, where $E$ is the identity matrix.

@AccidentalFourierTransform spaces around the & and the slashes!

heather

@AccidentalFourierTransform =P

709

A: What are imaginary numbers?

Let's go through some questions in order and see where it takes us. [Or skip to the bit about complex numbers below if you can't be bothered.] What are natural numbers? It took quite some evolution, but humans are blessed by their ability to notice that there is a similarity between the situati...

^this answer connected something else in for me...

"$90^∘$ rotation to be multiplication by $i$"

this is a 90 degree rotation matrix, right?

AccidentalFourierTransform

exactly

heather

@AccidentalFourierTransform huh, that's interesting

AccidentalFourierTransform

the point is, you can think of rotations in abstract

and you can represent them with matrices, but also with complex numbers

heather

16:31

okay, question: when watching that 3Blue1Brown video, it represented imaginary numbers with rotations...does $i$ equal that matrix as well as the square root of -1?

(if that makes any sense)

ACuriousMind

@heather Well, $\mathrm{i}$ can't really "equal" that matrix - one is a number and the other is a matrix, they can't be equal in the formal sense of the word.

The matrix is closely related to the imaginary unit and in some sense equivalent, but not equal

heather

hmm. okay.

AccidentalFourierTransform

^ that. We could say that our $I$ is equivalent to $i$. They are different objects but represent the same concept

heather

wait, they can or can't be?

$I=i$ or $I\neq i$?

AccidentalFourierTransform

$I\neq i$. The left-hand side is a matrix, and the right hand side is a complex number

you can write $I\sim i$ or something like that

heather

16:35

oh, okay.

AccidentalFourierTransform

but not $=$

heather

gotcha.

AccidentalFourierTransform

its rotations all the time, but different ways of expressing them mathematically

heather

i love how higher level math makes lower level math make so much sense

AccidentalFourierTransform

definitely, yes

its a steep learning curve

but its gets much more interesting with time

and in the end everything fits together

anyway, Im not sure what else Daniel wanted us to discuss

I guess he'll have something else to add when he comes back

heather

16:41

@AccidentalFourierTransform, @ACuriousMind, thank you both for your help

AccidentalFourierTransform

@heather don't mention it :-)

DanielSank

17:09

@heather do you understand all the things now?

heather

@DanielSank i do think it makes sense, yes =)

DanielSank

@AccidentalFourierTransform I mostly just thought it would be useful for heather to see rotation matrices, how they compose, and how they relate to trig identities.

The stuff about complex numbers is the obvious next step, as you guys already discussed.

It's no accident that the eigenvalues of $R(\theta)$ are $\exp(\pm i \theta)$.

AccidentalFourierTransform

@DanielSank also, what $\mathrm e^{I\theta}$ means. As, the matrix exponential and all that. Maybe something about De Moivre's formula? or somethign about infinitesimal generators of rotations? =P

There are many interesting things we could say but I don't want to overwhelm her

heather

this all sounds quite interesting

DanielSank

It is.

heather

17:18

@DanielSank $\exp(\pm i\theta)$?

what does that even mean?

DanielSank

$\exp(x) \equiv e^x$

It's just notation.

I use it for inline formulas to avoid the fact that superscripts are small and hard to read.

For example, $$e^{\frac{1}{2 \cos(x)}}$$ is harder to read than $$\exp \left[ \frac{1}{2 \cos(x)} \right]$$

AccidentalFourierTransform

especially when it doesn't compile properly

DanielSank

^ :-|

heather

ah, okay

$e^{\pm i\theta}$...that looks similar to $e^{i\pi}$

DanielSank

yep.

Of course. It's just the case where $\theta = \pi$.

heather

17:24

oh, true

interesting

DanielSank

But you don't really know about eigenvalues yet, right?

heather

sort of

eigenvectors are those that stay on the original vector's span during a linear transformation

and the eigenvalue is by how much they are stretched

if i remember right

DanielSank

Wow. You do know about them

Yeah that's exactly right.

The rotation matrix $R(\theta)$ has eigenvalues $\exp(\pm i \theta)$.

What's interesting about that, to me, is that when you multiply a complex number by $\exp(\pm i \theta)$ you rotate that complex number by $\theta$ about the origin :D

heather

huh, that's interesting

DanielSank

Yeah this is what @AccidentalFourierTransform was talking about earlier.

Any complex number can be written as $z = |z| e^{i \theta}$.

in other words, the magnitude multiplied by a phase factor.

AccidentalFourierTransform

17:31

but we like the unitary gauge where $\theta\to 0$.

DanielSank

So, if you multiply this by a pure phase... $$e^{i \phi} z = |z|e^{i \theta} e^{i \phi} = |z| e^{i (\theta + \phi)}$$

heather

@AccidentalFourierTransform unitary gauge?

DanielSank

He's being silly.

heather

what else is new =P

AccidentalFourierTransform

1 hour ago, by ACuriousMind

Like all cats, it seems the Fourier transformed version is also a bit of a troll ;)

DanielSank

17:32

hahahahaha

Emilio Pisanty

@DanielSank well, maybe you're using the wrong basis for your analysis, then, and you'd be better off using $(\hat{\mathbf e}_x\pm i \hat{\mathbf e}_y)/\sqrt{2}$ for rotations about $\hat{\mathbf e}_z$ instead.

Physics Meta

0

Q: Usage of some special formatting commands

What is the difference between This and this ? When are they supposed to be used ?

support

DanielSank

@EmilioPisanty indeed

heather

@EmilioPisanty, sorry, but your edit to the meta question doesn't really improve readability and it bothers grammar lovers (the space between the end of the sentence and question mark).

thanks

Swapnil Das

17:57

Hi.

Late Nights are wonderful, you get a lot of silence, lot of that.

AccidentalFourierTransform

welcome welcome

mochacat

18:29

oh hi friends

heather

@mochacat, hello

Secret

18:50

[Division by zero] current progress. Outlook good

shaded entries of the same color must have the same numbers

green=associativty verified

DanielSank

@Secret what is this?

Secret

Some abstract algebra hobby investigation I am doing. That is a composition table, otherwise known as a multiplication table. It tells you what you get when you pick any two elements and add or multiply them together

more details here:

$Mathematics$ Zero term algebra

All discussions on the ongoing project of algebraic structures...

Currently trying to make it at least left distributive and associative

otherwise that will suggest division by zero is not associative

vzn

19:22

Inside the Experimental Search for Dark Matter / dailybeast

heather

@vzn, hello

vzn

20:17

@heather hello

Sir Cumference

20:47

Howdy

You know, I just realized. @DanielSank is the only person I know here who uses his real name.

Oh wait, except heather

AccidentalFourierTransform

and me

Sir Cumference

Fourier Transform is the worst name for a child

AccidentalFourierTransform

I use me mothers surname though

yeah, I had a very hard childhood

Sir Cumference

So your name is Fourier, and last name is Transform?

AccidentalFourierTransform

thats why Im a physicist ;-)

yep

AND I WAS AN ACCIDENT!

Sir Cumference

20:51

We only learned about fourier transforms a few days ago, and I'm already forgetting what they are...

AccidentalFourierTransform

God help us!

mochacat

my real name is mochacat. my parents are from swaziland

Sir Cumference

I remember telling my parents I wanted to be a mechanic because I thought it referred to people like @Qmechanic

AccidentalFourierTransform

@Qmechanic g'evening

Qmechanic

@AccidentalFourierTransform : Good evening

heather

20:58

hello again, everyone

@SirCumference, how do you know "heather" is my real name...? =P

Sir Cumference

@heather Plot twist!

mochacat

heather is just a stage name so nobody knows that shes a heater

Sir Cumference

Heather reveals she's really 0celo

heather

it's actually 35h23t0-3r9

@mochacat uh...nope

for the record, i'm being sarcastic here =)

mochacat

so you really are a heater

this belongs in the world-building SE

heather

21:05

no, us 3r9-ers are an advanced processor chip.

mochacat

the singularity has begun

heather

developed to ::short circuits::

i'm sorry, what were we talking about?

@mochacat what? No, just the clone wars.

mochacat

i should leave before you take me off the grid

man everyone on the great outdoors SE is basically bear grylls

heather

lol

mochacat

i'm on this thread about removing fishhooks from humans and some guy suggested using fresh spiderwebs to seal the wound

DanielSank

21:12

@SirCumference yeah. I'm legit.

I did that so that if folks want to decide how much to trust my answers, they can google me and find out that I'm a real scientist.

So far there's only been one occasion where I regretted having my name here.

mochacat

engineer at boeing or electrical engineer at google?

when was that?

DanielSank

21:26

@mochacat huh?

@mochacat o_O

Sir Cumference

@DanielSank Which is?

heather

@mochacat 0_0

Emilio Pisanty

21:56

this is golden:

Wicked Bible

The Wicked Bible, sometimes called Adulterous Bible or Sinners' Bible, is the Bible published in 1631 by Robert Barker and Martin Lucas, the royal printers in London, which was meant to be a reprint of the King James Bible. The name is derived from a mistake made by the compositors: in the Ten Commandments (Exodus 20:14), the word "not" in the sentence "Thou shalt not commit adultery" was omitted, thus changing the sentence into "Thou shalt commit adultery". This blunder was spread in a number of copies. About a year later, the publishers of the Wicked Bible were called to the Star Chamber and...

AccidentalFourierTransform

mochacat

22:23

just lost the game in world-building SE

heather

@mochacat i've heard of that

but i decline to play it

because it bothers me to no end

Bible errata

Throughout history, printers' errors and peculiar translations have appeared in Bibles published throughout the world. == Manuscript Bibles == === The Book of Kells, circa 800 === The genealogy of Jesus, in the Gospel of Luke, has an extra ancestor at Luke 3:26 (the second name on this illustrated page). The reason for this error is that the transcriber of the Book of Kells read "QUI FUIT MATHATHIAE" as "QUI FUIT MATHATH | IAE," in which he considered IAE an additional individual (and so he added another QUI FUIT), rather than the Latin ending of "Matthew." Matthew 10:34b should read "I came not...

Sir Cumference

@EmilioPisanty Huh...

History is interesting

dmckee

23:38

@EmilioPisanty Every time I think of it, I wonder if the "bugger alle this" bible described in Good Omens (which also describes the instance you link to above) is a real thing. But not enough to actually do any research on the matter.

Thankfully someone else (or likely several someones) have already done it for me: en.wikipedia.org/wiki/Bible_errata

Hmm... alas there is no reference to it outside of Pratchett and Gaiman's book. ::sigh::

heather

nice hat @dmckee

Zero term algebra

Transcript for

$Mathematics$ Zero term algebra