Sandbox (Test/Try New Stuff Here!)

new stuff

Test

Hello. This is the sandbox.

Since Sonnhard lives in the playground, everyone should come here to escape Sonnhard.

~ Sonnhard.

new stuff – really new

Well, apparently we can.

Ahaan's physics diagrams to be tested here.

I'm new to these so forgive me for the really horrible diagrams.

So apparently dragging and dropping images does not work $-$ anybody else having that problem as well? Well, let's try to plot stuffages. Also, it seems I can't add forced new-lines. Darn.

Random thing

Ok, so if you need a good laugh, search this up. http://www.brainjet.com/random/4427/20-images-that-make-absolutely-no-sense?til=d-dy-4427#slide/3

This is a template that can be used to help in writing your posts. It contains quick examples of the most useful coded features Expii has to offer. Title

You can bold and italicize words, or even both! Heading 1

Use $\LaTeX$ to write math expressions like $1+1=2$. Learn more here.

Draw graphs and geometric diagrams!

Include videos, with annotations!

Or we can simply pause our explanation:

Click on the writing guide for more detailed information.

Template!

Or, we can simply pause our show:

LaTeX creates math expressions like $e^{i \pi} = -1$. Learn more here.

Include videos, with annotations!

Draw graphs and geometric diagrams!

Finally, you can always add pictures by dragging-and-dropping them right here!

I am testing things out in the Sandbox.

I enjoy petting cats and stargazing.

I also enjoy reading xkcd.

Test post on October 10 - after Biology release.

IMAGES (find le chat)

Just trying a new project:)

General Guideline:This is a sample post use the elements in the source code to get what you want the viewer to see.Delete un-necessary stuff for your post.

Title

Heading of Section

Formatting your answer to make it look $great$ requires many components like:

bold and italicized

bold

italics

strike-through

These basic formatting skills are helpful for any answer.

Basics of $\LaTeX$

The easiest way to learn $\LaTeX$ is by viewing others' code and output side-by-side! (This post is relatively long, so if you click "info" to see the source code here, it may be helpful to use Ctrl+F: there will always be "searchable" tags or keywords.)

$\LaTeX$ is not hard only a little bit different than your MS-Word (WYSIWYG-"what you see is what you get") which means you need to write it little differently.

The most essential thing to know is use double-dollar sign. Writing "double-dollar"(without quotes) between two dollar signs will give you $double-dollar$ which is is why $\LaTeX$ is beautiful.Try to use it in your posts to make them look $great$.

Now lets learn to ask a question in your post(please do even write answers)

Now about some basic stuff to write math.Really?

Simple operators are easy but what about something $\sqrt{3}$ or $3^{n}$ or maybe even $\frac{3}{5}$ ?

What you would type is

```  for  $ \sqrt{3}$

$3^{n}$

for $3^{n}$

$\frac{3}{5}$

for $\frac{3}{5}$

Lets Draw some Graphs

Draw graphs and geometric diagrams like this on $Expii$

With

<plot>
plot y(x) = x*x
</plot>

Happy drawing!

Embed A Video

Include videos, with annotations!

Wanna know more click here.

Click on writing guide for more information.

This is a template that can be used to help in writing your posts. It contains quick examples of the most useful coded features Expii has to offer. Title

You can bold and italicize words, or even both! Heading 1

Use $\LaTeX$ to write math expressions like $1+1=2$. Learn more here.

Or we can simply pause our explanation:

Click on the writing guide for more detailed information.

Template!

Or, we can simply pause our show:

LaTeX creates math expressions like $e^{i \pi} = -1$. Learn more here.

Include videos, with annotations!

Draw graphs and geometric diagrams!

Finally, you can always add pictures by dragging-and-dropping them right here!

testing testing 1 2 3 testing testing 1 2 3 testing testing 1 2 3 primes are cool

asdfqwerwerwerwerwerwerwerwerwerwerwerwerwe

Test Setup

$\boxed{CD = \Delta z}$

$\sin(\beta) = \frac{\Delta z}{AC} \Rightarrow \boxed{AC = \frac{\Delta z}{\sin(\beta)} }$

$\sin(\alpha + \beta) = \frac{CB}{AC} \Rightarrow CB = AC * \sin(\alpha + \beta) \Rightarrow \boxed{CB = \frac{\sin(\alpha + \beta)\Delta z}{\sin(\beta)}}$

$\cos(\beta) = \frac{AD}{AC} \Rightarrow AD = AC * \cos(\beta) \Rightarrow AD = \frac{\Delta z}{\sin(\beta)} * \cos(\beta) \Rightarrow \boxed{AD = \Delta z \cot(\beta)}$

By AAA congruence, $\triangle ABE \cong \triangle CDE$:

$\angle ABE = \angle CDE$

$\angle AEB = \angle CED$ (opposite angles)

And therefore, $\angle EAB = \angle ECD = \alpha$.

$\cos(\alpha) = \frac{\Delta z}{CE} \Rightarrow \boxed{CE = \frac{\Delta z}{\cos(\alpha)}}$

$\sin(\alpha) = \frac{ED}{\frac{\Delta z}{\cos(\alpha)}} \Rightarrow \boxed{ED = \Delta z \tan(\alpha)}$

$AE = AD - ED \Rightarrow AE = \Delta z \cot(\beta) - \Delta z \tan(\alpha) \Rightarrow \boxed{AE = \Delta z (\cot(\beta) - \tan(\alpha))}$

$\cos(\alpha + \beta) = \frac{AB}{AC} \Rightarrow AB = AC * \cos(\alpha + \beta) \Rightarrow \boxed{AB = \frac{\Delta z \cos(\alpha + \beta)}{\sin(\beta)}}$

$\sin(\alpha) = \frac{BE}{AD} \Rightarrow BE = AD * \sin(\alpha) \Rightarrow \boxed{BE = \Delta z \cot(\beta) \sin(\alpha)}$

Crash- The Summer Set Take me on the edge I wanna be the panic in your head I wanna say the words that turn you red Don't wanna pretend Oh no no

If I could start a war I'd wanna be the calm before your storm I'd wanna be the one you're dying for Again and again Oh no no

If I'm going down, then you're coming with me

Hold on tight Won't leave you high and dry Pull back your hair and ride And baby, you take the wheel to kiss me or crash me I can't breathe Turning the red to green You'll be the death of me And baby, you've got the wheel, so kiss me or crash tonight

Hands against your jeans I wanna feel the spaces in between The writing on the walls that no one's seen And sleep in your dreams Oh no no

Hanging by a thread But I feel your windows opening Throwing rocks into your bed Throwing rocks until you let me in

Hold on tight Won't leave you high and dry Pull back your hair and ride And baby, you take the wheel to kiss me or crash me I can't breathe Turning the red to green You'll be the death of me And baby, you've got the wheel, so kiss me or crash tonight

Hear them talking Back seats dancing Backwards all the while Turning ashes into fire

If I'm going down, then you're coming with me If I'm going down, then you're coming with me

Hold on tight Won't leave you high and dry Pull back your hair and ride And baby, you take the wheel to kiss me or crash me I can't breathe Turning the red to green You'll be the death of me And baby, you've got the wheel, so kiss me or crash tonight

This renders badly...

$$\int_{0}^{1}$$ $$\int_{0}^{1}$$

But if we remove the curly braces, it looks fine...

$$\int_0^1$$ $$\int_0^1$$

Alternatively, if we add a new line, it also looks fine...

$$\int_{0}^{1}$$ $$\int_{0}^{1}$$

Same thing happens with sums...

$$\sum_{0}^{1}$$ $$\sum_{0}^{1}$$

Or any stacking of subscripts and superscripts:

$$x_{a}^{b}$$ $$x_{a}^{b}$$

$ blah blah blah

blah blah blah$

$\pi$

$x

y$

test

$ blah blah blah

blah blah blah $

$$blah blah blah = blah blah blah$$

$$\begin{align} x \end{align}$$

test - new explanation - edit

INCOMPLETE

That's why I'm posting this in the sandbox (if the sandbox is not the place for people to pick up on others' articles, feel free to delete this). I feel, just as all of you, the obligation to share math with everyone, but my obligation starts with geometry and combinatorics, as in the footsteps of mathematical history. If anyone finishes the following article, I'd be delighted to see them post it under their own account and the smiles on students' faces when they discover this and much more.

The AM-GM Inequality

(Sorry, no Cauchy-Schwarz yet, but just as a side note, this is general AM-GM. I may do Cauchy-Schwarz separately because it may be more natural for some to discover one inequality before the other. Also, if you click on an answer that's marked wrong, you should still continue down the pauses for the feedback of that answer. The fact that it's marked wrong just means that there is another answer choice that directly leads to the next step.)

I don't recommend knowing what AM-GM stands for beforehand or trying to figure it out - it spoils the fun of starting from scratch. However, you can skip a bunch of stuff if you want; maybe you know the preliminaries but have not learned anything about AM-GM.

What number system do you think we're going to deal with most?

Let's start with the most simple. That's ...

Natural numbers.

Let's see if we can extend this. We try ...

Integers.

Let's try to extend it to ...

Rational numbers.

We try ...

Real numbers.

The next one is ...

Complex numbers.

Now let's return to $x\ge0$.

What is a simple change you can perform on $x\ge0$ in order to make it hold all the time? We consider ...

Operation(s). What's a simple number of operations?

We consider one operation. Left or right hand side?

Does it matter for addition?

No. We're just going to arbitrarily put the operation on the left for addition.

Where should we try multiplication?

It stays the same.

What do we like to do with $x^r\ge0$? What do we like to do with negative exponents?

We like dealing with positive exponents. How could we change it to a positive exponent?

It's just a matter of the definition. What we said earlier is equivalent to $\frac{1}{x^{|r|}\ge0$. What does this mean?

What do we want to do with such inequalities?

We don't like fractions, so we multiply, but be careful because ...

Be careful because of signs. What do we get?

$x^{|r|}\ge0$ corresponds with $1\ge0$, and $x^{|r|}\le0$ corresponds with $1\le0$. What do you notice ...

about the latter?

It's a contradiction! It doesn't really make sense to consider a negative exponent when our goal is simplicity, does it?

$x^2\ge0$. True?

What is $x^2$?

$x^2=x\cdot x$. If we want to find an inequality with two $x$'s, then what can we consider first?

We consider one $x$. What are some cases?

We compare $x$ with something else. Should it involve $x$ or be a real number?

How do we compare $x$ and 0?

We can say $x\ge0$ or $x<0$, or something similar.

If $x\ge0$, then what can we say about $x\cdot x$?

$x^2\ge0$. If $x<0$, then what can we say about $x\cdot x$? (If it's obvious, then just skip the following.)

What do we like to deal with, positive or negative?

Positive. What is $x$ in terms of something positive?

$x=-|x|$. What is $x\cdot x$? (This, though easy, will all be done in one step, so don't assume the next step is only going to show one transformation.)

$x\cdot x=-|x|\cdot-|x|=(|x|)^2\ge0$.

How can we make this interesting or powerful?

Think of some things to which a powerful inequality could be applied.

Introduce more variables. Let's first introduce ...

Two variables, $a$ and $b$. The square of a real-valued function of $a$ and $b$ is nonnegative.

What's a simple function on $a$ and $b$?

Nothing to move to any side. We can only expand. We get ...

$a^2+b^2+2ab\ge0$. How powerful is this?

Test something.

Most likely, you are going to test which type of numbers?

Positive numbers. Don't avoid $(a+b)^2\ge0$, but for now let's explore the other one.

$(a-b)^2\ge0$

What can you do with this inequality?

Nothing to move to any side. We can only expand. We get ...

$a^2+b^2-2ab\ge0$. Great. Let's extend this to ...

This is a 2-variable inequality, so we consider ...

3 variables, $a,b,c$. What would the inequality be?

When we go from 2 to 3, $a^2$ becomes ...

$a^3$. And similarly, $b^2$ becomes $b^3$.

When we go from 2 to 3, $2ab$ becomes ...

Watch out, there might be more things to change than you think.

$2ab$ becomes $3abc$. So now our inequality is ...

$a^3+b^3-3abc\ge0$. Anything missing?

What inequality do we already know?

$(a-b)^2\ge0$. Is this all we need?

This seems a bit arbitrary, so we add ...

$(b-c)^2\ge0$ and $(c-a)^2\ge0$. We get

$(a-b)^2+(b-c)^2+(c-a)^2\ge0$. What can we do with this?

Nothing but expand. We get ...

$2a^2+2b^2+2c^2-2ab-2bc-2ca\ge0$. This is equivalent to ...

$a^2+b^2+c^2-ab-bc-ca\ge0$. What were we trying to do?

Factor $a^3+b^3+c^3-3abc$. What do we do if we're hopeful?

We divide $\frac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ca}$. Divide completely. What do we get?

Before you move on, note that you can check your answer an easily.

$\frac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ca}=a+b+c$. So how do we write this?

$a^3+b^3+c^3-3abc$ factors as $(a+b+c)(a^3+b^3+c^3-ab-bc-ca)$. How do we finish?

Recall our inequality from earlier.

$a^3+b^3+c^3-ab-bc-ca\ge0$, so sometimes $a^3+b^3+c^3-3abc\ge0$. When?

We multiply by $a+b+c$, so if $a+b+c\ge0$, then $a^3+b^3+c^3-3abc\ge0$. However, it would still be nice if we can ... (choose another answer choice)

generalize

Wait a sec, we did too much work changing from 2 variables to 3 variables...

@ So consider the other answer choices. When we change from 2 variables to 3 variables, we are slightly ...

generalizing. Don't worry, we will go through the same steps in generalizing, so choose that answer choice.

What made it hard to go from 2 to 3 variables?

What was the first thing we did in the conversion?

We converted $a^2$ to $a^3$. How can we avoid this?

Just have $a$. And instead of $b^2$ in $a^2+b^2-2ab\ge0$ or $b^3$ in $a^3+b^3+c^3-3abc\ge0$, just have $b$. What do we get?

We do a substitution. What do we substitute?

$a$ as $\sqrt{a}$, $b$ as $\sqrt{b}$. How would we prove our inequality this way (not a different proof, just substitute)?

$(\sqrt{a}-\sqrt{b})^2\ge0$, so $a+b-2\sqrt{ab}\ge0$. Is there anything different than our proof before the substitution?

We wanted the conclusion to be different. What about the condition?

When is the inequality defined? $(\sqrt{a}-\sqrt{b})^2\ge0$ requires what?

$a,b\ge0$. What's the 3-variable version?

$a+b+c-3\sqrt[3]{abc}\ge0$. Any final manipulations before we come up with the general version?

Let's try to shorten it. Any ideas?

What is the real number that's an identity in the inequality?

$a+b+c\ge3\sqrt[3]{abc}$. You are encouraged to try proving this first, however, we will consider another manipulation before. It makes the inequality very easy to remember.

If you are finished exploring the generalization of the above form, continue, and our manipulation, the only obvious one left, will make the number of variables evident on both sides.

$\frac{a+b+c}{3}\ge\sqrt[3]{abc}$. What is the 2-variable case?

$\frac{a+b}{2}\ge\sqrt{ab}$. What would the general version be?

$\frac{a_1+a_2+\dots+a_n}{n}\ge\sqrt[n]{a_1a_2\cdots a_n}$ for any positive integer $n$ and nonnegative $a_1,a_2,\dots,a_n$. Guess why it's called AM-GM.

It's the arithmetic mean geometric mean inequality for $n$ variables! Let's get a feel for what happens with $n$ variables.

What's the $n$-element set $a_1,a_2,\dots,a_n$ with the simplest AM and GM?

Should we stick in a real number for any of them?

No, that would be arbitrary. A simple set would just be to set them all equal. That is, the case when there exists $a$ such that $a=a_i$ for all $i$. Does AM-GM hold?

$\frac{\underbrace{a+a+\dots+a}_n}{n}=a=\sqrt[n]{\underbrace{aa\cdots a}_n}$, so yes.

How can we use this simple case to prove AM-GM?

What do you think about $a$?

Have we determined it?

We want to determine $a$.

When you're stuck on algebra, it always helps to look at it from a non-symbolic perspective.

We have that the arithmetic mean geometric mean inequality is true for a simple case and we want to change it, maintaining its truth, to the general case.

What are you going to change?

What parts are there of the inequality?

We have the arithmetic mean and the geometric mean. Do you want to change both?

Would you rather choose one or two things?

Which sounds simpler?

If you think that we can't change only one thing, then think again.

Why or why not can we leave one part of the inequality the same?

Think of $a$ ...

We can determine it based on one side of the inequality.

Which part of the inequality do you want to determine it from?

The right may work, but there is only enough room to try the left. What do we have?

$\frac{a_1+a_2+\dots+a_n}{n}=\frac{\underbrace{a+a+\dots+a}_n}{n}$. Simplify this statement.

$a=\frac{a_1+a_2+\dots+a_n}{n}$. What do we want to do?

DRAFT STOPS HERE

If you want to finish this, then go ahead! I was planning to let students discover the algorithmic approach.

test

test

test tes tes test

Test Post!

doop