text:
6:00 AM
T -> Ea Eb Tα
Tα -> ? E : E
ε
E -> T

halp idk what to make html desktop look like

i made the latter optional :D
@MᴀʀsUʟᴛᴏʀ wat is the HTML desktop?

@Downgoat // TODO remove Greek characters from code

@Downgoat ???
@Phrancis greek characters are good

@MᴀʀsUʟᴛᴏʀ If you speak Greek

6:01 AM
@Downgoat I haven't started yet, that's why I'm looking for a design
@Phrancis plus for maths/formulae

@MᴀʀsUʟᴛᴏʀ sorry, I've become so used to every piece of code I see to be borked I just assumed it was broken
@Phrancis no you have to use greek letters if you want to look smart

@Downgoat Now that is honest :D

For example here is a basic equality with normal letters:
a = bc - a
now here it is with greek letter:

λ  = Δθ - λ

;_;

6:03 AM
Hello everyone!

λ //half-life 3 confirmed

@DrGreenEggsandIronMan Hello @DrGreenEggsand WHAT

@charredgrass wow nice super smart equation 10/10
@DrGreenEggsandIronMan Hello!
@Phrancis XD

@Phrancis what WHAT?

@Downgoat Basically, a desktop made with HTML (pretty much only for ace code editor/tinymce atm)

6:04 AM
@DrGreenEggsandIronMan IRON MAN!? ;_;

@MᴀʀsUʟᴛᴏʀ that sounds scary

@charredgrass how?

@MᴀʀsUʟᴛᴏʀ well for starters you cannot do blurs and transparency. you don't want it looking like ubuntu

Oh, yeah. I changed it as a joke. Just like every name I've ever had on this site.

@Downgoat Ubuntu is good :(

6:05 AM
@DrGreenEggsandIronMan ;_; DJ McGoathem wasn't serious name??? D: D: D: y u do dis

@DrGreenEggsandIronMan 30 days of shame

@MᴀʀsUʟᴛᴏʀ I agree.

@MᴀʀsUʟᴛᴏʀ I agree but the desktop looks like crap

Did I just see "Ubuntu" and "good" in the same sentence? :P

@Downgoat At default settings
also the HUD
but I rethemed it so it looks much better
i.e. no taskbar thing, I use docky

6:06 AM
@Doorknob ubuntu == good unity == bad.
I use ubuntu + i3.

@DrGreenEggsandIronMan s/ good// ;)

cheddar> 1
Syntax Error: Unexpected token at 1:0
1 | 1
| ^
i think im done now
bai

rip Downgoat

@Doorknob you added extra slashes. 0/10, poor golfing.
If you were using V, you wouldn't even need the first slash.

@MᴀʀsUʟᴛᴏʀ i3wm.org

6:09 AM
^
I'm very thankful I've never had to use an Intel i3.

@charredgrass ;_;

@DrGreenEggsandIronMan ew tiling window manager

llama@llama:...code/ruby/slaka$git push fatal: The current branch master has no upstream branch. To push the current branch and set the remote as upstream, use git push --set-upstream origin master llama@llama:...code/ruby/slaka$ pls
git push --set-upstream origin master [enter/↑/↓/ctrl+c]
Counting objects: 3, done.
Delta compression using up to 4 threads.
Compressing objects: 100% (2/2), done.
Writing objects: 100% (3/3), 1.44 KiB | 0 bytes/s, done.
Total 3 (delta 0), reused 0 (delta 0)
Ah, I love this. :P

@Doorknob mathjax plugin is going crazy on that

(This being this.)
@Downgoat Haha, that'd be the prompt, I guess

6:12 AM
@Doorknob that's amazing.

It does tons of other stuff, too.
llama@llama:~$pacman -Syu error: you cannot perform this operation unless you are root. llama@llama:~$ pls
sudo pacman -Syu [enter/↑/↓/ctrl+c]

@Doorknob I'm tempted to leave in issue saying it does not fix the command "print an ASCII art goat"

Why'd you alias it?

llama@llama:~$print an ASCII art goat an ASCII art goat huh doesn't work for me 6:14 AM @DrGreenEggsandIronMan I like pls better. :P $ print an ASCII art goat
-bash: print: command not found

llama@llama:~$echo$SHELL
/bin/zsh

Haha, ninja'd!

D:

6:15 AM
Psychic ninja.

@Downgoat zsh > bash since zsh is great and does all things
oh-my-zsh = jquery for zsh

Is there a way to generate a SSH key for github that only has access to one repo?

@MᴀʀsUʟᴛᴏʀ i cannot brew install zsh so it's not that good
Deploy keys =\= ssh keys

@Downgoat That's your package manager's fault

6:17 AM
@Downgoat but they are SSH

wait really? huh

> Click Add deploy key. Paste your public key in and submit.
So github keeps my public key and I have to encypt my private key? :/
idk I'm setting up an auto-deploy script from travis but if travis's secure enviorment variables muck up I don't want people to get RW access to the cheddar website

Why did I even get on TNB in the first place? I'm so tired...

What time is it where you are?

6:20 AM
s/'m so tired/ don't have enough coffee

12:20. Which isn't even that late, but I played a ton of soccer earlier.

@Doorknob 9:22 here.

wait, if we're using vim for sed replacement jokes can I do:
d3wa don't have enough coffee

@Downgoat I love coffee, but at midnight? That's ridiculous.

f'C...

6:22 AM
^

@Doorknob ?

Find quote, change to end of line

@Downgoat "e don't have enough coffeeven get on TNB..."

Change is basically "d" + "i" together.

oh, am I supposed to use c so I don't need a?

6:23 AM
Which is an oversimplification.

idk how 2 vim anymore ._.

wwsto
:P

I get that joke!
He's saying you're bad at spelling.

oh
Da don't have enough coffee
there fixed

d$ is D. 6:25 AM @Doorknob golfier (and hacky): <C-a>sto @DrGreenEggsandIronMan @Doorknob ಠ_ಠ @Doorknob oh. TIL @DrGreenEggsandIronMan heh, nice Da at start of line is S. (or C) No "S" is cc. @DrGreenEggsandIronMan <C-a>s is same as xi? @DrGreenEggsandIronMan Yeah, so if he's at the start of the line, it'll have the same effect @Downgoat <C-a> jumps to the next number and increments it 6:27 AM oh yeah @Downgoat not even remotely. s is xi, or cl. nvm was looking at just the s part there were no numbers or digits in the document i was editing It doesn't have to be over one, just before it. It jumps to the number as a side effect. Neat vimgolf trick. I've always thought "s" is kinda pointless, since it's just synonyms for "c" but I suppose it's good for golfing. Also ^^ I use s and S regularly I also use s instead of c in visual mode, just because it's homerow 6:29 AM I should post that as a vim-golf-tips answer. The <C-a> trick. s is c<right arrow key> right? > arrow key pls s is cl, yes @Doorknob im sorry, ive been using vim not like an idiot for only like 2 weeks >_> @Downgoat ಠ_ಠ @Downgoat Map your arrow keys to nop :P (Only somewhat tongue-in-cheek; I've seen people do that) 6:31 AM That's fine, we all start somewhere. I used vim like an idiot for a long time. @DrGreenEggsandIronMan they are like 100 vim motions, give the gaot a break :P Actually, when I got extremely comfortable in vim was right around the same time @Doorknob suggested I track my vimrc on git. I learned a ton because of that. Ty! @DrGreenEggsandIronMan "used vim like an idiot" means using it inefficiently, right? not saying that vim is for idiots? @DrGreenEggsandIronMan :) what is the opposite of l, I've looked through :help motions and can't find it >_> 6:33 AM h hjkl is left/down/up/right @charredgrass when I said it it meant "only knowing commands a and x and w and q" ah ok thanks Try typing vimtutor in your terminal. Super useful, it's how I first learned vim 2 whose brilliant idea was it to make l go right and not left Look at where hjkl are on your keyboard ;) 6:34 AM @Doorknob oh, I've been using: Jul 2 at 0:40, by Dr Green Eggs and Iron Man Mandatory reading for every vimmer: http://blog.carbonfive.com/2011/10/17/vim-text-objects-the-definitive-guide/ I was going to post a caret, but vimtutor is an extremely good tip. They deserves a star instead. vimtutor is quite useful yeah @Doorknob E492: Not an editor command: vimtutor @Downgoat You probably should learn about the basic motions first before text objects @Downgoat In your terminal, not vim I know only the vital commands: 6:35 AM @charredgrass Yeah, pretty much. Stuff like staying in insert mode forever, using arrow keys, the mouse, etc. @Doorknob I learned about some motions $ 0, aw w, b, etc. and what like v does inside the things

i to Insert mode, Ctrl-C to exit Insert mode, :w to write, and :q to quit.

no :wq? that was like the first thing I learned

I learned it as :x :|

like, I knew :wq before I even knew how to insert :P

6:36 AM
There is a :wq.

ZZ ftw

The rest I look up in Vim's built in help.

Oh, another pro tip for vim: github.com/alols/xcape

@Downgoat :x is actually not the same thing as :wq.

I have caps bound to esc when tapped, and ctrl when pressed with another key.

6:37 AM
They're very very similar though.

@DrGreenEggsandIronMan ._. what's the difference?
docs say ZZ == :x

Sorry, I meant :wq. They're not exactlythe same.

0

We all know the standard English alphabet, and how to print it in code. For instance, in Python: >>> import string >>> string.ascii_letters 'abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ' But what about other alphabets? We don't have something to output this easily in python: ΑαΒβΓγΔδΕ...

@Downgoat Golfier, though. No need for a <cr>. :P
Alright, have to go now. Nice vimming with you :)

it's probably a good idea to learn how to spell interpret when writing documentation about my interpreter

6:38 AM
@Doorknob goodbye knob of the door.

:w will write no matter what. It can also be redirected. :x only writes if there are unsaved changes, and you can't redirect it AFAIK.

@charredgrass i wrote entire cheddar interpreter spelling enviornment as enviorment. 0/10 very difficult to fix.

@Doorknob agreed. Bai!

@DrGreenEggsandIronMan so :!touch . | x is the same?
@DrGreenEggsandIronMan Goodbye Dr. Ham jam

I'm not leaving (yet).

6:41 AM
oh :|

Spelled interpret wrong in my file name, gg

I have done that too many times
> ΑαΒβΓγΔδΕεΖζΗηΘθΙιΚκΛλΜμΝνΞξΟοΠπΡρΣσΤτΥυΦφΧχΨψΩω
@charredgrass 10/10 most genius mathematical expression ever

@Downgoat It's not a math expression. o_o

@Downgoat I don't think so. Think of :x as save and quit. :w is like "Output the text in the buffer (and by default, redirect it into the file we have open)"

yes it is @zʏᴀʙiɴ101, little do you know, I assigned values to those variables

6:43 AM
@NewMainPosts @charredgrass idk, either this challenge is a dupe of the ASCII one or it's just boring as there is no pattern between the unicode points of the greek letters. Maybe move it into the sandbox?
@zʏᴀʙiɴ101 it was a joke

@Downgoat CMC: Prevent Downgoat from defining joke.

@Downgoat Yeah, I think you're right. I'm going to think of a better greek letter challenge in the Sandbox

@zʏᴀʙiɴ101 please stop.

@DrGreenEggsandIronMan Please stop thinking I'm annoying and should leave chat forever.

/(?:([☢♫❤]).*)/ may be the weirdest regular expression I've ever written

6:52 AM
@zʏᴀʙiɴ101 I didn't say that, and AFAIK, nobody else did either. We just want you to act a little more mature about the text you type.

@charredgrass what is that even for

@MᴀʀsUʟᴛᴏʀ my JavaScript-based emoji esolang

7:13 AM
9

Introduction Kipple is a stack-based, esoteric programming language invented by Rune Berg in March 2003. Kipple has 27 stacks, 4 operators, and a control structure. Stacks The stacks are named a-z and contain 32-bit signed integers. There is also a special stack, @, to make outputting numbers...

new bounty

Doorknob pushed to Babble! \o/

I love how my keyboard randomly switches to Chinese sometimes. Makes for some very interesting errors when I don't notice.
Although, I really should pay more attention to that

7:30 AM
@charredgrass Accidentally pressing windows+space?

Probably. It's not a keyboard I'm familiar with, I often press the wrong keys
so sometimes I'm just typing and it turns out 螮蒐尸火土尸口

> npm install --save npm
@Downgoat 10/10 javascript ^

7:59 AM
oh my god, you can include stuff like ★ in file extensions
my custom language now uses .★js as its extension
4

@charredgrass This is gold !

@charredgrass oh my golly, it's discouraged
ohai @Katenkyo ^^'

Yeah,but it's so... pretty !
@zʏᴀʙiɴ101 Ohaio ! How are you?

I'm fine, thanks for asking ^^
I should try answering using 68k assembly. :P
Or 68k machine code, 'cuz that's golfier.

You must be masochist if you want to use ASM!
3

8:09 AM
I am! 'cuz I don't know what I'm doing!
It can "Hello, World!"?
With some effort, it can.
It can input?
With some effort, it can.
It can add? Yup. It can primality check? Erm... it should be able to.

@zʏᴀʙiɴ101 it can do anything higher-level languages can do since they compile to it

Ah k :)

Is zyabin becoming Sp?

@Doorknob Sp knows 68k assembly? o_O

8:13 AM
ah k

Oh no. I stole Sp's catch phrase six times. o_o

@zʏᴀʙiɴ101 if assembly and machine code were not programming languages by our definition, we would have some trouble finding languages to compete in ^^

Hello World in 68k machine code (EASy68k): 30 bytes.

we're going to need to see that code

00000000: 303c 000d 43f9 0000 1010 4e4f ffff ffff  0<..C.....NO....
00000010: 4865 6c6c 6f2c 2057 6f72 6c64 2100       Hello, World!.
Ohh, a NO slipped into that code.

8:23 AM
It's asking you to stop torturing it

brb submitting answer

No, it's your conscience saying "No, don't write in ASM!"

@charredgrass Write in ASM, then convert to machine code, it's much golfier!
Welp, this one was a shameful plug.
from plugs import shameless
Exception: not a chance
Welp, I can't make a shameless one. Well, you can see on the "Hello, World!" question.

is var_dump a valid output source for PHP answers?

@Quill Don't know this function, but from the name I would say it depends

If there's a strict output format ( or just "you should only output an integer representing x") I'd say no, otherwise yes

hii
today it's my birthday \o/
3

U+1F382 hbd

@TùxCräftîñg Happy birthday! You're now 1 year wloser to your death !
4

._.

8:47 AM
(Yes, I always find the words people want to hear)

._____________________.

@Katenkyo you're now 1 second closer to your death
oh no, another second closer
oh no!!!! several seconds closer!

@charredgrass Oh nooooo! I'm loosing so much time!

Happy birthday. Fun fact: there exists a number system in which its your 10 birthday!

 What does he have doctor?
It's his time, he's losing it too fast. I'm afraid he's running out of it.

8:59 AM
lol someone just deleted their answer in front of my eyes, 2 seconds after it was posted

._.

Ah, it's completely horribly wrong, that would be why
It was my lucky String question, the function took a long as a parameter...

._._._._.

and returned a long, wtf

Hey a long is almost good you never know if 32 will fit in regular int..

9:23 AM
CMC:

9:41 AM
hi @flawr
and all

0

This year's UEFA Euro 2016 is over and besides a couple of negative headlines there has been a very positive surprise as well – the Iceland national football team. Let's draw their national flag. It's been discussed in the Sandbox, whether we need yet another "Draw that flag"-question, and th...

@flawr I was wondering if you might have any interesting in codegolf.stackexchange.com/questions/84592/… ?
I am hoping the answer is yes :)

Looks too complicated for me

hello guys

I have trouble understanding the question
@Lembik I recommend -ing all code-ish parts for readability
@Lembik Do we exclusively consider binary strings?

10:01 AM
codegolf.stackexchange.com/questions/85141/… I posted a TI-BASIC answer, fuck
leaving, see you guys

Mathematica's builtins will never fail to amaze me
Why do they have a builtin for outputting nation's flags?Oo

@Katenkyo MMA has a builtin for everything that is known.

except strings?

@zʏᴀʙiɴ101 Does it have a builtin that makes coffee?
1 hour ago, by Katenkyo
@TùxCräftîñg Happy birthday! You're now 1 year wloser to your death !
Just saw I have a typo in this ¤_¤

@Katenkyo We don't have a nearby mod that can eleven the typo out.
Only Doorknob, who's inactive.
@Katenkyo Also $_$

10:15 AM
@zʏᴀʙiɴ101 I would not like to distrurb a mod just for a typo on the starboard, I'm dumb, just let everybody know it :p

reading a published paper about an algorithm..

@aditsu On which algorythm?

Is it strange that in two days I have no comments on my sandbox post?

Let S be a string of n characters s̝͇̘͎͕̘̣̿ͧ̋͛͐ͩ͡t̴͙͎̪͚͈̙͊ǫ̱͇̰͖̤̾͗r̳̰͛͑̓̅̓ͭe̜d̷̲̪͎̭̍ͅ i̗̰͖̗̼͍͔͍̓͌̃ͪ͆ͣ̿̊ń̴̠͔̝̬͓̼̟ ̡͍̪͔ͪͩ͛ͨ̂̑̎͜ã̸̛̤͗͑͊n̖̞̲͔̮̉ͪͩ̍̔̽͟ à̸̢̛͈͓̳̘̲͞r̸̴̟͎̮̪͍̙̺̫̼̝̥̳̜̙̟̀͜r̸̪͈̱͍͔̞̱̬̘̤̫̠̻̀́͢a̷̧͖̮̮͍̫̜̫͍̕͡ͅý̵̛͘҉̥͉͓͉̟͎̳̗̗̫̼‌̲̣̥͇͍̼̖͎ ̻̫͈̤̜͈̹͙̥[̴̡̛̤̥͔͉̞͇͔̖̹̥̻̺̕0͎̠͍̘̝̯̘̮̫͈͔́͝͞.̷̨̧͔̻̭̠̥͍̟̦̼̫͚͇͝ͅ.̸̢̀҉̘̰͙̱͍̼ǹ̨̝̤̦͜͞͡‌̱͎͍̳̭͍̖ ̨̞͉̟̤͜͝−͓̦̼̱̬͖̤̣͜ ̶̟̟̠͙̰͝1̛̩̼̤̼̱̱̯̮̹̳̫̳̜̭̟̤̀͞͞]̡͖̦͔͎͈̬͎̙̮̗̀͡ͅ
why can't they write with human words? :/

all words known are human words el oh el

10:21 AM
well.. I mean a sequence of symbols that has some kind of discernable meaning

Can I cheat by compressing the output in a zlib-compressed string and outputting it? ;) — zyabin101 19 mins ago

isnt it basically the principle of bubblegum? Except for method of compression :P

wtf does it mean for something to have "higher priority" when comparing 2 things? is it smaller or greater than the other thing?
(in the context of sorting)

think priority queue

so.. smaller then?

10:27 AM
I would say so

I'm waiting for this comment to be answered and me getting a LaughOverflow. ;)

ugh, I think it might actually mean greater

@Quill Seems legit formatted...

are you folks always that inquisitive when somebody new answers?

10:33 AM
also, each pdf comes with a different notation :/

@Quill As an avid user with R installed, I tested the contraption.
> m=rep(" ",33);for(i in 1:6){n=ifelse(m=="=",m,sample(c(" ","="),33,T,c(6-i,i)));m=n;cat(n,"\n",sep="")}
=   =              =  =   =
=  =  ==  = =   = == ==  ==  = =
= ==  ==  = =   = == == ========
= == ===  === = ================
=================================
=================================
> m=rep(" ",33);for(i in 1:6){n=ifelse(m=="=",m,sample(c(" ","="),33,T,c(6-i,i)));m=n;cat(n,"\n",sep="")}
=       =     = =     =
==  = ===     ==     === =  ===
==== = ======= === = ======= ===`
A cursory examination seems to actually limit the spectrogram to 2-6 beats, and there doesn't seem to be a 1 beat column.
But about randomness, it should be okay.
So, experienced users, vote, whether valid or not.
As the one who started the vote, I abstain.

@flawr Hi... we don't just consider binary strings but I think all optimal answers are binary
@flawr that's a good idea... thanks
@aditsu hi
@flawr if anything is unclear please let me know

About my comment, I didn't get an answer in 30 mins, so I deleted the comment.

@Lembik hi

@aditsu any progress on your secret mission :) ?

10:43 AM
see what I wrote above..

that you are reading a published paper?

yeah and it's not making any sense

which paper?
we are here to help :)

hmm.. should I keep it secret and risk wasting way too much time.. or should I reveal it and risk having somebody else solve it first? :/

no one else will solve it first :)
tell all!

10:46 AM
@aditsu Yes.

@mınxomaτ it wasn't a P ∨ ¬P
well fine, it's about suffix arrays
reading multiple documents and web pages, none of them makes sense

oh ok I know about them
@aditsu let's start at the start... do you understand what a suffix array is?
or is it linear time construction you don't get?

@Lembik do you know and understand the linear algorithm?

@aditsu yes

wow, ok.. (also HOW???!!)

10:51 AM
there are several different linear time construction algorithms but I assume you are referring to "DC3"
@aditsu it's just divide and conquer

never heard of "DC3", I'm talking about SA-IS

oh I don't know that one
SA-IS seems more complicated although I see there is a description here zork.net/~st/jottings/sais.html
I would stick to DC3 though .. the one described in the lecture notes I pasted

oh, is DC3 the one that groups substrings of 3 characters?

@aditsu yes.. well it splits the string into 2/3 and 1/3
it's much simpler

it's referred to as KS in the document I'm reading

10:56 AM
ok
notice that there is even source code for the algorithm at the back of the paper! Section A

so the main document I'm talking about is ge-nong.googlecode.com/files/…
@Katenkyo ^

interesting!


text:
# Nate has 5 green marbles and 3 red marbles. Select the ratios that compares the number of red marbles to the total number

###### Question:

Nate has 5 green marbles and 3 red marbles. Select the ratios that compares the number of red marbles to the total number of marbles. Mark all that apply. 3 to 8
5 to 8
3:8
5:3
5/8
3/8

### What is the difference between the words spray and strength?

What is the difference between the words spray and strength?...

### Pls help me with this question

Pls help me with this question $Pls help me with this question$...

### Which of the following is best associated with a mt air mass? a relief from warm humid summer conditions

Which of the following is best associated with a mt air mass? a relief from warm humid summer conditions in the us northeast b hot, dry summer weather in the us southwest c cold, dry conditions in winter in the us southeast d flooding rains in winter along the southern part of the us pacific coast...

### At the acquisition date of an active investment, when the cost of the shares acquired exceeds the underlying book value, the

At the acquisition date of an active investment, when the cost of the shares acquired exceeds the underlying book value, the investor is required to amortize any excess that is attributable to separately identifiable assets have an indefinite life. Which of the following is a separately identifiable...

### A chromatogram has three bands with Rf values of 0.25, 0.47 and 0.82. Which value corresponds to the material with the highest

A chromatogram has three bands with Rf values of 0.25, 0.47 and 0.82. Which value corresponds to the material with the highest molecular weight?...

### 1.How much momentum will an object of mass 5kg transfer to the floor if it falls from a height of 2m?

1.How much momentum will an object of mass 5kg transfer to the floor if it falls from a height of 2m?...

### Will give brainliest

Will give brainliest $Will give brainliest$...

### Find the area of the figure

Find the area of the figure $Find the area of the figure$...

### Select the lines that represent functions

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### Ahistorian has made the claim that too much is being made of the cold

Ahistorian has made the claim that too much is being made of the cold...

### Who are you guys voting for. I’m voting Coochie man 2020

Who are you guys voting for. I’m voting Coochie man 2020...

### JK is equal to 2:2:3 If HK=49, find IJ

JK is equal to 2:2:3 If HK=49, find IJ...

### Which expression shows the sum of the polynomials with like terms grouped together? (8x+32–822)+(4y-52)

Which expression shows the sum of the polynomials with like terms grouped together? (8x+32–822)+(4y-52) (bx+4y+(-822)] +32+(-57) bx+4y+ 3+ (-82²)+(-52)] (8x+(-822)] +4y+ [32+(-52)] 8x+4y+ (-822) + (32+(-52)]...

### Aword's denotation is? ? i also need with a word's connotation is? ?

Aword's denotation is? ? i also need with a word's connotation is? ? $Aword's denotation is? ? i also need with a word's connotation is? ?$...

### Cotx(tanx-1)=0 solve over [0, 2pie)

Cotx(tanx-1)=0 solve over [0, 2pie)...

### What type of plate boundary is shown at location B in the diagram below?

What type of plate boundary is shown at location B in the diagram below? $What type of plate boundary is shown at location B in the diagram below?$...

### Which was not a way that charlemagne promoted learning and education in europe

Which was not a way that charlemagne promoted learning and education in europe...

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Prove that $4\tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right)= \frac{\pi}{4}$

Prove that $$4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)=\dfrac{\pi}{4}.$$

I was wondering if there was a shorter solution than the method below?

Below is my attempt using what I would call the standard approach to these kinds of problems.

The expression on the left hand side is equivalent to $$\tan^{-1}\left[\tan \left(4\tan^{-1}\left(\dfrac{1}{5}\right)\right)-\tan^{-1}\left(\dfrac{1}{239}\right)\right]\\ =\tan^{-1}\left(\dfrac{\tan(4\tan^{-1}(\frac{1}{5}))-\frac{1}{239}}{1+\frac{1}{239}\tan(4\tan^{-1}(\frac{1}{5}))}\right)\;(1).$$

We have that $$\tan(4\tan^{-1}(\frac{1}{5}))=\dfrac{2\tan(2\tan^{-1}(\frac{1}{5}))}{1-\tan^2(2\tan^{-1}(\frac{1}{5})}\;(2)$$

and that

$$\tan(2\tan^{-1}(\frac{1}{5}))=\dfrac{2\cdot \frac{1}{5}}{1-(\frac{1}{5})^2}=\dfrac{5}{12}\;(3).$$

Plugging in the result of $$(3)$$ into $$(2)$$ gives $$\tan\left(4\tan^{-1}\left(\frac{1}{5}\right)\right) = \dfrac{2\cdot \frac{5}{12}}{1-(\frac{5}{12})^2}=\dfrac{120}{119}\;(4).$$

Pluggin in the result of $$(4)$$ into $$(1)$$ gives that the original expression is equivalent to $$\tan^{-1}\left(\dfrac{\frac{120}{119}-\frac{1}{239}}{1+\frac{1}{239}\cdot\frac{120}{119}}\right)=\tan^{-1}\left(\dfrac{\frac{119\cdot 239 + 239-119}{239\cdot 119}}{\frac{119\cdot 239+120}{119\cdot 239}}\right)=\tan^{-1}(1)=\dfrac\pi4,$$

as desired.

• Basically you have proven that $$\tan\left(4\arctan (\dfrac{1}{5}) - \arctan(\dfrac{1}{239})\right)=1=\tan\left(\frac\pi4\right).$$ I don't think that using Taylor series is going to help you here – User Nov 16 '19 at 22:52
• Whoever told you to solve it using Taylor series should be your deadly enemy. – WhatsUp Nov 16 '19 at 22:57
• @WhatsUp thanks. – user718615 Nov 16 '19 at 22:58
• What weirds me out the most about this is that nobody else on the site, at least from what I could find, had ever asked how to prove Machin's Formula. – URL Nov 16 '19 at 23:22
• See this thread for professional mathematician's take at it. – Jyrki Lahtonen Nov 17 '19 at 6:18

We can also use

$$\arctan(u) \pm \arctan(v) = \arctan\left(\frac{u \pm v}{1 \mp uv}\right)$$

to obtain in four steps

$$\frac{\frac15 - \frac1{239}}{1 + \frac1{5\cdot 239}}=\frac{239-5}{5\cdot 239+1}=\frac{234}{5\cdot 239+1}=\frac9{46} \to$$

$$\to \frac{\frac15 + \frac9{46}}{1 - \frac15\frac9{46}}= \frac7{17} \\\to \frac{\frac15 + \frac7{17}}{1 - \frac15\frac7{17}}= \frac2{3} \\\to \frac{\frac15 + \frac2{3}}{1 - \frac15\frac2{3}}= 1$$

• Very nice. My best compliments. :-) I'm amazed how you remember all these equalities. – Sebastiano Nov 16 '19 at 23:22
• @Sebastiano Thanks! I've encounered it recently in another different context. It is a very useful identity indeed. – user Nov 16 '19 at 23:26

A slightly faster variant of the same computation using the identity $$\tan^{-1} u \pm \tan^{-1} v = \tan^{-1} \frac{u \pm v}{1 \mp u v}$$ can be performed by observing that in the special case $$u = v$$ $$2\tan^{-1} u = \tan^{-1} \frac{2u}{1-u^2}.$$ Consequently, we iterate $$g(u) = 2u/(1-u^2)$$ twice for $$u = 1/5$$ to obtain $$4 \tan^{-1} \frac{1}{5} = \tan^{-1} g(g(\tfrac{1}{5})) = \tan^{-1} \frac{120}{119}.$$ Now we apply the original formula to obtain $$4 \tan^{-1} \frac{1}{5} - \tan^{-1} \frac{1}{239} = \tan^{-1} \frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{(119)(239)}} = \tan^{-1} 1 = \frac{\pi}{4}.$$ In all, we used three steps instead of four.

It is also worth noting that when $$u, v \in \mathbb Q$$, we can write $$\tan^{-1} \frac{p}{q} \pm \tan^{-1} \frac{r}{s} = \tan^{-1} \frac{ps \pm qr}{qs \mp pr}.$$ If we think of each rational as being represented by an ordered pair, which in turn is an element of the complex numbers, e.g. $$u = p/q$$ has the representation $$z = q + pi$$, and we define the function $$T(z,w) = \tan\left(\tan^{-1} \frac{\Im(z)}{\Re(z)} + \tan^{-1} \frac{\Im(w)}{\Re(w)}\right),$$ then $$T(z,w) = \frac{\Im(zw)}{\Re(zw)}.$$ In fact, the inverse tangent identity is simply a consequence of multiplication in the complex plane: $$\arg(zw) = \arg(z) + \arg(w).$$ I leave the details of this relationship as an exercise for the reader.

From the above, we may then regard Machin's formula as a statement about the existence of a nonzero real number $$\rho$$ such that $$(5+i)^4 = \rho(1+i)(239+i).$$ What is this number?

Shortest proof:

$$(5+i)^4(239-i)=114244+114244i.$$

Taking the arguments,

$$4\arctan \frac15-\arctan\frac1{239}=\frac\pi4.$$

Note that the computation avoids the fractions and immediately generalizes to other Machin-like formulas (https://en.wikipedia.org/wiki/Machin-like_formula#More_terms).

To perform the computation by hand, consider

$$(5+i)^2=24+10i\propto12+5i,$$

$$(12+5i)^2=119+120i,$$

$$(119+120 i)(239-i)=(119\cdot239+120)+(120\cdot239-119)i\propto 1+i.$$

(After simplification by $$119\cdot239$$, we have $$120=239-119$$.)

As advised by Maximilian Janisch, you should use the $$\tan x$$ formula rather $$\tan^{-1}x$$: $$\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right) - \tan^{-1}\left(\dfrac{1}{239}\right)\right]=\tan\left[\dfrac{\pi}{4}\right] \iff \\ \frac{\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]-\frac1{239}}{1+\tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]\cdot \frac1{239}}=1 \iff \\ \tan\left[4\tan^{-1} \left(\dfrac{1}{5}\right)\right]=\frac{120}{119} \iff \\ \frac{2\tan\left[2\tan^{-1} \left(\dfrac{1}{5}\right)\right]}{1-\tan^2\left[2\tan^{-1} \left(\dfrac{1}{5}\right)\right]}=\frac{120}{119} \iff \\ \frac{2\cdot \frac{2\cdot \frac15}{1-\frac1{5^2}}}{1-\left[\frac{2\cdot \frac15}{1-\frac1{5^2}}\right]^2}=\frac{120}{119} \iff \\ \frac{\frac5{6}}{1-\frac{25}{144}}=\frac{120}{119} \ \checkmark$$


text:
Opened 2 years ago

Closed 2 years ago

# Create a new class for Parity-Check Codes

Reported by: Owned by: richmond major sage-7.6 coding theory linear code, sd75, rd3 dlucas, nthiery Tania Richmond David Lucas, Julien Lavauzelle N/A 1aa4b4e (Commits) 1aa4b4e7333d96fc39836bee509fee97cfe28dfc

A linear parity-check code of length n and dimension k over GF(q), with n = k+1 is the set of all words, formed by the checksum at the last position of all k first symbols modulo q.

This ticket proposes a new class for parity-check codes along with an encoder for it.

### comment:1 Changed 2 years ago by richmond

• Branch set to u/richmond/parity_codes

### comment:2 Changed 2 years ago by richmond

• Commit set to b77de0ff1050b206b78e8b127f8dce32c0ff6c68

Hello,

I implemented a new class for parity-check codes over GF(q), with generator matrix (identity matrix concatenated with a ones column) and straightforward (minus sum of message symbols added at the end of the message) encoder classes. I also implemented a method for minimum distance.

Setting to needs-review.

Tania.

New commits:

 52b8b8a adds relative_distance method b77de0f Added new file that introduces the new parity-check code class. Initializes code class, generator matrix encoder class, straightforward encoder class and minimum distance method.

### comment:3 Changed 2 years ago by richmond

• Authors set to Tania Richmond
• Description modified (diff)
• Status changed from new to needs_review
• Summary changed from Create a new class for Parity Codes to Create a new class for Parity-Check Codes

### comment:4 follow-ups: ↓ 5 ↓ 11 Changed 2 years ago by dlucas

Hi Tania,

• You forgot to add your module (Python file) to the index of documented modules for sage/coding, which means no html/pdf documentation file will be generated for parity_check_code.py. You have to add sage/coding/parity_check_code to sage/doc/en/reference/coding/index.rst.
• Once done, you'll notice building the documentation fails because there's already a reference named [W] in /sage/schemes/hyperelliptic_curves/invariants.py. So, you have to rename your [W] to something else ([W88]?)
• I would add a small definition of a parity-check code in the module docstring (where you wrote "A simple way of detecting [...] even."). What about:

" A parity-check code of dimension k over F_q is the set:

\{(m_1, \dots, m_k, -\Sigma_{i=1}^{k}) m_i \mid (m_1, \dots, m_n) \in F_q\}"?

## ParityCheckCode

• __init__: I think you forgot to remove the docstring. Here it does not make a lot of sense to write "An abstract linear code object should never be created.". I usually omit

docstring for __init__ as I consider there's already some doc in the class docstring.

• __init__: I would add or no base_field.is_field() to the sanity check line 73. Indeed, for now if one passes anything but a field, the error message says AttributeError: 'anything' object has no attribute 'is_finite' which is not the most informative error message. BTW, it's also possible to create a Parity-Check code whose dimension is 0 or less. I know #21326 will fix that soon, but I guess a simple check on dimension can't hurt, right? And in any case, there should be at least something like if not isinstance(dimension, (int, Integer) raise ValueError("dimension has to be a Python int or a Sage Integer"). Otherwise, if the user passes anything but an int/Integer, it will fail with some weird error.

## ParityCheckGeneratorMatrixEncoder

• Doctest lines 179-183 calls the straightforward encoder instead of the generator matrix encoder.
sage: C  = codes.RandomLinearCode(10, 4, GF(11))
sage: codes.encoders.ParityCheckCodeGeneratorMatrixEncoder(C)

works while it should fail.

• generator_matrix: I'm quite sure l.246 should be G = G.augment(vector([field.order()] * k)). Because for now, the encoding does not work:
   sage: C = codes.ParityCheckCode(GF(5),7)
sage: E = codes.encoders.ParityCheckCodeGeneratorMatrixEncoder(C)
sage: m = vector(GF(5), [1, 0, 4, 2, 0, 3, 2])
sage: sum(E.encode(m))
4 #should be 0

## ParityCheckCodeStraightForwardEncoder

• I know we already talked about it, but I still think StraightforwardEncoder is not the most straightforward name... What about ParitySymbolEncoder? Note that I am not adamant with this name change though.
• encode: I think you can do it in one line with return vector(self.code().base_field(), message.list()+[-sum(message)])

## Encoders

• Last one: lines 404/405, you chose "ParityCheckGeneratorMatrixEncoder" and "ParityCheckStraightforwardEncoder" for the encoders string names. I'd choose "GeneratorMatrix" and "Straightforward" instead, as the user has to type those names when they use C.encoder('str_name'). See the names I chose in grs.py for instance.

Apart from those small remarks, it looks good. Great job for your first code class :)

David

### comment:5 in reply to: ↑ 4 ; follow-up: ↓ 6 Changed 2 years ago by nthiery

• __init__: I think you forgot to remove the docstring. Here it does not make a lot of sense to write "An abstract linear code object should never be created.". I usually omit

docstring for __init__ as I consider there's already some doc in the class docstring.

Just to clarify: __init__ methods, like all other methods, should have a docstring. But it can be made into a bare minimal, with a single doctest constructing an object and running generic tests on it; typically:

def init(self, ...):

""" Initialize this parity code

TESTS

sage: C = ParityCode?(...) sage: TestSuite?(C).run()

"""

Great job for your first code class :)

+1!

Cheers,

Nicolas

### comment:6 in reply to: ↑ 5 ; follow-up: ↓ 10 Changed 2 years ago by dlucas

• __init__: I think you forgot to remove the docstring. Here it does not make a lot of sense to write "An abstract linear code object should never be created.". I usually omit

docstring for __init__ as I consider there's already some doc in the class docstring.

Just to clarify: __init__ methods, like all other methods, should have a docstring. But it can be made into a bare minimal, with a single doctest constructing an object and running generic tests on it; typically:

def init(self, ...):

""" Initialize this parity code

TESTS

sage: C = ParityCode?(...) sage: TestSuite?(C).run()

"""

Oops. Well, I guess I should open a ticket to fix my __init__ then, because all my classes look like:

class ClassName:
r"""
Short description

Long description

EXAMPLES::

....
"""

def __init__(self, ...):
r"""
TESTS::

....
"""

BTW, doctest line 383 in parity_check_code.py should run unencode_nocheck, not unencode. sage --coverage complains about this... But not about missing docstrings in my __init__ blocks.

David

### comment:7 follow-up: ↓ 9 Changed 2 years ago by dlucas

• Reviewers set to David Lucas

Hi,

I just noticed something: according to the diff, you also implement relative_distance in this ticket. I'm quite sure it's because you were working on #21315 at the same time. Could you please remove the related lines from linear_code.py?

David

### comment:8 Changed 2 years ago by git

• Commit changed from b77de0ff1050b206b78e8b127f8dce32c0ff6c68 to 0b0d2f82b033532d2084219b1ee1d6f0c63f45e0

Branch pushed to git repo; I updated commit sha1. New commits:

 0b0d2f8 Correcting mistakes. Two errors are still waiting for correction: ParityCheckCodeStraightforwardEncoder.unencode_nocheck and ParityCheckCodeGeneratorMatrixEncoder.__init__

### comment:9 in reply to: ↑ 7 Changed 2 years ago by richmond

Hi,

I just noticed something: according to the diff, you also implement relative_distance in this ticket. I'm quite sure it's because you were working on #21315 at the same time. Could you please remove the related lines from linear_code.py?

David

Hi,

True! I removed this method.

Tania.

### comment:10 in reply to: ↑ 6 Changed 2 years ago by richmond

• __init__: I think you forgot to remove the docstring. Here it does not make a lot of sense to write "An abstract linear code object should never be created.". I usually omit

docstring for __init__ as I consider there's already some doc in the class docstring.

Just to clarify: __init__ methods, like all other methods, should have a docstring. But it can be made into a bare minimal, with a single doctest constructing an object and running generic tests on it; typically:

def init(self, ...):

""" Initialize this parity code

TESTS

sage: C = ParityCode?(...) sage: TestSuite?(C).run()

"""

Oops. Well, I guess I should open a ticket to fix my __init__ then, because all my classes look like:

class ClassName:
r"""
Short description

Long description

EXAMPLES::

....
"""

def __init__(self, ...):
r"""
TESTS::

....
"""

BTW, doctest line 383 in parity_check_code.py should run unencode_nocheck, not unencode. sage --coverage complains about this... But not about missing docstrings in my __init__ blocks.

David

Hi,

I changed unencode by unencode_nocheck, but it gives me an error in tests. I do not understand exactly where does it come from, so if you can take a look on it please...

For description and test, I just followed the toturial: http://doc.sagemath.org/html/en/thematic_tutorials/structures_in_coding_theory.html I also took inspiration from others classes in coding theory (like Hamming code or GRS code). I hope it is correct like that.

Tania.

Last edited 2 years ago by richmond (previous) (diff)

### comment:11 in reply to: ↑ 4 Changed 2 years ago by richmond

Hi Tania,

• You forgot to add your module (Python file) to the index of documented modules for sage/coding, which means no html/pdf documentation file will be generated for parity_check_code.py. You have to add sage/coding/parity_check_code to sage/doc/en/reference/coding/index.rst.
• Once done, you'll notice building the documentation fails because there's already a reference named [W] in /sage/schemes/hyperelliptic_curves/invariants.py. So, you have to rename your [W] to something else ([W88]?)
• I would add a small definition of a parity-check code in the module docstring (where you wrote "A simple way of detecting [...] even."). What about:

" A parity-check code of dimension k over F_q is the set:

\{(m_1, \dots, m_k, -\Sigma_{i=1}^{k}) m_i \mid (m_1, \dots, m_n) \in F_q\}"?

## ParityCheckCode

• __init__: I think you forgot to remove the docstring. Here it does not make a lot of sense to write "An abstract linear code object should never be created.". I usually omit

docstring for __init__ as I consider there's already some doc in the class docstring.

• __init__: I would add or no base_field.is_field() to the sanity check line 73. Indeed, for now if one passes anything but a field, the error message says AttributeError: 'anything' object has no attribute 'is_finite' which is not the most informative error message. BTW, it's also possible to create a Parity-Check code whose dimension is 0 or less. I know #21326 will fix that soon, but I guess a simple check on dimension can't hurt, right? And in any case, there should be at least something like if not isinstance(dimension, (int, Integer) raise ValueError("dimension has to be a Python int or a Sage Integer"). Otherwise, if the user passes anything but an int/Integer, it will fail with some weird error.

## ParityCheckGeneratorMatrixEncoder

• Doctest lines 179-183 calls the straightforward encoder instead of the generator matrix encoder.
sage: C  = codes.RandomLinearCode(10, 4, GF(11))
sage: codes.encoders.ParityCheckCodeGeneratorMatrixEncoder(C)

works while it should fail.

• generator_matrix: I'm quite sure l.246 should be G = G.augment(vector([field.order()] * k)). Because for now, the encoding does not work:
   sage: C = codes.ParityCheckCode(GF(5),7)
sage: E = codes.encoders.ParityCheckCodeGeneratorMatrixEncoder(C)
sage: m = vector(GF(5), [1, 0, 4, 2, 0, 3, 2])
sage: sum(E.encode(m))
4 #should be 0

## ParityCheckCodeStraightForwardEncoder

• I know we already talked about it, but I still think StraightforwardEncoder is not the most straightforward name... What about ParitySymbolEncoder? Note that I am not adamant with this name change though.
• encode: I think you can do it in one line with return vector(self.code().base_field(), message.list()+[-sum(message)])

## Encoders

• Last one: lines 404/405, you chose "ParityCheckGeneratorMatrixEncoder" and "ParityCheckStraightforwardEncoder" for the encoders string names. I'd choose "GeneratorMatrix" and "Straightforward" instead, as the user has to type those names when they use C.encoder('str_name'). See the names I chose in grs.py for instance.

Apart from those small remarks, it looks good. Great job for your first code class :)

David

Hi,

Thank you for this long but useful comment. I made changes you asked except in ParityCheckGeneratorMatrixEncoder. Indeed, if we suppose that a word is a codeword only when the sum of all its coodinates, then G = G.augment(vector(field,[- field.one()] * k)). It was my mistake because I use to work with binary codes so there is no -.

Tania.

Last edited 2 years ago by richmond (previous) (diff)

### comment:12 Changed 2 years ago by jlavauzelle

Hi Tania,

A first general comment. When you push a new version of your code, you must take care if a new release has been done. For instance, your code is written under release 7.4.beta1, but we're now under 7.4.beta5. So you need to merge your version with the latest beta. New releases are announced here: https://groups.google.com/forum/#!forum/sage-release

• Your push seems to remove relative_distance() method in linear_code.py. Now, this method is on the latest release, so you need to keep it.
• l.9 : m_k instead of m_n
• In ParityCheckCodeGeneratorMatrixEncoder, the constructor won't raise any error as it is claimed in the doctest. Generically, you can run sage tests (./sage -t --long src/sage/coding/parity_check_code.py) to see if your doctests are well written.
• Then you'll also see that codes.RandomLinearCode(n, k, F) is deprecated. Use codes.random_linear_code(F, n, k)instead.
• l.199 to 210 : it seems that you only copy paste the _latex_ function of the code, whithout replacing the code by the encoder.
• This is the same for the _repr_ method (only the doctest).
• The two previous remarks also hold for ParityCheckCodeStraightforwardEncoder (lines 297 to 320).
• You need a unencode_nocheck() method in ParityCheckCodeStraightforwardEncoder.

Best,

Julien

### comment:13 Changed 2 years ago by jlavauzelle

• Branch changed from u/richmond/parity_codes to u/jlavauzelle/parity_codes

### comment:14 Changed 2 years ago by jlavauzelle

• Commit changed from 0b0d2f82b033532d2084219b1ee1d6f0c63f45e0 to 1aa4b4e7333d96fc39836bee509fee97cfe28dfc
• Milestone changed from sage-7.4 to sage-7.6
• Reviewers changed from David Lucas to David Lucas, Julien Lavauzelle

Hi,

I fixed what I proposed and solved the merge conflict. Maybe it still need a quick review because I did many (minor) changes.

Julien

New commits:

 62abcd7 Fixed merge conflict. 1aa4b4e Fixed doctests. Shorter output. Generator matrix encoder inherits from the generic one. Fixed encoders. Cleaned the code.

### comment:15 Changed 2 years ago by dlucas

• Status changed from needs_review to positive_review

Hello,

Tests pass, doc compiles, I agree with your changes, giving the green light.

David

### comment:16 Changed 2 years ago by jsrn

• Status changed from positive_review to needs_work

Codes should no longer print in the way the current implementation here does.

### comment:17 Changed 2 years ago by jsrn

• Status changed from needs_work to positive_review

Sorry, I was being too fast. Everything is allright here.

### comment:18 Changed 2 years ago by vbraun

• Branch changed from u/jlavauzelle/parity_codes to 1aa4b4e7333d96fc39836bee509fee97cfe28dfc
• Resolution set to fixed
• Status changed from positive_review to closed
Note: See TracTickets for help on using tickets.


text:
#### 题目列表

r>0

p is 75 percent of r.

Quantity A:50 percent of p

Quantity B:25 percent of r

From the cars manufactured in 2005 indicated in the table, one car is to be selected at random from the model A cars and one car is to be selected at random from the model B cars.

Quantity A: The probability that the model A car selected will be one that required repairs in its first year

Quantity B: The probability that the model B car selected will be one that required repairs in its first year
x>0

Quantity A:$\frac{x}{x+1}$

Quantity B:$\frac{x+1}{x+2}$
In the xy-plane, P and Q are two points on the line with equation y=2x+1. The x-coordinates of P and Q are 2 and 6, respectively.

Quantity A:The length of line segment PQ

Quantity B:9

In the circle with center G, arc MKN has length $\frac{8}{3}π$.

In the circle with center J, arc PQR has length $\frac{16}{3}π$.

Quantity A:a

Quantity B:b
y+2<0

Quantity A:$\frac{|y|}{y}$

Quantity B:-2

On the number line shown, a and b are the coordinates of two points.

Quantity A:ab

Quantity B:-ab
A small company made a profit of $9,000 last quarter. The dollar amount of the profit was distributed equally among the 4 divisions of the company. If each division distributed its dollar amount of the profit equally among the people in the division, how much more money did a person in a division of 6 people receive than a person in a division of 10 people?$_____
In a social group 62 members enjoy dancing and 28 members enjoy hiking. If each of the 75 members in the group enjoys dancing or hiking, how many members enjoy both dancing and hiking?
On each of the first 15 days of April, Aaron spent 5 percent of his monthly income for April. If at the end of the 15th day of April, he had $1,250 of his monthly income remaining, what was his monthly income for April? A school stadium contains 2,000 seats. For a certain game, student tickets cost$3 each and nonstudent tickets cost $6 each. What is the least number of nonstudent tickets that must be sold so that the total ticket sales will be at least$8,400 ?
If r, s, and t are consecutive positive integers and r< s< t, which of the following statements must be true?

Indicate all such statements.
26, 29, 32, 33, 36, 37, 41, 42, 48, 50

The 10 numbers in list M (not shown) are determined by multiplying each of the 10 numbers in the list above by 5 and then subtracting 20 from each product. How many times greater is the range of the numbers in list M than the range of the numbers in the list above?

_____times greater
In which of the following years was the number of foreign-born residents greatest?
From 1960 to 1980, the total population increased by approximately what percent?
The ratio of the number of foreign-born residents in 1900 to the number of foreign-born residents in 1980 is closest to which of the following?
The sides of a triangle have lengths 7.5, 11, and x. If x is an integer, what is the least possible value of x ?
When the integer n is divided by 24, the remainder is 21. Which of the following must be a divisor of n ?
What is the value of the smallest positive integer n for which $6^n$ is a multiple of 24 ?
If $5^{x^{2}-2x}=\frac{1}{5}$, what is the value of x ?

25000 +道题目

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