fun math

Why on earth is matrix multiplication NOT commutative ? – An Intuition

One is commonly asked to prove in college as part of a linear algebra problem set that matrix multiplication is not commutative. i.e If A and B are two matrices then :

But without getting into the Algebra part of it, why should this even be true ? Let’s use linear transformations to get a feel for it.


If A and B are two Linear Transformations namely Rotation and Shear. Then it means that.

Is that true? Well, lets perform these linear operations on a unit square and find out:

(Rotation)(Shearing)

(Shearing)(Rotation)

You can clearly see that the resultant shape is not the same upon the two transformations.

This means that the order of matrix multiplication matters a lot ! ( or matrix multiplication is not commutative.)

Have a great day!

3

Science Math and Simpatico

————————-
Math is also his weakness. Brainstorm is happy to exploit this.

—————–
for the wonderful @gunthatshootsennui​, because it was his idea. :’D

Happy Valentines Day! :D
and for those who are single, like me, Happy Single Awareness Day!
—————-

s&s1: Law of Universal Gravitation
s&s3: Couples and Moments
s&s4: The Benefits of Hugging
s&s5: The Second Law of Thermodynamics
s&s6: Power Law

The Promised No-study SAT Tips

I saw that a lot of you wanted these~ Disclaimer: You still have to know English and the basics of math for these. This goes especially if you’re not a native speaker - your English needs to be at a pretty good level.

General:

  1. Read. A lot. Whenever you see a text that’s at least a paragraph or two long, take time to practice skimming. If you’re bored and have a little time, take something, for example a food wrapper, and try to find occurrences of a word (for example “Acid” for food) as quickly as possible. Hard mode: look for synonyms.
  2. Practice filling out the answer sheet. This is a massive time-sink for a lot of people, so you should practice to eliminate it. Print out an example answer sheet and try filling out the circles quickly and accurately without distracting yourself a lot. Hard mode:Try doing it while not focusing only on the circles - look away or start thinking about the next question.
  3. Check. A lot. The main goal of this strategy is to leave yourself enough time when you’ve filled out an answer for each question when you’re calm, know the questions and can focus on checking. Try and go through the questions, thinking, “This question tests this and that.” If you have the time, look at each answer and identify the error in it (harder for the math questions, but loads of fun if you can do it).
  4. Think in patterns: Whenever you’re stuck on an example question, don’t just check the answer. Try and understand how the person found it, if this question is similar to others you have seen. The SAT only uses a few different types of questions, there will rarely be something to surprise you if you know the common patterns.
  5. Rest: The SAT is a very demanding exam. Give your brain time to relax - my advice would be not to do anything mentally strenuous the day before the test. Also, something I found out from competitions - bring chocolate. The sugar in it helps your brain work better and shrug off tiredness and eating it will draw blood away from your brain, effectively hibernating it for the break to conserve energy. Also, it’s just a really tasty snack!

Writing:

  1. Use the right format for the essay. There are a lot of easy points for using the four/five paragraph system. Introduction, Reason 1, Reason 2, Conclusion. Begin each paragraph with a topic sentence and follow up with a story from your life or a book/movie to illustrate it. This way, even without using fancy vocab or grammar, you can get the points for structure and critical thought. Now just try not to make any obvious spelling mistakes and call it a day!
  2. Try to quickly find an argument for the essay. They don’t actually rate how intelligent your argument is. So, take a minute or two, breathe deeply, and no matter how stupid your idea is, write it out. (You might still want to take caution with sensitive topics, especially if you’re an international. A dumb mistake I made in my first sitting was bashing on American charity - that definitely did not endear me to the proctors.)
  3. Paragraphs: You have to have experience reading - look at how the topic never changes abruptly. Insert sentences that link what’s written before and after the gap. Final sentences of paragraphs shouldn’t raise more questions.
  4. Sentence questions: Skim through the questions. Try to answer most of them, the first thing that comes to mind, and fill out the answer sheet immediately. Chances are, if it sounds good to you, it’s the correct choice. Do this quickly, then try and do the paragraphs. After you’ve done this, go back to the questions and start checking.
  5. They usually test for a few broad topics. Identify if each sentence fits one of the patterns and answer accordingly. For the others, try and think what error they might want you to make. If you know you have the time, look at each answer in turn and identify the mistake in it. The most common ways for you to change a sentence would be:
  • Fragments: Try and see if each clause has a subject and a verb. Example: “In the dim light, making his way through the cave.” -> “In the dim light, he makes his way through the cave.”
  • Subject-verb agreement: Make sure that the subject is the one actually doing the action and singular/plural match. Example: “Gathering stones, the river was blocked by the men.” Did the river gather stones? No.
  • Consistency: Make sure that something introduced one way is always referred to like that (don’t switch out ‘one’ for ‘you’ or ‘they’). Make sure there are no extra linkers (”Since I was there, but he went too.”). Check if any verbs change tense when they shouldn’t. Don’t compare apples to oranges (”His homework was as good as John.” -> “As good as John’s”).
  • Adverb or adjective? If it describes a verb, it has a ‘ly’. Example: “She winked playful.” -> “She winked playfully.”
  • Singular or plural? Make sure not to refer to a plural object in singular. “Pandas, numbering in the hundreds now, is an endangered species.”
  • Prepositions, linkers, all the small words Sadly, you’ll have to know how they’re used.

Reading

  1. Word fill: Note the answers that obviously don’t make sense. Mark the one of the others that sounds best to you (in the answer sheet, too!). If you don’t know one or more of the words, aim for simplicity. After you’ve quickly answered all of the reading questions, come back to these. Look at the relationships between the gap and the sentence - are you looking for a positive or negative word? Antonyms or synonyms to something before? Try and guess what unknown words mean. This way, you will probably be able to eliminate all the wrong answers.
  2. Reading comprehension: You are not tested for understanding the text. Keep this in mind. What you are actually trying to do here is quickly find synonyms. If the question asks for “Was Anna’s family a) warm b) cold c) the spawn of Cthulhu?”, chances are that the text contains “Anna’s relatives acted chilly.” or something like that. Read the first question. Skim the text until it comes to that topic, then look for synonyms of the answers. Don’t make deductions! If you come across a ‘general message’ or ‘tone of the author’ question, skip it and answer it at the end of the text. The other questions will be in the same order as the answers are mentioned in the text. Checking: If you have time, look at each answer and try to see what in the text could mislead somebody to make that mistake.

Mathematics

  1. Calculator use: My advice would be to not bring a complex graphing calculator. They just slow you down. Try and do most operations by hand, then use the calculator only for, well, calculations.
  2. Basic topics to know: You are expected to be familiar with how to rearrange equations (ab=1 is the same as a=1/b) and solve linear and quadratics; cosine and Pythagorean theorems; number representations of lines and their intersections; median, mean and mode.
  3. Solve like a crab! One of the best things I learnt in “Fun Math” classes was that problems are solved more easily if you work from the answer back. Try and see what you would need (in terms of information) to find the answer. Then look back to the text of the problem - is what you need there? In most SAT problems, it is, or you can easily find it.
  4. Visualise: Especially for distance or geometry problems, make a small chart of what’s happening. Make lines for the distances the cars traveled or draw that pesky cylinder. Try and see in your mind how different elements move and which stay the same.

I guess this is all that I can say for now. Of course, this is my strategy so it might not work for everyone or it might not work without practice, so don’t think it’s a miracle solve-all. I’m always open for questions about ideas or specific problems, just write an ask~ And good luck to all future test-takers!

Try studying trigonometry before knowing how to do basic geometry.
Loving someone else and being loved by them before you know how to love yourself is just as confusing and frustrating.
—  You can know all the formulae but will you know how to apply them?
n roots of unity

When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the n roots of unity. i.e

As is common knowledge z = 1 is always a solution.


Multiplication as a transformation

Multiplication in the complex plane is mere rotation and scaling. i.e

Now what does finding the n roots of unity mean?

If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.

We just need to find the complex numbers that do this.i.e

This implies that :

And therefore :

Take a circle, slice it into n equal parts and voila you have your n roots of unity.


Okay, but what does this imply ?

Multiplication by 1 is a 360o / 0o rotation.

When you say that you are multiplying a positive real number(say 1) with 1 , we get  a number(1) that is on the same positive real axis.


Multiplication by (-1) is a 180o rotation.

When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis 

The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.



Multiplication by i is a 90o rotation.

Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation. 

This applies to -i as well.

so on and so forth,

Have a great day!

❉ = fluff / ✥ = angst / ✱ = comedy / ✺ = 😏

admins favorites = 🍑 / 🥝


JIN

YOUR WORSE THEN NAMJOON

i don’t mind people finding out

pancakes

you ruined my pickup line

unrelated topic, did you like what you saw ?

okay, i give you permission to kill him, but please wait until Thursday 🥝

guess who loves you

DO YOU NOT WANT ICE CREAM ???


i take you out one time and suddenly i’m the local delivery guy ?? ❉ & ✱


YOONGS

i am not cute

i don’t want to talk to you anymore

come help me out here ?

if it makes you feel any better ur a great chair

roasting you is like the best part of my day

i need lady stuff

what happend to the yoongi who doesn’t take shit from anyone ? ❉ & ✥

are you texting me song lyrics ?

come to the studio and sit on my lap ✺ & ✱

text me, call me, email me, send a letter; for all i care it could be by pigeon

i promise i only type like this when i like the person i text

completely giving up, starring me.

he’s just not hitting anything other then my knees

i was thinking about making out with you all day

not the pillow part you one celled creetin

lets get another doggo

thank god, i’d thought you’d never leave me alone

yeah being as short and mean as you is a real gift.


HOBI

fuck the fuck off

can i have like five of you ?

ur so pretty

have fun with math

but hobi wants some love ❉ & ✱

that’s cuz im trying to communicate with u

jung hoseok you’ll be the death of me

i wanna kiss your face every time i see you too

you know i support everything you do right ?

where did all that wisdom come from ?

stop talking to the small man

series : big bro hobi

i’m disowning you.

BOYS WILL EAT YOU ALIVE


JOON

you still up ?

yeah i can rap like the entire thing

right, and what does your mom call you ?

sure, have fun with my bras

that’s because you don’t speak Korean

i see you’ve had a few drinks ❉ & ✱


CHIM

ur such a loser

but you dont like shopping

the disrespect in this one is strong

it was supposed to be a surprise

you mean my singing ? ✱ & ❉ 

he’s just jaelous his dongsae had a child before him

very funny babygirl ✱ & ✺

we’ll go wherever you want

yoonmin : you’re like a candy bar jimin, half sweet half nuts.

you want me to come over ? we can cuddle


TAE

Hey ! i bought you ice-cream

the inventor of snapback look™

tae don’t you think the pillow wall is a bit of an overkill ?

thank god at least one of you know how to make a move

you’re not disappointing anyone by eating breakfast and lunch ❉ & ✥

did you do the english assignment ?

it’s just like 99.999% your fault

vkook : THIS IS A SHARED DORM

you’ll send me letters everyday right ?

are you telling me you’ve been smart this whole time ?

it’s in my job description


JK

wassup dudette

ayo ladies and gentleman

send nudes

what are you doing right now ?

look how cute we are

i just have a strong disliking to your personality

i like our bickering

babe i need your opinion ✱ & ✺

i’ll buy you food for a week if you tell me i’m your favorite

you could pay me a million and i still wouldn’t consider it

i didn’t want to leave your side that night because i wanted you to feel safe

stop kicking my chair

what does oppa mean ?

i miss every little thing you do

i’ll give you three breadsticks and a goat if you give me jimin

in your dreams

jk he’s not lord voldemort

i didn’t think this one through

good job, guk 


2 OR MORE

yeah and i’m just implying that i want to take you out on a date

what are you an owl ?

jin just made the worse recipe ever and i doubt he put ANY effort into it

i don’t like going outside

absolute soul mate has 16 letters and so does fuck jungkook asap

you’ve both got the same mental age

you are pure evil

i’m so whipped, this is incredible

for now i’ll have your lovely messages to look forward to ❉ & ✥

dont mess with me park ✱ & ❉

i like to send nudes ok ? if that’s my biggest flaw i think i’m ok.

if you were any thicker you’d look at the weather app outside

don’t start with me pindick

i’m not related to you anymore

we don’t want him

he really wanted to be in the picture

softie.

i can’t believe you saw mini nochu

i gain 3k for every shirtless picture

the power of jjang jjang man

next time you plan on rapping verse1 invite me

i’ll tell them to stop bothering you

okay, but if it attacks my dog i’m giving it to jimin ✱ & ❉

CRUSH CANCELLED ✱ & ❉

way to ruin the mood loser ✱ & ✺

dont act so smug you little shit

tell him to drop dead, with my compliments

you use that meme when you get nervous ✱ & ❉

was i your second choice ? ✱ & ❉

he’s a little small, but he’ll do ✱ & ✺

bring me some vodka and condoms ✱ & ✺

cheating series :

hold on imma go beat his ass

i told him the only thing smaller then him is his dick

this bitch gonna die tonight

i’m going to jail when i see his ass

when u gotta kill your hyung

dumped him so hard he probably got whiplash

losing a member series : 

we bangtan 6 now 🍑

how is it you’re the smartest and dumbest person i’ve ever met

YOU LOST HIM DIDN’T YOU

gang au series :

you act like you’ve never been shot before !

but did you die ?

is everyone alive and not in jail ?

mafia gangs dont have facebook pages.

teasing members series :

confess or i’ll set your mattress on fire

namjoon i will call your mother

so when are y'all getting kids

VIDEOS

footage of me at the club

footage of me at a house party

footage of me in the kitchen

footage of me with friends

why i shouldn’t be allowed to edit part 1

why i shouldn’t be allowed to edit part 2

The simple harmonic oscillator

Anonymous asked: Please explain the intuition of solving the SHM equation.

Okay Anon! Here you go, this is my rendition.


The problem

You have a mass suspended on a spring. We want to know where the mass will be at any instant of time.

Describe the motion of the mass


The physical solution

Now before we get on to the math, let us first visualize the motion by attaching a spray paint bottle as the mass.

Oh, wait that seems like a function that we are familiar with - The sinusoid.

Without even having to write down a single equation, we have found out the solution to our problem. The motion that is traced  by the mass is a sinusoid.


But what do I mean by a sinusoid ?

If you took the plotted paper and tried to create that function with the help of sum of polynomials i.e x, x2, x3 … Now you this what it would like :

By taking an infinite of these polynomial sums you get the function Since this series of polynomial occurs a lot, its given the name - sine.

I hope this shed some light on the intuition of the SHM equation. Have fun!