geometric proof

‘Ello, studyblr! Maths (or Math, whatever) can be quite / extremely / obscenely frustrating at times and very simple at others and it’s sort of hard to find a bridge between that. You can’t exactly cram for it either or study in under a week for finals so after flunking my first term epicly, I came up with a strategy of sorts and now my grade’s an A+ (yay!!). Anyway, thought I’d share ‘cause this shit could have saved me a lot of tears.

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Typically you loved college. It was light years better than high school. There was none of the drama and no cliques, you got to take classes on things you were interested instead of what was required by the state, and your professors were generally funny and interesting. Until you encountered The Class. At some point in their experience with higher education everyone has a class they flat out hate with a professor they can’t stand, and you were currently experiencing that. You were fairly certain your professor hated you and because of that you were afraid to go to office hours for help, it seemed like whatever you tried you kept getting bad grades and by the time you realized how much you hated the class it was too late to drop it. 

About halfway through the semester with your grade at a low C and a paper worth a large percentage of your grade that you were struggling with, you began to worry that you would fail and have a black mark on your transcript forever. Luke watched you stress about The Class multiple times, though when he asked what was wrong you gave a vague answer of “Just school stuff”. He’d never been to college, and you figured he wouldn’t understand. One day you were feeling a little worse than usual and when he asked if you were okay the floodgates opened. You spilled about everything that was going on while Luke listened and held you when you started to cry despite your best efforts not to. 

After talking about it you actually felt a bit better and that night you slept better than you had in weeks and the next morning you went off to your classes for the day (which thankfully did not include The Class) in a good mood. When you returned from your classes, your apartment looked very different from when you’d left it. It now resembled an art museum, pictures in frames occupying every spare inch of wall and Luke clad in black from head to toe resembled an art student standing among this. 

“What’s all this?” You asked confused. 

“The national museum of your achievements.” Luke spread his arms ta-da style. Still confused you stepped closer to the picture closest to you, realizing it wasn’t a picture at all. It was a spelling test from the first grade that you’d gotten an A on. Your handwriting was terrible, but spelling had never been your strong suit so when you’d brought home the test with a gold star on it your mom had put it on the fridge. You thought that she had thrown it away after awhile, but she must have kept it hidden around somewhere. “I called your mom and asked for any old papers or anything that she kept and she had tons. And I found the English papers you’ve kept since we’ve been living together and had it all framed.” Luke went on. 

You were touched. You knew that framing could be expensive, and you couldn’t imagine how much work finding all the papers and later hanging them up must have been. You threw yourself into Luke’s arms. “Thank you. This must have taken all day.” 

It probably had but Luke didn’t say so. “I thought you needed a reminder or how smart and amazing you are. And how you can do anything you put your mind to.” He whispered into your hair. You pulled back from the hug enough to thank him again and kiss him, letting the kiss go on a little longer than you normally would. When you broke apart, Luke offered you his arm like a proper Victorian gentleman. “Would you like the grand tour?” 

Luke led you through your entire apartment, where pretty much every good grade you’d ever gotten hung on the walls. You couldn’t believe that your mom had saved all this stuff and that Luke had took the time to sort through it all. The first A you’d received on a college paper, a paper you’d kept because of your professors nice comments that reminded you of why you loved writing so much, hung above the TV. The A on a chemistry exam on balancing chemical equations from high school, something you’d found impossible at first, was in the kitchen. Another high school exam from math on geometric proofs was in the hallway. Luke had even framed your lame awards from elementary school for learning cursive or getting an A in social studies-the ones they handed out on the last day of school so no one left out. Your high school diploma-which had probably been collecting dust in at attic of your parents’ house until now-was framed and in the place of honor above the bed. 

“Feeling better?” Luke asked once you’d circled your entire apartment and seen grades and exams you’d completely forgotten about until now. 

“Lots. Thank you.” You said again, kissing him again. 

That night you started your paper for The Class, making actual progress this time, determined to earn another A for the collection. 

HOW MATH GOES: a process


yes, numbers, yes, good.

first grade:

okay. adding the numbers. still good.

subtracting the numbers. ive got this

second grade:

multiplying the numbers. this is going to take some practice. i have to memorize what now?

fourth grade:

long division? I GOT THIS

sixth grade:

numbers can be negative? 

seventh grade:

the fuck? when did letters get into the math problem? idk if i can do this? ummm help?

ninth grade:

pshhhhh letters and numbers piece of cake, about this geometric proof though. 

tenth grade:

whooo. finally. back to the numbers and letters thing. wait, now you’re saying numbers can be imaginary?

omigod mathhh whhhyyyy

eleventh grade:

sin? cos? tan? the FUCK IS THIS

fking precalculus this is the worst

twelfth grade:

derivatives the what?




derive the derivative

related rates? geometry and derivatives at the sAME fuCKING tiME! 

now do it backwards




anonymous asked:

Hello, I've been researching about the 584 tritype and I have to admit I'm pretty much confused. Could you point some differences between 5w4 8w9 4w5 sp/sx and 5w6 8w7 4w3 sp/sx? And in terms of cognitive functions, which MBTI type do you think has the highest correlation with each tritype? Thank you.

5w4 vs. 5w6:  
5w4s have their withdrawn 4 wing (hornevian) while 5w6s have a compliant 6 wing. Because of this, 5w4s are more withdrawn, focussing on their own thoughts and ideas and emotions rather than the external world. 5w6s are more focussed on the external world and it’s investigation. The 4 wing is in the emotional realness group (harmonic) along with the 6. The 4 wing seeks someone to see them and validate their identity, while the 6 wing wants the support of a person but ultimately needs to be depended on as well. 4s keep others interested by limiting access, playing “hard to get,” and some of this is reflected in the 5w4. 6s are committed and reliable in order to gain the support of others but also strive to maintain their own dependence in fear of being without support, and some of this is reflected in the 5w6. Examples of 5w4s vs. 5w6s are:

Elon Musk (5w4) vs. Mark Zuckerberg (5w6)
John Nash (5w4) vs. Euclid (5w6)
Albert Einstein (5w4) vs. Stephen Hawking (5w6)
Nikola Tesla (5w4) vs. Charles Darwin (5w6)

If you compare any of those two my point should become apparent. 5w4s tend to value the beauty in science, and take a more scattered and creative approach to it (Musk sat down and programmed a sci-fi video game when he was 12 just because, Nash would frequently write his work on windows or make elaborate serial-killer style connections with newspaper articles and such, Einstein romanticized the universe and it’s beauty and found equations to represent it, and Tesla would invent things in his head and forget to write them down). 5w6s take a more planned and systematic approach (Zuckerberg has always had a grand vision of Facebook being a giant database of information and has taking systematic steps to develop it into such a thing, Euclid invented the axiom in order to invent the geometric proof, Hawking’s goal has always been the investigation and understanding of the universe in a very systematic way, and Darwin observed wildlife and took detailed notes before he understood his theories or drew any conclusions).  

8w9 vs. 8w7: 

Eights with a Seven wing tend to be more expansive extroverted and openly aggressive than those with the Nine wing. They are more likely to be sensation seekers and are generally more overtly ambitious than those with a Nine wing. Eights with a Seven wing especially tend to relish intensity of experience. Conversely, Eights with a Nine wing hold more of their energy in reserve and exhibit more of a grounded, even stubborn quality. They are generally less obviously volatile than Eights with a Seven wing but can slip just as radically into open aggression when pushed.

This summarizes the difference between 8w9 and 8w7 pretty well. Because 8 and 7 are both in the aggressive triad while 9 is in the withdrawn triad, 8w7s are more aggressive and “I’ll do whatever the fuck I want” than 8w9s. 8w9s present a more grounded and withdrawn energy and get aggressive during conflict rather than being aggressive all the time. 

4w5 vs. 4w3: 

4w3s are theatrical, dramatic, and effete. Compared to 4w5s they are generally more ambitious and competitive, and place a greater emphasis on appearing beautiful, desirable, and elite. They are said to be divas and aristocrats as their three wing transforms their sources of shame and defectiveness into art and expression, an aloof presentation that incorporates conventionally desirable elements into their style. 4w5s have a harsher edge than 4w3s and are the true outsiders of the enneagram. They tend to be more intellectual and introspective. They are more likely to philosophize their inner reality. Many 4w5s have an unflinching “this is me so deal with it” persona that’s harder and crustier in comparison to 4w3s. 

Basically, think theater kid vs. tumblr feminist (the stereotypes). Because of their aggressive 3 wing, 4w3s are more demanding and outgoing in their ambitions. They generally demand attention. 4w5s have their withdrawn 5 wing so they tend to be more oriented towards knowing themselves. Both types are self-centered, but 4w3s are focussed on their identity and ambitions while 4w5s are focussed on their identity and their introspection. 4w3s look outward to find their identity while 4w5s look inward. 

MBTI correlations:
Generally the 584 tritype is home to INTJs and INTPs. The other types I would say are reasonably common are ISTP and ENTJ (in that order). I don’t think going through the likely types for every tritype is going to be of much help to you and would take me a very long time, as there are 27 tritypes and 6 possible orders within each tritype which means I would have to do 162 of them. That’s too much information and much of it will be useless. I’ll stick to trends (not rules, just statistical trends):

An 8 fix that high usually points to high Te or inferior Fe. 5w4s are more commonly INTJs or INFJs but INTPs are common as well. So blindspots (when you’re Sp/Sx or Sx/Sp you have an So blindspot) are more common in thinkers than in feelers. 4w3 is generally correlated to Fi, it’s less common for Fe users to have a 4 in their tritype and they’re more likely to have a 4w5 than a 4w3. 8w9 is generally more aux Te than dom Te but it can go either way. Overall:

5w6 8w7 4w3: Most likely an INTJ 
5w4 8w9 4w5: Most likely an INTJ but INTP isn’t unlikely       

However, you shouldn’t base your enneagram type on MBTI-enneagram trends. Any MBTI type can be any enneagram type. They are two different systems.

5w6 8w7 4w3: Much more aggressive and focussed on the external world because of the two aggressive wings of 8w7 and 4w3 and because of the compliant wing of 5w6. Very common for INTJs. 
5w4 8w9 4w5: Much more grounded and focussed on their internal world because of the three withdrawn wings of 5w4 8w9 and 4w5. Very common for INTJs and INTPs. 

History of Quadratic Equation part 2

Much of the knowledge built up by the old civilizations of Egypt and Babylonia was passed down to the Greeks, who, in turn, gave mathematics scientific form.

Between about 540 and 250 BC, the ancient Greeks, represented by Pythagoras, his followers the Pythagorean, and Euclid, gave strict geometric proofs to algebraic problems, using lines and ares for numbers and products. ( in the image you can see how they understood quadratic equations)

The Greeks had considerable difficulty in solving cubic equations since their practice of treating algebraic problems as problems of geometry led to complicated three-dimensional constructions.

Quadratic equation part 3

In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation as follows: “To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.” This is equivalent to the formula we have today to solve the equation.

The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process. He also described the method of completing the square and recognized that the discriminant must be positive, which was proven by his contemporary ‘Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution. While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions as well as irrational numbers as solutions. Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation. The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.