#euler's formula

[jijjmoon]

3.14 Happy pi day!

so, umm I came back earlier than I thought. I can draw if I draw at daytime. I usally drew at night but drawing at daytime is better than not drawing.

[aedislumen]

Another question because why not..

Your favourite math equation?

I admit that there have been a lot of equations I liked so much in the past that I wanted to have them tattooed (Shroedinger's, Dirac's, Maxwell's etc).

While my fondness for them has not vanished, it's only Euler's identity that I might consider to get on my skin (I don't think I will ever do it, because I don't like the idea of having ink under my epidermis...)

Despite the name, the identity isn't present in any of Euler's works, in fact, its name derives from the fact that it comes from a special case of Euler's formula, that of x = π).

Euler's formula:

If θ = π, one gets the identity:

I like it a lot because it ties together the most important constants and numbers related to mathematics:

e is the Napier's constant (aka Euler's number), an irrational number, which is present in logarithm and exponential functions. It's fundamental to study growth and decrease in physical processes. it's often approximates as 2,7.

i is the imaginary unit, defined as such: i^2 = -1. Its formulation led to the introduction of imaginary numbers, that allowed some "impossible" equation to be solved. (Note that they are solvable in the domain of imaginary numbers, not in that of real ones)

π is the ratio between the circumference and the diameter. Like Napier's constant, it's an irrational one and is present pretty much everywhere, both in math and the real world.

1 is the neutral element for the operation of multiplication, meaning that any number multiplied by 1 gives back the number.

0 is the neutral element for the operation of addition: a number plus 0 gives as result that number. It's also used to define negative numbers.

[anumberofcatschilling]

Both parts are at least somewhat related to each other. I'll clarify for any FAQ's later when I have the spoons.

Transcript:

A number of calculations for a general solution for a linear second-order differential equation with constant coefficients. The assumption here is that for coefficients a, b, and c in the differential equation, b-squared is less than 4ac, resulting in complex roots for the quadratic. The root in the upper two quadrants of the Complex Plane is used to get constant coefficients alpha and beta, and then raising e to the power of alpha plus i times beta results in two solutions that solve the differential equation because of Euler's formula. The general solution and its first derivative are written with constant coefficients C1 and C2.

Would I be rebuked for throwing the above (the general solution and its derivative) into a matrix & getting to rref for C1 and C2?

Also, I've got Autism & ADHD, and I'm aware that Autism can be more disabling than how I experience it, but the amount of control I have over my environment (complete control over the lights in my dorm & freedom to go wherever outside of class) means my ADHD is what fucks me over 75% of the time. Differences in severity aside, why do I get the impression that the general public would look at Autism + ADHD and think the Autism would be more disabling than the ADHD?

[oopsallsticks]

Current Family Photo 🧡🖤🤍

I love every one so far -B

[ryanthel0ser]

I am now believing that Vincent just wrote all of his calculus notes on a wall in the basement for reference because he too was frustrated flipping through his notes like I am

[river-gale]

STOP naming things after euler. i have nothing but respect and admiration for leonhard euler and his contributions to the beautiful field of mathematics but TOO MANY THINGS ARE NAMED AFTER HIM. please. enough. it's getting so confusing

[gemkun]

anonymous said : hey ratio what's your favorite mathmatical theory to ponder?

      ⸻       ❝   my   eponym   —   the   golden   ratio.   the   divine   proportion   that   approximates   to   1.618033987   ,   represented   by   Φ.   ❞   typical   ,   perhaps   even   predictable   ,   but   there   is   no   disputing   its   application   across   many   ,   if   not   all   ,   disciplines.   nor   it   being   the   pinnacle   of   aesthetic   perfection.   harmonious   in   its   simplicity.   ❝   it   is   a   notorious   theory   ,   achieving   its   recognition   as   a   unifier   of   mathematics   with   artistry.   creating   a   bridge   between   what   once   was   often   thought   to   be   dichotomous.   ❞

[blondiest]

okay kill me with a rock if i reappear here before tomorrow i'm serious

[natural-blogarithm]

i got to show proof by induction to my summer camp students today, it was very exciting

[psycherprince]

not to be That Guy but love is literally when your fiancee looks over in bed to see you've gotten a pen and paper and started doing calculus for funsies and finds it endearing rather than offputting

[wannadewar]

omg we've hit the point in the season for playoff math i'm about to go bury my head in the sand and scream bc isn't it worse this year since the east AND west look tight

[evenant]

Se Euler não fez, eu fiz

Eu não sei como foi que Euler inventou ou calculou $\latex e^{it}$ mas De Moivre já havia provado a sua fórmula e, como de hábito para a época, deve tê-la comunicado a outros matemáticos de renome, como Euler. Se Euler não usou a fórmula de De Moivre para definir , eu o estou fazendo aqui.

[storymaker14]

Bizzaro Right Triangles

Okay, we all know that, if you have a right triangle with sides a and b and hypotenuse c, that means a^2 + b^2 = c^2, right?

So you can have 3-4-5 right triangles, and 5-12-13, and 7-24-25, etc.

And technically, a 1-i-0 right triangle follows this pattern, too, which has made the rounds in the math meme community.

But I found something better and weirder. I found a whole family of better and weirder.

I'm gonna skip a bit of the beginning and start here:

cos x = (e^ix + e^-ix) / 2

That follows from Euler's formula; if you need a walkthrough, I can provide, but I want to start with the above. So then, it follows that:

cos (i*lnx) = (x + 1/x)/2 = (x^2 + 1)/2x

Which means that an angle of (i*lnx) radians in a right triangle will have an adjacent side of (x^2 + 1) and a hypotenuse of 2x. By Pythagoras, then, the opposite side must be (x^2 - 1)*i.

But please note: if x is odd, then the lengths of all three sides will be even, and thus can be divided by 2.

Which means -- are you holding onto your hats and shoes?!? -- means that the proportions of the right triangle with angle i*ln(3) radians... are 5-4i-3.

And for i*ln(5) radians, they are 13-12i-5.

And so forth, with all the familiar Pythagorean triplets sqrt(2n+1)-n-(n+1) showing up, just with one side imaginary and the hypotenuse and remaining side swapped -- so, (n+1)-n*i-sqrt(2n+1). They still fulfill Pythagoras, every single one.

Which I think is, pardon my directness, fucking terrific. But just as a little bonus, please note that this means the triangle with angle i*ln(2) radians, the proportions are 5-3i-4, which is just delightful IMHO.

[abiwaif]

Visualization of Euler's formula. ✍️

Euler's formula shows the deep connection between complex numbers and trigonometry. This means that when you take an exponential of a complex number, it combines circular motion (cosine and sine) with growth (the exponential function).

[askviktor]

what are your craziest most ridonkulous most astonishing hear me outs. i have a list

- the quadratic formula

- the hexcore

- the lesbian flag

- bicycle chain

- pink rhinestone cowgirl hat

- jellyfish haircut

- alternate rhyme scheme

- lady macbeth

- sheila birling

Ah, I have become familiar with this concept. You wish to know the people, objects, or concepts with which I would like to have intercourse, yes?

I am rarely sexually attracted to people, so creating such a list will prove difficult. Perhaps I will list things to which I have a deep attraction, in some form.

- The Hexcore

- Euler’s Identity

- Painlessness

- Chronomancy

- The Riemann Hypothesis

Eh, this is the best I can do.

[ducktoonsfanart]

Scrooge McDuck as Julius Caesar and the March Ides and Donald Duck as an ancient Greek mathematician (Pythagoras and Archimedes) and Pi Day - Fearsome Four (Bushroot, Liquidator, Quackerjack and Megavolt) from Darkwing Duck and March and April holidays - Duck comics, Darkwing Duck and Duckverse - My style

Even though I'm late and don't arrive on time, I still decided to draw something related to the international days that are celebrated during March and early April. So I drew a few drawings related to that. The first drawing I drew shows Donald and Scrooge dressed from the classical period (the time of ancient Greece and ancient Rome) related to March 14th and 15th, or according to the ancient Romans, the days of the Ides of March. Ides, or in Latin Idus, means the middle of the month, and on that March 15th in 44 BC, before the birth of Christ, a terrible event occurred in the then Roman Republic when corrupt senators led by Brutus and Cassius killed the Roman dictator and leader Gaius Julius Caesar, which put the Roman state in danger and a new civil war. In the end, the senators who killed Caesar were punished and after more than 15 years, the civil war ended with the victory of Caesar's grandson-nephew Octavian Augustus, who declared himself emperor, and Rome became the capital of the Roman Empire. Scrooge McDuck is here as Gaius Julius Caesar and it's part of my Duckverse in History, luckily he wasn't killed, but he got a lot of knives in his salad like Caesar's salad. By the way, Caesar's salad was only created in 1924, named after the Italian chef Caesar Cardini who worked at Caesar's restaurant in Tijuana, Mexico, so it has nothing to do with Gaius Julius Caesar, but that's how a meme related to it was created. You can imagine Scrooge's reaction to that. "Beware of the Ides of March!" By the way, if you are more interested in Scrooge as Caesar, look here:

Donald Duck is here as an ancient Greek mathematician like Pythagoras who gave the Pythagorean theorem regarding a triangle and how it is calculated. "The sum of the areas of the two squares on the legs (a and b) equals the area of ​​the square on the hypotenuse (c)." You can see the given formula in the drawing. By the way, Pythagoras is one of the greatest mathematicians of all time and lived in the 6th century BC and came from the Greek island of Samos. He was an excellent geometer and was a lover of music, and he also founded a secret order related to that. More precisely, I was inspired by the cartoon "Donald in Mathmagic Land" from 1959 in which Donald goes to the land of mathematics and then goes back in time to Ancient Greece and meets Pythagoras and becomes a member of his society. March 14th is the International Day of Mathematics because of the infinite number Pi, which is a Greek letter in their alphabet and is often used in mathematics and physics. "It is defined as the ratio of the circumference to the diameter of a circle or as the ratio of the area of ​​a circle to the square over its radius." The number is actually 3.14159…It is also called Archimedes' constant by the ancient Greek mathematician Archimedes from the third century BC who lived in Syracuse, now Sicily. The number Pi was introduced there by William Jones and popularized by Leonard Euler in the 18th century. On that March 14th, 1879, physicist Albert Einstein was born, so that day is also celebrated as Einstein's Day. That's why I drew Donald wearing ancient Greek clothes with the letter Pi on his T-shirt, and wearing a green laurel wreath.

The other two drawings are related to Darkwing Duck, or rather the villainous team of the Fearsome Four consisting of Bushroot, Liquidator, Megavolt and Quackerjack, certainly a great team and I drew them also related to international days. Dr. Reginald Bushroot wears a Leprechaun hat and a four-leaf clover, which also marks St. Patrick's Day, which is celebrated on March 17. Bud Flud (Flood), or Liquidator, presents his product of bottled plastic water, which marks International Water Day, which is celebrated on March 22. Water is certainly important to us, and plants and water definitely go together. And finally, Quackerjack and Megavolt together while Quackerjack holds a toy tooth that winds up, and they mark International April Fool's Day, which is celebrated on April 1. I haven't drawn them in a long time, and they deserve it so I drew them, and they are also one of my favorite characters.

I hope you like these drawings and that you like these characters, especially from Darkwing Duck and that you like these ideas, and feel free to like and reblog this, just please don't copy this without mentioning me and without my permission. Thank you! And I apologize for the long post. Also happy holidays celebrating this that I tagged.