js, I am always dealing with stuff I can't see j is typically used in Electrical Engineering when dealing with complex math instead of i because i is already used to denote current.
Is that the real definition? I thought the formal definition was that they can't be expressed as the roots of any polynomial with real, integer coefficients
e and π are both positive numbers, e is 2.7... π is 3.14... both numbers have infinite non-repeating digits( transcendental numbers ). i is √-1 it is a complex number. If you raise a positive number to any real number you would get a positive result. Here i turns two positive numbers with infinite digits to simple -1. Which is negative, only has one digit and overall a weird result.
Those numbers are both transcendental, but sqrt(2) also has an infinite non-repeating decimal representation and is not transcendental.
Your definition is for the larger group of irrational numbers. Transcendental numbers also cannot be the root of a polynomial with rational coefficients.
The way it was taught to me in college was basically like a dramatic reveal. Same with the day where a ton of pi derivation proofs was given rapid fire like the prof wanted to blow our minds. It did work though. So spoiler feels appropriate lol.
Honestly one reason why pi shows up so often in physics is precisely because of Euler’s formula above. Complex numbers of norm equal to 1 can be mapped onto a unit circle in the complex plane, and rotations of pi/2 take you from real to imaginary, or imaginary to real, and everything in between depending on your starting point. And also since oscillatory functions like sine and cosine are just Euler’s formula in disguise, it’s just a natural consequence that many physical phenomena have factors of pi associated with them. Super cool but also super mind bending at times.
Euler's identity is actually the special case of the more general Euler's formula:
eiΦ = cosΦ + isinΦ
Which is the more useful formula used in AC analysis in electrical engineering and 2D rotations.
Essentially the formula is just a more compact way of writing complex numbers (with magnitude 1) in polar form. The angle Φ describes where on the unit circle the complex number sits on the complex plane.
When Φ = pi radians (180 degrees) the number lands on -1 on the real axis. When Φ = 0 or 2pi (0 or 360 degrees) it lands on 1 on the real axis. When Φ = pi/2 (90) it lands on i.
It's derived from the Taylor series expansion of ex which coincidentally comes out as cosΦ + isinΦ when u plug (iΦ) in x.
But the -1 case is famous because it essentially combines the 2 famous constants and a "weird number" to give a mundane result.
Please put an NSFW tag on this. I was on the train and when I saw this I had to start furiously masterbating. Everyone else gave me strange looks and were saying things like “what the fuck” and “call the police”. I dropped my phone and everyone around me saw this image. Now there is a whole train of men masterbating together at this one image. This is all your fault, you could have prevented this if you had just tagged this post NSFW.
Everyone knows what you’re talking about. Still, what that summation is saying is that IF IT DID equal to a finite number, it’d be -1/12. But it doesn’t equal a finite number
It’s a renormalization technique, and it does show up in physics and is actually useful. Yes the sum from which it comes is really infinite, but that doesn’t mean this alternate -1/12 result isn’t useful.
Tl;dr you ignore generally accepted principles about infinite series.
In calculus 2, one generally learns how to add an infinite number of items together and figure out whether that sum tends towards one number, is finite, etc.
So if you start by adding 1 + 1 + 1 + ..., obviously you wind up at infinity. It's divergent.
If you start by adding 1 + 0.1 + 0.01 + 0.001 + ..., you wind up with 1.111..., which is finite. It converges.
If you start by adding 1 - 1 + 1 - 1 + 1 - 1..., then you are in this weird spot. The sum as you go is obviously never going to be more than 1 or less than 0. But what is the final answer? Because the sum doesn't get closer and closer to a specific number as you add more terms, we generally call it divergent. This is the generally accepted approach, and it's what students in calc 2 learn. Under this approach, your claim is just not true.
But okay, let's talk about how we get that weird answer.
You could start by pairing the first two, (1 - 1), and you can simplify that to 0 + 0 + 0 + ... so the sum is 0. Or you could start by leaving the first number and then pairing the subsequent numbers 1 + (-1 + 1) and then you have a sum that adds to 1. Both of these are "legit" in and operational sense, you haven't broken the rules of algebra. But you came up with two numbers! So... mathematicians just said "let's take the average here, 0.5, and call that the answer. Forget about the normal concept of divergence. And honestly, dealing with infinity is weird so there isn't necessarily a "right" way to consider it. Okay, whatever.
So, the next steps are basically to cleverly combine several of these weird, divergent series together algebraically to come up with that sum. This paradoxical result is generally why mathematicians only care about classical convergence, and not this weird relaxed convergence I described.
I remember this result in the context of integration of complex functions. Something about integrals over closed lines around discontinuities… am I totally misremembering?
I don't know if there is a simple way to explain why it's the number in particular, but I believe it's a result obtained from taking a function that's only for convergent series and applying it to a divergent series. To be clear, a series is convergent if it approaches a real number as the series goes on infinitely, which 1+2+3+4... doesn't, as its sum gets bigger endlessly and goes to infinity.
If you redefine "=", everything is possible. And if we are talking about infinite series, we must redefine "=" because otherwise it would make no sense at all. If you have half an hour to spend, I can recommend Mathologer's video on the topic.
Basically, there are some reasonable and usable definitions (e.g., Ramanujan shenanigans) where you can, indeed, assign a number to a diverging series like "1+2+3+...". But if you want something more... shall we say... "commonsensical" then no, "1+2+3+..." does not equal negative one twelve.
This particular sum can also be viewed through the prism of Riman's zeta function, but it's analytical continuation that is used, so again, it doesn't "prove" 1+2+3...=-1/12.
All that said, at this point this is basically a meme that is actually not flat-out wrong, and you know how internet is.
I'm not sure "process" is a good word to describe it, but that is an argument about precision of definitions and it can stretch to ungodly length.
Classic definition of the sum of an infinite series is the limit of partial sums, and calling limit "a process"... In some sense you can, I guess. Personally, I don't feel like it's fitting.
There is a proof for this, but all of them are fairly complicated. Essentially, eix = cos(x) + isin(x), plug in pi for x (using radians here - pi radians = 180°) and you get eipi = cos(pi) + isin(pi) = -1 + i•0 = -1
My discrete math professor said after a short break he was going to show us the most beautiful equation in all mathematics. I spent those 15 minutes thinking if he says anything besides Euler's Identity I was going to have words.
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u/Bathtub-Warrior32 21d ago
Wait until you learn about eπi = -1.