The PROOF: e and pi are transcendental

K, I made it through Level 1 no problem - the geometric series in question is 1/(b+1), 1/(b+1)^2... with a common ratio 1/(b+1) and therefore the infinite sum of the series is 1/b, skipping algebra.
Level 2 was ok until the end bit, where I needed Marty's proof, which I just about understood, but took a little thinking to understand why we were just allowed to choose m as we please.
Level 3: so far, so good. We can demand a = 0 as if it were not 0, then we would have e(b+ce+de^2) = 0, so the latter quadratic factor would need to be 0, but this cannot be the case as e is not the root of such an equation, per Level 2. At least I think that logic works out.
Done with Level 3, though I'm concerned that the small bits could be 0, of course.

Could there be a base system for which e and π could be represented as an integer if so i think they would be prime

May I suggest you change the background into a more elegant and less intrusive black board?

Have you covered conways surreal numbers? I would love to see that if you have

Yes, philosophically, {0 somehow approaches 1, via♾} on the x-y axis! We know this, because 1approaches 2. ⛑😎

If .9999..... Equals 1 why doesn't 3.14159265359... equal 4

敬

I was looking at Mapquest, saw these symbols in California near an old star Fort That’s a current military base, with Lat an long that is 33N 117 S which added are 6 9, which is strange most sacred places are built at lat an long that added equal 3,6,9,but others can be at 5,7,8, sacred Numbers.. as I was looking around the area there is a what looks like mounds of earth that have the symbol E * Pi (symbol).. thought it was pretty fascinating. No idea how new or old the mound image is.. just cool

Engineered math. 😊

by the time it's level 7, i may run mad

All your efforts that go into making these concepts accessible for non-mathematicians like myself, is much appreciated. Thank you!

You sure did a good job showing that .999 repeated doesn't equal one. Since it's described by the infinite series for 9/10^x with x starting at 1. So any point you stop it equal ((10^x)-1)/10^x plus some number smaller than 1/10^x

(don't wanna mentioned, Lucas numbers are rounded-up.. decimal)
From Tribonacci Ratio :
t. = 1.839 286 755 2
( t +1)/t = t (t -1) = 1.543 689
Find that : (π/e)^3 ={ t (t+1)}=( t+1)/t

Okay so what I learned from this is that contradictions are solvable by an pseudo-imaginary not-contradiction. Those not-contradictions are necessary for solving things that are known to exist.
therefore, there must be a math that includes those not-contradictions as not-contradictory. And said math must be contradictory to the standard integer-centric math.

”Complex demons” sounds like manifestations of PTSD, or something 😅.

Reading Marty’s proof that e is not a quadratic irrational finally made it click in my brain, where the ”alternating +/-” -series for 1/e comes from: Simply from substituting -1 for x, in the ”e^x” -series, and the fact that the powers of x alternate, in parity. 😌👍🏻

I like how honest the channel is to its mathematical reasoning. They simplify the concepts but not at the cost of giving incorrect or incomplete reasoning.

When I saw the gamma function, I wonder if this could be expressed as a Laplace transform and then the shifting has a natural application. I know what I’ll be doing tonight!

When I was doing my PhD in functional analysis in the 70s, I ran across a short 1 paragraph proof that Pi was transcendental written in Spanish with no mathematical symbols or equations in the proof. I spent a couple of days learning enough Spanish to understand and verify it.

Definitely worth your 200 hours, thank you!!

above teeth

The Mandelbrot Set in the thumbnail looks like a tardigrade (water bear)! 😂🙃👍

I would love to see dark background like @3blue1brown and @standupmaths

for practical reasons , the perfect curved line of the circle has been considered irrationnelle . but from 7/36 came e .

the irrational part of pi=4* i * ln(i/sqrt(2)+1/sqrt(2))+3

On page 4 of Marty’s proof of the transcendence of e and pi (towards the middle of the page):

For the inequality involving δ_k, would it not be more accurate to say that |δ_k| is less than the expression on the right? (And later on, that |δ_k| goes to zero as p goes to infinity, rather than δ_k?)

In general, δ_k is complex valued, and there’s no such thing as an ordering of the complex numbers (or at least, not a total order that respects the field structure of C… Though you could always go with the lexicographic order induced by R and see what happens 🤓)

The exclamation marks in the Taylor expansion of e = exp(1) mean "I am really excited about this beautiful sum!!!", but also the factorial.

A simpler proof that e is irrational is to show that 1/e is irrational, using its alternating series expansion.

Hey that's me

Teaching Assistants need to show how and why e-Pi-i 1-0-infinity Singularity-point Apature of universal omnidirectional-dimensional logarithmic resonance is revealed in the Camera Obscura, vertices at holes in the nuclei->apex of i-reflection inclusion-exclusion vortices, conic-cyclonic coherence-cohesion sync-duration holography, is this made-of-making elemental spin-spiral phase-locked holographic time-timing presence of nothing floating in No-thing Relativity. Hyperfluid slippery Conception.
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The symbolic representation of reciprocation-recirculation '"String Vibration", a Tensor by default, is a transverse trancendental sync-duration holography dimensionality in the Singularity-point Aether/Apature of entangled point-line-circle e-Pi-i 1-0-infinity instantaneous quantization potential oscillation, ..parallel coexistence crystallisation cubics, spheroidal nodal-vibrational emitter-receiver log-antilog density-intensity scaling and resonance prime-cofactor frequency positioning.

See Steven Strogatz, Veritasium, 3BLUE 1BROWN channels for various developments of the theme of sync-duration=> holography.

Brady Haran has a "body of video work" that can't be trancended, Oxford Mathematics, Unzicker's Big Flash concept plus the combination of string nodal-vibrational transverse trancendental logarithmic time-timing inherent in wave-particle collapse, Susskind's Singularity-point positioning etc, ..it's brought to a peak trancendence in this video.

And everything collected and collated has to be demonstrated by Huygens Optics, the transverse Lensing orientation-observation resonance shapes of density-intensity frequency-amplitude and alignment of entangled 1-0-infinity focus in the log-antilog picture-plane containment of. Cross-sectional holographic images.

Okay so hear me out, I strongly believe that the people who say that there are more numbers in-between 0 and 1 than are between 0 and infinity in the whole numbers are wrong. I believe this because when you consider something such as pi for example, for any decimal expansion which you could cut off would have a corollary in the whole numbers which could literally just be the number flipped over the decimal place. You can construct larger and larger numbers like this ad infinitum until you have an infinitely long number (for which you don't know the first digits) but you do know the last digits, much like how currently we don't know the last digits but do know the first digits for pi. I seriously don't see how this argument is refuted. In my perspective, the same logic for constructing a number this way is fine. We tend to prefer precision to decrease from large to small, such as only knowing the first few digits of a measurement. But I think this does not actually exist as a real symmetry in the universe, as numbers would have defined values even if we cannot measure them. For this reason I think cantor's diagonal argument is flawed because it is asymmetrical without justification. You can instead construct symmetrical numbers across the decimal place which would effectively counter the argument, since then the numbers are truly infinite in size. People don't like this argument because it goes against intuition of numbers needing to be defined by their largest scale first. I believe that numbers of infinite length with only known ending digits are still valid whole numbers so long as they do not have any nonzero part beyond the decimal point, which I believe is the definition of a whole number. I am convinced that cantor's argument could be used to likewise show that there are more whole numbers than decimal numbers if you simply flipped how you structure the argument. In fact you could define numbers which are simply the decimal-flipped version of transcendental numbers over the decimal place. If you have thoughts LMK and if you can explain why my logic doesn't work for modern mathematics please LMK. This is something which has really irked me for quite some time.

the infinite fraction directly proofs that e is not the solution to any quadratic equation with integer coefficients.

I wouldn't reproduce it. But I got a blueprint fot the backbone of the proof.
Which is a success for the team of authors.

E and pi cant be trancentental. This is an illusion if you belieeeeebe in infinity. Tho trancentental forms are holographic maps and are completly different thing that one may think.
e^IPi-1=0. Ponder about it. This what time is. Time is not what they claim. One can step outside of time. You do it everyday. But you cant grasp it, because noone told you, that it is possible. Intuition works out of time. Intelligence is not intuition. Different things.

Related: go check New Calculus. Infinite sums are delusional.

Yesterday I was feeling a little shortchanged as a rapidly balding middle-aged man. Mathologer makes me proud to be a member of the cue ball community.

Proof by contradiction:
1. assume false statement
2. handwave
3. contradiction
QED lol.

I have to make stuff up for a job.

e is irrational, granted because the non-repeating decimals go to infinity However, I don't understand how a repeating decimal that repeats to infinity is rational and an irrational number with nonrepeating decimals to infinity is irrational. They both go to infinity. It seems like an arbitrary distinction.

This is the best math video I've ever seen, including your other ones.

I am very amazed by how well digested the proof of transcencence of e is here. Amazing video 👏👏 Just before I found the video, I had dedicated my evening today to reading the proof in Hardy and Wright's book on transcendence number theory, which is the book from which the screenshot at the beginning of the video is taken, and although I got to understand the individual steps, I couldn't make sense of where the definitions came from. This video has been definitely very helpful to understand that, they really captured the idea. Even though they don't discuss the same proof to that in Hardy and Wright's book, one can guess that the idea beneath is the same. Brilliant!!

Were watching a trancendental proof and being explained factorials....

Can a mathematician cut a perfect pie in half? Where would they make the cut?