Please don't ever stop making maths videos. I truly appreciate your explanation (history and reasoning). Love your work and truly admire what you are doing.
@@granvillebarraclough8846 Why though? because at day 1 MA he'd be bound to be sick of it already or have moved on to another area of studies altogether? hehehe :D Sorry, I had to reply, even 2 years late :P I love the videos as well, gotta say.
+HalcyonSerenade Kinda reminds me of that scene in Monty Python And The Holy Grail, where King Arthur got annoyed by a scene and then ran in front of the camera and screamed "Cut that out! Cut that out!".
I tried to eyeball what the y=1/x -like function might be, and the closest I can seem to get is y=((e-1)^2)/(x-1) + 1. It matches the asymptotic behavior of x^y = y^x and passes through (e,e), but it doesn't perfectly match. The true function has something to do with the fact that ln(x)/x has two different x values that give the same y value for any positive y less than about 0.368, but for the life of me I can't figure it out.
@@davidiv4915 it's pretty satisfying if you sub in x=e to both equations. a^x = (e^(1/e))^e = e^(e^e) = e^1 = e loga(x) = log(e^(1/e))(e) = e (because of the line above)
I remembered when I first watched this video. It was like 2 years ago and I barely understand what’s going on. Now I watched it again after the previous 2 years and I found myself turned super focused. I understood everything Thanks for your explanation
e^(i*pi) = -1 is very useful in electrical engineering phasor notation and complex impedance. It essentially allows equations that would need calculus to solve to be easily solved algebraically.
e's decimal expansion is random, so the likelyhood of finding a four integer sequence in it that coincides with the year of the attack by the Spanish Armada is equal to 1.
@@chanderule605 Even though e is not a normal number, it's digits are still random. While you can't find any sequence of any length in it, you are pretty much guaranteed to find every four digit sequence, also the year of the Spanish Armada.
My Business Calculus professor at the University of Southern Maine was a big Euler fan. All of the classrooms Mainville used had a bit of graffiti; the Euler number, its square and its cube (all written out to five or six places) were written above the blackboard in magic marker. Since we used natural logs in the class, it was like an authorized cheat sheet during exams.
Thank you! There are lots of discussions of e on UA-cam, but this is by far the most lucid I have seen. I learned calculus 55 years ago, never used it in real life (I became a soldier and later a lawyer) and am returning to re-learn it at my own pace. This video has been a tremendous help in my understanding of this important concept.
Is not it such a treat? It is analogous to discovering classical music later on in life, when you were once exposed to it as a child, but strayed from it to listen to contemporary music in the interim.
e^(i*pi)+1=0 not only brings together the most important constants, but also the basic arithmetic operations addition, multiplication and exponentiation.
@@sticks9757 e , π and i(√-1 if you don't know) are the most iconic constants (except i) in Math. So in this equation they powered e to the power of π multiplied by i and added them with 1 which is equal to 0 (bizzare actually. How e, π and i are really complex numbers in math somehow combine into making a simple number which is -1) So in summary all the iconic constants(except i) in 1 equation.
This is my new favorite channel; it's so well-explained, interesting and fun even when dealing with complex topics, and yet somehow really adorable and nerdy at the same time.
I legitimately believe this will help when I try to do a college calculus class again. No one bothers to explain what’s going on with new material. When you understand why you use symbols and how, it makes it easier to solve things. Thank you.
What a lovely title. Captures the essence of this video quite well. Fully encapsulates and proves the quote by the "not-so-well-known-outside-his-circles" Victor Tran, "A letter says an entire video!" :)
Merlin Brennt But this is a mathematical channel. As soon as I saw the title, I knew that they were going to do something with Euler's constant. :) But I suppose the logarithmic graph and James Grime on the thumbnail helped too :)
e is such an interesting number. I came up with the strangest function that approached e as x approached infinity. y = 1/((x^(x+1)/(x+1)^x) - ((x-1)^x/x^(x-1) It’s pretty cool. ex: (56^57 / 57^56) - (55^56 / 56^55) Take a reciprocal of this number and be amazed
e shows up in very interesting places. x^y=y^x is an unusual graph that shows 2 lines, one linear and one curved, intersecting at the point (e,e). the graph of x^x also contains some e-related information; the minimum value for f(x) is at x=e^-1, or decimal 0.3678... so the lowest point in the graph of x^x is (e^-1,(e^-1)^e^-1)) or (0.3678,0.6922). It all has to do with exponential and logarithmic growth.
As this video approaches pi million views, it occurs to me that I would love for Numberphile to revisit Euler's other number -- gamma -- from a mathematician's perspective. I enjoyed Tony's video on the subject, but I would be very interested to learn more about the mathematical underpinnings of gamma, including its connection to the Gamma and zeta functions, and given that James has discussed Riemann before, he seems to be the logical choice for such a video. I've been waiting for 3Blue1Brown to do something on gamma tying in with their videos on Riemann and the prime number spirals too. Maybe because we know so little about it, but I find gamma to be a far more interesting number than either pi or e... which are still the second and third most interesting numbers, with the golden ratio coming in at #4.
Thank you so much for this, I’m learning logarithms right now, and I couldn’t understand what the natural logarithm was about and wanted to know. This helps a lot!
Thanks for this enthusiastic explanation! :) I find the use of e very useful in complex analysis, where we can write e^(it) instead of cos(t) + i sin(t).
Which of those expressions would be easier to evaluate? I suppose I could plug in some dummy values and figure out myself but im lazy. Personally I think the latter expression but I like how the first expression is more compact.
f(x) = 0 is not synonymous with x=0, but y=0. At any point on the function, the output (y) is equal to 0, the slope is 0, and the area under the curve is 0.
Beautiful video! I now have a neat idea of what Euler's constant. Nearly at the end of the video, you mentioned that if we don't use Euler's constant we will have nasty ratios introduced into the equation, which will make it more complicated. Can you pl. show one such example? The lecture will be complete with that. And again, thanks a ton for this video!
If you're taking the integral or derivative of an exponential function like 2^x, you would have to divide or multiply by the natural logarithm of 2. with e^x, ln(e) is one so dividing or multiplying by it doesn't need to be written.
+marche45 Here's how I think about your comment. Zero could be considered a trivial multi-variable function, with a well-defined gradient. That gradient is zero. If you say, "0 does not have a gradient", doesn't that mean you're implicitly saying that 'having the value zero' is the same thing as 'not having a value'? Little bit pedantic, but that's the spirit of this thread anyway.
I'm a complete mathematical moron, and yet I've always been fascinated by higher level maths. The idea of calculus, even geometry is just amazing. I've never in my life found anybody who could get me through a college level algebra course. I get close, but have never been able to finish it. I feel as though I've always sat outside the sun, watching other people achieve and understand this. I'm a sort of "Flowers For Algernon" guy. In any case, this is really interesting.
Is this for e^x? Or all functions? Both don't work because n^x=0 is undefined except for n=0, and there are plenty of functions where the gradient at y=0 is not 0? or am I misinterpreting the idea? Haha
great video. Thumbs up of course. Euler's formula is incredible - raising an irrational number to the power of another irrational number multiplied by an imaginary number giving a negative integer - mind blowing.
I know it's been around for years, but I gotta say this is my favorite video this channel has done. Why? Because I didn't understand this number before, but with the way it was explained, the number is beautiful. It really makes me wonder what other everyday events could generate new mathematical constants in the future.
My biggest problem with math is that it drives me crazy to just plug numbers into formulas without knowing the significance of that number and why it needs to be in said formula. e has been driving me crazy for a while now and none of the explanations I'd gotten satisfied me until seeing this, so thank you.
two numbers being irrational doesnt guarantee that their sum will be irrational too. You may be correct in this specific case but your explanation isn't correct
James Grime is baaaack! This is what I love about Numberphile; you made me not only understand maths, but you made me realize I ENJOY maths. I had no idea what e was about before this video, yet I knew about Euler's identity and that it was supposed to be this awesome beautiful formula. Now I understand WHY it's so famous; even if James is "a bit jaded" to it. :) For some of us, seeing it illuminated in the proper light for the first time... it is beautiful.
5:50 Do you add 1 to the denominator and then divide by 2 or do you add 1+1/2? I'm confused to as to how the sequence of the operations is with such a fraction. Can anyone help me?
@@EbonyPope It's an infitie fraction so there is no end. It's more like the 2nd option you have. You would basically say when you want to stop writing down the fractions and then solve for it going backwards from down up.
Actually, any function y=a*e^x has that property - thanks to the constant factor rule - with the special cases of a=1 (y=1*e^x=e^x) and a=0 (y=0*e^x=0)
Pointing out "y=0 has that property, too" works very often for exactly these reasons though. It is a polynomial with point symmetry. It is a polynomial with reflection symmetry. Thus I've often heard solutions of "everything is zero" or sometimes "is one" as trivial cases, thus I was brainwashed into finding general solutions more attractive, especiall if they, like here, also include the trivial one :)
Ulkomaalainen brainwashed? you mean taught? nontrivial solutions are more attractive because they're actually useful, it's not just something we do because they look cooler
They already talked about all the numbers, so now that there are no more numbers to talk about, they have to start with letters. Otherwise, there wouldn't be any content, so I am ok with letterphile.
James forgot the fact that e^iπ + 1 = 0 also has the 3 basic arithmetic operations: Addition, multiplication, and exponentiation (raising to a power); as well, as equality; and nothing else, apart from the constants: e, i, π, 1, and 0; which James already mentioned. Not to mention that it’s *_e_* that gets raised to the power; just like in the basic function of e: f(x) = e^x. It’s also very simple and easy to remember, and beautiful in that way. 😌
@@cpotisch As is the presence of e, i, π, 1, and 0; so, if you include them, there’s no reason not to include the other stuff; and, if you don’t include the other stuff, there’s no reason to include the constants, either.
@@PC_Simo all the other stuff was already included because it’s literally in the identity. You are living in a different reality if you didn’t hear him say “that its e that gets raised to the power”. He objectively did.
7:27 is truly explaining this. It's confusing when others talk about e like its some universal constant worked into all of nature, which is strange. Here he shows that you *could* use other terms, its just *easier* to use e, since it's where area and tangent coincide.
"The number "e" is the "natural" exponential, because it arises naturally in math and the physical sciences (that is, in "real life" situations), just as pi arises naturally in geometry." I include this quote from purplemath because you are mistakenly focused on the least important part of the video. We don't use e because it is easier, rather, e is necessary because it and no other number captures continuous growth.
For those who are wondering why e to the pi*I equals -1, it is basically because of an infinite series that represents sine (if you ever hear an engineer say they approximate sine of x as just x, x is the first term of the infinite series. -x^3/6 is the next term, and as you add an infinite number of subsequent terms the limit equals sin(x).) We learned the proof in my calc two class so that was fun
I love this video. Great at explaining e for the layman, yet completely technically correct and inspiring. Highlights both the importance and significance/practical usability of e as well as the pure beauty of e, and mathematics in general. But for the love of all that is holy don't say that "i is the square root of -1"
It would be true if he was takig the square root of -1, but that would be i and -i We must start teaching the true definition of i and the true definition of complex numbers
Please don't ever stop making maths videos. I truly appreciate your explanation (history and reasoning). Love your work and truly admire what you are doing.
If they stop making math videos they would stop making videos, the channel is numberphile
BA in Maths I presume.
eeeeeeeeeeeeeeeeeeeeeeeeeeeeee
@@granvillebarraclough8846 Why though? because at day 1 MA he'd be bound to be sick of it already or have moved on to another area of studies altogether? hehehe :D
Sorry, I had to reply, even 2 years late :P
I love the videos as well, gotta say.
"Don't put that in the video"
_Puts that in the video_
they had ONE job
absolute M A D L A D
+HalcyonSerenade
Kinda reminds me of that scene in Monty Python And The Holy Grail, where King Arthur got annoyed by a scene and then ran in front of the camera and screamed "Cut that out! Cut that out!".
r/madlads
Savage
If you graph x^y=y^x, you get two simultaneous graphs. One is x=y, and the other is some variation of y=1/x. They cross at the co-ordinate (e,e)
Nice
I tried to eyeball what the y=1/x -like function might be, and the closest I can seem to get is y=((e-1)^2)/(x-1) + 1. It matches the asymptotic behavior of x^y = y^x and passes through (e,e), but it doesn't perfectly match. The true function has something to do with the fact that ln(x)/x has two different x values that give the same y value for any positive y less than about 0.368, but for the life of me I can't figure it out.
@@adamkotter6174similarly the graphs a^x and loga(x) intersect at coordinates (e,e) when a = e^1/e
not really sure why that is but it’s neat.
@@adamkotter6174 that "about 0.368" number is actually 1/e. I don't know why though.
@@davidiv4915 it's pretty satisfying if you sub in x=e to both equations.
a^x = (e^(1/e))^e = e^(e^e) = e^1 = e
loga(x) = log(e^(1/e))(e) = e (because of the line above)
Math is just all about who can pronounce Euler the most correct
The football team at one of my hometown high schools was the “Oilers.”
I have a tattoo of Euler’s identity and just now found out it’s pronounced “Oy-ler”
the edmonton hockey team is called the oilers (don't even ask me about hockey alr)
Germans: You dare challenge us mortals
@@PMT17 fax
Main Theme - e
Main Theme - e
Main Theme - e
YTSunny Main Theme - e
Main Theme - e
Main Theme - e
I remembered when I first watched this video. It was like 2 years ago and I barely understand what’s going on. Now I watched it again after the previous 2 years and I found myself turned super focused. I understood everything
Thanks for your explanation
I can relate... I was 13 when I watched it... Now I'm 16 and I understand this lol. This is what I study loool
One might say you had continuous interest
@@MrInsideEye
Underrated comment.
You said the same thing on the Cardano's formula video
I’ve recommended Numberphile to various young students having trouble with maths. It is an excellent motivator for cultivating an interest in maths.
e^(i*pi) = -1 is very useful in electrical engineering phasor notation and complex impedance. It essentially allows equations that would need calculus to solve to be easily solved algebraically.
The complete Euler formula tho for every angle, not just the Euler identity everyone knows
You can thank Laplace and Fourier for that too!
It’s also used in Controls engineering, and vibrations
Also useful in digital signal processing.
It's also used in computer science for writing various algorithms and making them efficient wrt complexity
i love siivagunner
Equal-Chan Moment
Teacher: In what year did the defeat of the Spanish Armada occur?
Me: Euler or Gauss.
e's decimal expansion is random, so the likelyhood of finding a four integer sequence in it that coincides with the year of the attack by the Spanish Armada is equal to 1.
@@hammerth1421 WRONG
There's no proof of e being a normal number
So, unless you prove that or find the year itself, you can't say that
@@chanderule605 Even though e is not a normal number, it's digits are still random. While you can't find any sequence of any length in it, you are pretty much guaranteed to find every four digit sequence, also the year of the Spanish Armada.
@@hammerth1421 There's no proof of that other than "it's kinda likely"
And I dont think it's proven to be random either
HammerTh its digits are not random. Now I'll have to wait and see if you reply with what I expect you'll reply with.
My Business Calculus professor at the University of Southern Maine was a big Euler fan. All of the classrooms Mainville used had a bit of graffiti; the Euler number, its square and its cube (all written out to five or six places) were written above the blackboard in magic marker. Since we used natural logs in the class, it was like an authorized cheat sheet during exams.
"Everybody knows this anyway. If you don't, now you do."
not knowing the values of it to a few decimal points is crazy anyway😭 or just using a calculator but i presume it was a non calculator paper
1:29 Said no-one ever...
xD
Zahlenteufel1 I mean, maybe in sarcasm...
That was hillarious, I'm dying lol
Republican$ and Democrat$ in Congress say that all-the-time...
No, _everyone_ said this...
in a sarcastic manner.
Thank you! There are lots of discussions of e on UA-cam, but this is by far the most lucid I have seen. I learned calculus 55 years ago, never used it in real life (I became a soldier and later a lawyer) and am returning to re-learn it at my own pace. This video has been a tremendous help in my understanding of this important concept.
Your story is interesting
Is not it such a treat? It is analogous to discovering classical music later on in life, when you were once exposed to it as a child, but strayed from it to listen to contemporary music in the interim.
e^(i*pi)+1=0 not only brings together the most important constants, but also the basic arithmetic operations addition, multiplication and exponentiation.
And is the most tatooed formula among maths nerds.
Yet it works only because of some semi arbitrary definitions, such as the extension of powers of e along the imaginary axis.
@Clumsy Jester Exactly 👌🏻🎯😌.
8:58 when they say Infinity War is the most ambitious crossover in history
What?
I don't get it
Lol
@@sticks9757 e , π and i(√-1 if you don't know) are the most iconic constants (except i) in Math. So in this equation they powered e to the power of π multiplied by i and added them with 1 which is equal to 0 (bizzare actually. How e, π and i are really complex numbers in math somehow combine into making a simple number which is -1) So in summary all the iconic constants(except i) in 1 equation.
@@78anurag That's literally not what I asked
best title - 10/10
2.7182818284/2.7182818284
i r8 e/e
e/π
A solid 5/7.
Pride x humility = "I would not have called it g, I would have hoped that someone else would have called it g and I would have accepted that."
This is my new favorite channel; it's so well-explained, interesting and fun even when dealing with complex topics, and yet somehow really adorable and nerdy at the same time.
Watch ted ed
It isn't nerdy.
I can't believe Euler predicted the meme.
no he didn't
@@terrariaman8454 wooosh
Terraria Man r/woooosh
@@tipstyx1867 r/ihavereddit
You fool. The man made the oldest shitpost
December 31st 2019, 3:52 AM GMT
2.718.281 (e million) views
:)
Wow
What about e^e?
Today on February 71st,1828, 1:82 e'clock
01/01/2020 15:45pm.
@@HenryTheTodd hahahahahah
I legitimately believe this will help when I try to do a college calculus class again. No one bothers to explain what’s going on with new material. When you understand why you use symbols and how, it makes it easier to solve things. Thank you.
What a lovely title. Captures the essence of this video quite well. Fully encapsulates and proves the quote by the "not-so-well-known-outside-his-circles" Victor Tran, "A letter says an entire video!" :)
clik bait title
badman jones how tf is 'e' clickbait?
Prediator X makes ya curious. "what the heck? its just an e?? gotta look at this"
not enough to classify as clickbait though!
Merlin Brennt But this is a mathematical channel. As soon as I saw the title, I knew that they were going to do something with Euler's constant. :)
But I suppose the logarithmic graph and James Grime on the thumbnail helped too :)
Victor Tran oh you must be very smart!
Whats this, A Numberphile video, about a real number!? I never thought id see the day.
It also happens to be a complex number.
Thomas Eding um, no it isnt.
Complex numbers encompass the reals. It is complex with imaginary part = 0.
Thomas Eding e is totally on the complex plane.
All real numbers are in the set of complex numbers. If that seems odd, just consider also that all integers are in the set of reals...etc.
what is up with all these clickbait titles
VSeve I know right, they didn't even put the thumbnail in the video. Fockin clickbaiters😂😂😂
VSeve they need more arrows and circles
How is it clickbait? The video is about... e.
Andrew anon its an ironic comment
ApplicationBot Technically, since he obviously knew what he was talking about, it wasn't ironic, it was just a joke.
I've been trying to understand the relevance of this for days, but you made it so damn simple to understand. Truly the best channel for maths content.
The relation and usefulness of e, I, and trigonometry melted my mind.
8:58 Biggest crossover in history.
Heckk yea
e
No one cares
Sari Masu yea haha I look in subscriptions and was uploaded 3 second ago :D
e
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965435885668550377313129658797581050121491620765676995065971534476347032085
321560367482860837865680307306265763346977429563464371670939719306087696349
532884683361303882943104080029687386911706666614680001512114344225602387447
432525076938707777519329994213727721125884360871583483562696166198057252661
220679754062106208064988291845439530152998209250300549825704339055357016865
312052649561485724925738620691740369521353373253166634546658859728665945113
644137033139367211856955395210845840724432383558606310680696492485123263269
951460359603729725319836842336390463213671011619282171115028280160448805880
Liwo Jima you forgot a couple digits
It is a beautiful equation. I had it on my birthday cake when I was 18.
e is such an interesting number. I came up with the strangest function that approached e as x approached infinity.
y = 1/((x^(x+1)/(x+1)^x) - ((x-1)^x/x^(x-1)
It’s pretty cool.
ex: (56^57 / 57^56) - (55^56 / 56^55)
Take a reciprocal of this number and be amazed
great job!
What about this
Y^D=Y
So tell me What numbers the letters are ok
e shows up in very interesting places. x^y=y^x is an unusual graph that shows 2 lines, one linear and one curved, intersecting at the point (e,e). the graph of x^x also contains some e-related information; the minimum value for f(x) is at x=e^-1, or decimal 0.3678... so the lowest point in the graph of x^x is (e^-1,(e^-1)^e^-1)) or (0.3678,0.6922). It all has to do with exponential and logarithmic growth.
Nice work bro, I like these types of persons
Just use (1 + 1/x)^x like normal people...
Ok it's a joke, don't get offended
I think this is the first numberphile video I ever watched and I come back to rewatch it every once in a while. I just love it.
I mostly like how passionate and excited he is while talking about this math's stuff
this content is timeless thanks for all the enthusiasm and clarity
شكرا جزيلا على هذا الشرح المبسط واشكر كل من قام بالترجمة ...اتمنى المزيد 🌺🌷
Lord Farquad wants to know your location.
Owen Lewis you mean Lord MARKquad
Markiplier wants to know your location.
uhh...
Lord Farquad *E*
I can switch this between 2k likes and 2.1k likes imao
We're gonna talk about...
EEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEE
EEEEE
EEEEE
EEEEEEEEEEE
EEEEEEEEEEE
EEEEE
EEEEE
EEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEE
EEEEEE
EEEEEE
EEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEE
EEEEEE
EEEEEE
EEEEEEEEEEEEEEE
EEEEEEEEEEEEEEE
. eeeeeee
eeeeeeeeeee
eeee eeeee
eeeeeeeeeeeeeee
eeee
eeee eeee
eeeee eeee
eeeeeeeee
E and e is not the same
domjanabi yeah, one is capital, the other is lowercase
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEE
EEEEE
EEEEE
EEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEE
EEEEE
EEEEE
EEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEE
EEEEE
EEEEE
EEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEE
EEEEE
EEEEE
EEERE
EEEEEEEEEEEEEEEEEEEEEEE
EEEEEEEEEEEEEEEEEEEEEEE
VVV
VVV
VVV
VVV
VVV
VVV VVV
VVVVVV
As this video approaches pi million views, it occurs to me that I would love for Numberphile to revisit Euler's other number -- gamma -- from a mathematician's perspective. I enjoyed Tony's video on the subject, but I would be very interested to learn more about the mathematical underpinnings of gamma, including its connection to the Gamma and zeta functions, and given that James has discussed Riemann before, he seems to be the logical choice for such a video. I've been waiting for 3Blue1Brown to do something on gamma tying in with their videos on Riemann and the prime number spirals too. Maybe because we know so little about it, but I find gamma to be a far more interesting number than either pi or e... which are still the second and third most interesting numbers, with the golden ratio coming in at #4.
Thank you so much for this, I’m learning logarithms right now, and I couldn’t understand what the natural logarithm was about and wanted to know. This helps a lot!
I always understood e using limits in Calculus however this is a great, practical view that should be taught in class. Great video!
Crikey Bluey, there was a flamin' Ozzie 10c piece in there!
+picknikbasket ;)
A U.S. dime, as well :)
?H$lllllaaA
Thanks for this enthusiastic explanation! :)
I find the use of e very useful in complex analysis, where we can write e^(it) instead of cos(t) + i sin(t).
Which of those expressions would be easier to evaluate? I suppose I could plug in some dummy values and figure out myself but im lazy.
Personally I think the latter expression but I like how the first expression is more compact.
f(x)=0 also has that property... but not as useful.
Jay Chan isn't the gradient of X=0 infinity?
f(x) = 0 is not synonymous with x=0, but y=0. At any point on the function, the output (y) is equal to 0, the slope is 0, and the area under the curve is 0.
This video just made me extremely excited about math. For the first time, i actually enjoyed learning this.
pi > e
tau > pi > e
Gramm's number > e
3 > e > 2
colox97 tbh i before e except after C :)
τ > π > e > φ > √2 > 1/φ = φ-1
That's fascinating especially the bit where the area = the value = the gradient.
Beautiful video! I now have a neat idea of what Euler's constant.
Nearly at the end of the video, you mentioned that if we don't use Euler's constant we will have nasty ratios introduced into the equation, which will make it more complicated.
Can you pl. show one such example? The lecture will be complete with that.
And again, thanks a ton for this video!
If you're taking the integral or derivative of an exponential function like 2^x, you would have to divide or multiply by the natural logarithm of 2. with e^x, ln(e) is one so dividing or multiplying by it doesn't need to be written.
this is why you pay your mortgage every 2 weeks, and make one extra payment a year, and a 30yr mtg = ~15 yrs. the interest kills you.
xrisku yet with inflation that changes
It's still calculated p.a.
Many banks simply hold the partial payment until the complete monthly payment is received.
I don't think a bank gives you any credit for paying multiple times within the month.
e^x is not the only function to have the same area, value, and gradient. f(x)=0 also has that property.
MOOSEWHISKER so does 2 * e^x :O
That's true of all functrions on the form f(x) = ke^x, where k is a constant. Your is a special case where k = 0
Chrischo1997 finally someone who knows what there on about
That's trivial
+marche45
Here's how I think about your comment. Zero could be considered a trivial multi-variable function, with a well-defined gradient. That gradient is zero. If you say, "0 does not have a gradient", doesn't that mean you're implicitly saying that 'having the value zero' is the same thing as 'not having a value'? Little bit pedantic, but that's the spirit of this thread anyway.
I love how everything comes together, especially in the end..
smooth e
309 likes and still no answers
e
Nice mix.
And here we are, the letter e used to be a running jokes on siIva’s rips and now a lord farquaad meme.
So siIva is now a normie meme?
*TASTE E*
Does saying "Don't put this in the video" ever work?
we will never know because there can never be recorded evidence, because that would mean putting it in the video
badman jones Perhaps memoirs would recollect how a courteous editor obeyed the request?
Pablo Griswold you could put it in a separate video and thus you wouldn't be putting it in the video the person's telling you not to put it in
Why don't you ask Schrodinger?
Revan Sith'ari ahah imagine like a whole "dont put that in the video" compilation
The UA-cam video with the shortest title?
John Thimakis no vsauce holds that title with no title
John Thimakis yet it doesn't terminate
there's a vsauce3 video with an infinity symbol as the title... so no, not quite
e
John Thimakis Vsauce also has a video with no title, its about nothingness
I'm a complete mathematical moron, and yet I've always been fascinated by higher level maths. The idea of calculus, even geometry is just amazing. I've never in my life found anybody who could get me through a college level algebra course. I get close, but have never been able to finish it. I feel as though I've always sat outside the sun, watching other people achieve and understand this. I'm a sort of "Flowers For Algernon" guy. In any case, this is really interesting.
BRING BACK JAMES
-A big numberphile fan, August 2019
Happy e day! (2/7/18)
What about 27 January?
Woowowowowowowowowowowowowowowowowowoeowowowow
I late. Waiting for 27/18/28
The next is 2718-2-8
Oh my gosh... there can only be one e day a century🤤
y = 0
gradient is 0, area under the graph is 0
lol that's actually true xD
says person with a pony icon
Yeah but that function is boring and useless :(
Is this for e^x? Or all functions? Both don't work because n^x=0 is undefined except for n=0, and there are plenty of functions where the gradient at y=0 is not 0? or am I misinterpreting the idea? Haha
OriginalPiMan Nevermind, just got it. "f(x)->0". Mind you, this is kinda cheating seeing as the x value has no effect on the gradient anyway ;)
Great explanation. How have I never been taught this before, it took just 10 minutes and I totally get it.
I sexually identify as Euler's Number.
This comment made me laugh c:
So basically you make it easy to visualize growth
gunjeet singh I visualise growth in many ways ( ͡° ͜ʖ ͡°)
that killed me
but you are whale?!
great video. Thumbs up of course.
Euler's formula is incredible - raising an irrational number to the power of another irrational number multiplied by an imaginary number giving a negative integer - mind blowing.
"Euler, he works everything out!"
Well does he work out?
He's having a break now. Stay tuned for the next iteration of the Universe.
Does he even lift bro
He skipped everything but head day ... wait, that sounded kind of dirty!
@@Raptorman0909 It did didn't it?
@@Raptorman0909Calling it brain day would work
An absolute masterpiece, I have not seen a better explanation historical or logical before.
Well done.
I know it's been around for years, but I gotta say this is my favorite video this channel has done. Why? Because I didn't understand this number before, but with the way it was explained, the number is beautiful.
It really makes me wonder what other everyday events could generate new mathematical constants in the future.
+1
Congratulações. Excelent work: Dr. James Grime has the hability to be clear in teaching.
Five minutes in to math and chill and he gives you this look 5:18
the amount of passion and love for math makes the points go straight to ur heart and brain
"Don't put that in the video" *puts it in the video* typical!
We can not know how many times they actually say that, and it is actually not in the video. So you can't say if it is really typical.
Koen Looijmans ಠ_ಠ
I gave you 200
Holy cow... I finally understand e ... That was brilliant
3:51 let’s just take a moment to appreciate his hard work
My biggest problem with math is that it drives me crazy to just plug numbers into formulas without knowing the significance of that number and why it needs to be in said formula. e has been driving me crazy for a while now and none of the explanations I'd gotten satisfied me until seeing this, so thank you.
Could you please make a video on all relations between e and pi?
Because there are so many weird ones
***** That's pretty neat!
I'm a physics student btw
And there's a reason for that too?
A great mathematical reason of course ;)
Might there also be a relation like pi^i*e perhaps?
I wish I had such beautiful dreams...
But e and pi are some weird numbers indeed. And the fact that they are related in so many ways is even stranger!
i e pi (Weebl and Bob)
pi minus e is 0.42... meaning of life confirmed
StupidNedFlander that's what those '...' mean.
I wanted to like this comment but didn't because it has 42 likes.
Ostebrix s
and so what, 0.42.. is log5?? no I dont know what is 0.42
StupidNedFlander
It's called rounding
two numbers being irrational doesnt guarantee that their sum will be irrational too. You may be correct in this specific case but your explanation isn't correct
James Grime is baaaack!
This is what I love about Numberphile; you made me not only understand maths, but you made me realize I ENJOY maths. I had no idea what e was about before this video, yet I knew about Euler's identity and that it was supposed to be this awesome beautiful formula. Now I understand WHY it's so famous; even if James is "a bit jaded" to it. :) For some of us, seeing it illuminated in the proper light for the first time... it is beautiful.
value = gradient = area.... OMG! I have used e for many years, first time noticing this amazing thing!! Thank you for the tutorial!
Wow SiIvagunner
yea
5:50 Do you add 1 to the denominator and then divide by 2 or do you add 1+1/2? I'm confused to as to how the sequence of the operations is with such a fraction. Can anyone help me?
@@EbonyPope It's an infitie fraction so there is no end. It's more like the 2nd option you have. You would basically say when you want to stop writing down the fractions and then solve for it going backwards from down up.
Your thumbnail is mine now.
Makoren Why does your pic look so familiar
Essem because it's the same pic as in Ur identity card.
the function y=0 also has its value as its gradient and area underneath from negative infinity. just wanted to point that one out.
Actually, any function y=a*e^x has that property - thanks to the constant factor rule - with the special cases of a=1 (y=1*e^x=e^x) and a=0 (y=0*e^x=0)
well yeah, a*e^x is the solution to the differential equation y=dy/dx. but most people would consider y=0 to be a "different" function than the rest.
yup. I had previously said "most people" because the vast majority of Numperphile viewers have not received higher level math education.
Pointing out "y=0 has that property, too" works very often for exactly these reasons though. It is a polynomial with point symmetry. It is a polynomial with reflection symmetry. Thus I've often heard solutions of "everything is zero" or sometimes "is one" as trivial cases, thus I was brainwashed into finding general solutions more attractive, especiall if they, like here, also include the trivial one :)
Ulkomaalainen brainwashed? you mean taught? nontrivial solutions are more attractive because they're actually useful, it's not just something we do because they look cooler
I would never have understood this concept if it wasn't for this video.
Isn't this channel called "Numberphile"?
Not "Letterphile".
Kappa.
It is a number represented by a letter. Kappa
Kappa means he didnt mean it seriously
It's funny, since kappa is also a letter (κ).
Letterphile would end rather quickly, wouldn't it?
They already talked about all the numbers, so now that there are no more numbers to talk about, they have to start with letters.
Otherwise, there wouldn't be any content, so I am ok with letterphile.
1:29 "Wow, thanks a lot bank!" made me laugh more than it should have
I knew i wasn't alone😂
Great video. I nearly understood what he was on about!
"Wow thanks a lot, bank!"
😂😂😂🤭🤭
I really need to clean my monitor. It seemed especially Grimey during this video.
He seemed high in this video, like he'd just done an e.
Dr. Grime seems so genuinely happy when he explains Maths. Is there any way I can attend any of his lectures?
Go to Cambridge I suppose
Thank you sooo soooo much. I couldn't find a video that really explains the e. This is so helpful
I can smell the sharpie from here
7:44
CORRECTION
y=0 also has this property.
Hassan Akhtar You right I retract my statement
@@h_3795 so does y=C*0 for any constant C ;)
Well that's just y=0*e^x ;)
I believe 0^0 creates a wrinkle in this statement.
@@h_3795But you didn't actually say anything wrong. You were just offering a counterexample to a statement.
James forgot the fact that e^iπ + 1 = 0 also has the 3 basic arithmetic operations: Addition, multiplication, and exponentiation (raising to a power); as well, as equality; and nothing else, apart from the constants: e, i, π, 1, and 0; which James already mentioned. Not to mention that it’s *_e_* that gets raised to the power; just like in the basic function of e: f(x) = e^x. It’s also very simple and easy to remember, and beautiful in that way. 😌
He didn’t “forget” anything. All that is self-explanatory.
You seriously think that no one noticed that e is in e^ipi?
@@cpotisch As is the presence of e, i, π, 1, and 0; so, if you include them, there’s no reason not to include the other stuff; and, if you don’t include the other stuff, there’s no reason to include the constants, either.
@@PC_Simo all the other stuff was already included because it’s literally in the identity.
You are living in a different reality if you didn’t hear him say “that its e that gets raised to the power”. He objectively did.
@@cpotisch Well, I sure missed that.
@@PC_Simo Then that speaks to your own lack of mathematical prowess, because it was the most obvious thing I’ve ever seen.
I've just starting considering this number extensively because of calculus one. This is wondrous. Thank you for the video
e is also used for log or Ln bases like e^ln(2) or log based e.
4:58 I believe that "e" is officially the Euler number, & the Euler constant is γ ≈ 0.5772.
7:27 is truly explaining this. It's confusing when others talk about e like its some universal constant worked into all of nature, which is strange. Here he shows that you *could* use other terms, its just *easier* to use e, since it's where area and tangent coincide.
"The number "e" is the "natural" exponential, because it arises naturally in math and the physical sciences (that is, in "real life" situations), just as pi arises naturally in geometry." I include this quote from purplemath because you are mistakenly focused on the least important part of the video. We don't use e because it is easier, rather, e is necessary because it and no other number captures continuous growth.
For those who are wondering why e to the pi*I equals -1, it is basically because of an infinite series that represents sine (if you ever hear an engineer say they approximate sine of x as just x, x is the first term of the infinite series. -x^3/6 is the next term, and as you add an infinite number of subsequent terms the limit equals sin(x).) We learned the proof in my calc two class so that was fun
Such a beautiful and simple explanation, thank you!
My favorite game is *e*.
Dr. Awesome is it a videogame??
what is that?
I love this video. Great at explaining e for the layman, yet completely technically correct and inspiring. Highlights both the importance and significance/practical usability of e as well as the pure beauty of e, and mathematics in general.
But for the love of all that is holy don't say that "i is the square root of -1"
It would be true if he was takig the square root of -1, but that would be i and -i
We must start teaching the true definition of i and the true definition of complex numbers
i is DEFINED as sqrt of -1. and i^2 = -1
I can't listen to the intro without hearing the SiIvaGunner cover
You always have a Rubik’s cube in the background.
I want to see you solve that
I can do that
@@zerksez9963 ur mom cant
lol
You can learn how to solve a rubik's cube in 10 minutes. It's very easy
I was waiting for a video on e for soo long. Finally!
This is perhaps one of my favorite videos of all time. Thank you.
the thumbnail haunts me in my dreams