What makes the natural log "natural"? | Ep. 7 Lockdown live math
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- čas přidán 7. 05. 2020
- All about ln(x).
Full playlist: • Lockdown math
Home page: www.3blue1brown.com
Brought to you by you: 3b1b.co/ldm-thanks
Beautiful pictorial summary by @ThuyNganVu:
/ 1259288683489849344
Errors:
At minute 16, the sum should be written with a "..." to indicate going to infinity.
At minute 38, the exponent should have 1/(2s^2) instead of 1/s^2 for s to represent standard deviation.
At minute 54, an equal sign was mistakenly used in taking the derivative of x^3 / 3!.
At the end, it should be pointed out that the alternating series with x^n terms only converges for values of x between -1 and 1, so the values one can't be considered proven with values of x outside that range. Everything with the argument here is fine, as it only deals with the convergent input, but that fact should still be mentioned.
Related videos.
Calculus series:
• Essence of calculus
The sum giving pi^2 / 6:
• Why is pi here? And w...
The sum giving pi / 4:
• Pi hiding in prime reg...
• Fermat's Christmas the... (Mathologer)
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
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Video timeline (thanks to user "noonesperfect")
0:00:14 - Question 1
0:02:29 - Answer 1
0:06:27 - Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
0:12:01 - More examples of prime numbers in infinite series and their relationship with ln
0:17:25 - Question 2
0:19:20 - Answer 2 and explanation using ln
0:22:25 - Question 3 and families of curves
0:26:37 - Answer 3 and explanation
0:28:50 - Imaginary exponential
0:30:57 - Derivatives of exponential terms
0:37:21 - Why derivative of e^t is the same as that e^t itself?
0:41:21 - Question 4
0:44:12 - Answer 4 and explanation using Python
0:46:02 - Taylor Series for e^x
0:48:29 - Derivatives of polynomial terms/Derivatives of e^x
0:50:56 - Derivative of natural logarithm using graph
0:56:07 - Question 5
0:57:37 - Answer 5 and explanation
1:02:15 - Euler-Mascheroni constant
1:08:37 - Question 6
1:12:41 - Connecting dots to the familiarity of different expression in math
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“To see if you have been paying attention so far” this is the scariest thing to hear in math class
started to pay attention after this sentence, then i noticed the gorilla
@@user-uj9mq7vq5d what gorilla?
Lol nobody tell him!
@@user-uj9mq7vq5d what gorilla???????
No way. Those words mean it’s time to get it and watch my peers struggle :) There is a guy in a gorilla suit that walks by when you are focused on the numbers.
That riff on 69 was just about the smoothest damn thing I've ever seen.
isn't it true though?
Nice.
This dude is an absolute legend for the following reasons:
1. Amazing, well thought out, quality lessons.
2. Trolling the 69 trolls.
3. DAT GORILLA
Selective attention test fail. Doesn't work well here though because, as opposed to the original test (counting the number of ball passes between people), here the viewers are focusing on working out the problem on paper & pencil and not watching the screen.
@@ZomB1986 what are you on about?
@@SoumilSahu 18:02 gorilla!
@@alopexlagopus1488
Yeah! I saw it too!!!
@@randomman1000tweeny1 Oh, just checked it out. Makes sense now. I noticed the gorilla and counted 12 passes. Eh.
The planning of the '69' baffles the war strategists to this date.
69 ☠️☠️💀 broooo
27:40 I love how he just "happened to know" the value of ln(69). Not that he anticipated this or anything.
"as any mathematician will tell you -- a well known constant of nature!"
Well, he's clever enough to predict that sheeples will put in a """fun""" number.
@@MisterMajister you sound like fun at parties
@@MisterMajister why do people misuse that word so much...
@@hassanakhtar7874 Because it's a dumb word to begin with
0:14 Question 1
0:57 Python Program to find all primes in a range
2:29 Answer 1
3:02 Explanation (Mathematician logic to find prime nos. in particular range)
06:27 Prime nos. in Infinite Geometric Series (Basel problem) and their relationship with Natural logarithm
12:01 More examples of prime numbers in infinite series and their relationship with ln
17:25 Question 2
17:58 Can you spot Gorilla?
19:20 Answer 2 and explanation using ln
22:25 Question 3
22:46 Does family of curves depend on any particular mathematical constant?
26:37 Answer 3 and explanation
28:50 Imaginary exponential
30:57 Derivatives of exponential terms
37:21 Why derivative of e^t is same as that e^t itself?
38:10 Explanation using graph and limit
41:21 Question 4
42:32 Audience questions from tweeter
44:12 Answer 4 and explanation using Python
46:02 Taylor Series for e^x
48:29 Derivatives of polynomial terms/Derivatives of e^x
50:56 Derivative of natural logarithm using graph
56:07 Question 5
57:37 Answer 5 and explanation
1:02:15 Euler-Mascheroni constant
1:03:30 Alternate expression for infinite series
1:08:37 Question 6
1:09:41 Answer and Explanation
1:12:41 Connecting dots to familiarity of different expression in math / Genius way to approach solution
_Water sips_ : 18:48 , 18:54, 42:04, 56:36, 1:09:10
*Note: Changes were made according to new time-line, Let me know if any errors or changes you think need to be made. Thanks*
your nickname do you mean to say Grant Sanders is not perfect?
@@99bits46 ahm that is debatable issue lol
this is the best timestamp list.
*w a t e r s i p s .*
I tip my hat to you, Time Warrior.
Water sips🤣🤣🤣
26:40 only Grant could make a video targeted to high schoolers, get 69 as the answer to their favorite number, and with a straight face innocently deliver a flawless math fact as to why 69 is interesting which has absolutely nothing to do with why 69 is up there lol love this channel 😂
@@pigeonlove says the person who still has a queen
ikr so good xD
Yeah, as we all know the real reason was that thing he said in the other video about the digits or something.
He’s so nerdy and innocent I love this channel
365 aka 1 year likes
Thanks to everyone who joined! One thing I should add: For the manipulations at the end, all of that is only valid for values of x where the relevant series converges. So as x gets outside the range [-1, 1], it wouldn't necessarily follow that you get similar results.
Edit: Once again, bizarrely, after I cut out the intro animation so that the video actually starts where the lesson begins, despite a preview in CZcams's editor indicating otherwise, the end got chopped off. It's only a half-sentence with little significance, but if you're wondering what's going on, well, there you go.
Was the selective intention experiment used to gauge how many are focusing on the math?
Is there a live chat replay
Grant, thanks a lot for doing this, this series is awesome
I did, despite I’m still in elementary school.
Isn’t the constant at 1:06:35 the oily macaroni constant?
The gorilla was awesome!
Plus the way Grant was prepared for 69 was great. Completely trolled the trolls in the audience.
Hehe 69 likes
@@felixroux It was almost like he was nervous the vote was so close because he had this gold comedic analysis of the concept with 69 all prepared and in his head he was like "oh crap, 'i' almost won, all that preparation would have been for nothing!".
Where is the gorilla. I missed it
@@AdhiNarayananYR
At 18:02 ))
he always do :))
26:45 The man prepared for this eventuality by thinking of something beforehand that would yield the number. Well played.
I'm 50 years old.. I never got a chance to learn logs in high school in Iran in 1987 because Saddam was lobbing missiles at Tehran and schools closed.. logs always remained an unlearned topic for me and much later here in the US and throughout college I never truly grasped the concept, particularly e and natural logs.. I've watched your videos many times and I keep watching them and I love them because finally and after all these years I'm getting it, thanks to your beautiful explanation and these awesome demo tools. Thanks a million!
🥺🤝🙌❤️
А сейчас Иран предоставляет России атакующие дроны, которыми они убивают Украину...
awesome!! you go!!
If you like the joke about the logorithm neperien, maybe you'll like this one:
Q. What does the B in "Benoit B. Mandelbrot" stand for?
A. Benoit B. Mandelbrot
That's clever! I'll have to remember that one...
Exponentielle et logarithme vont à une fête. Logarithme s'amuse avec les autres mais exponentielle reste dans son coin. Logarithme lui dit, "pourquoi n'essaie tu pas de t'intégrer ?". Exponentielle répond : "Je m'intègre, je m'intègre, mais il ne se passe rien."
Logarithme et exponentielle font du bateau. Soudain logarithme dit "mince, on dérive". Exponentielle lui répond : "je m'en moque".
@Sai Sasank yup, is turtles all the way down...
and so on, and so on, and so on ...
An infinite number of mathematicians walk into a bar. The bartenders ask what they want to drink. The first one walks up to the bartender and says, "Give me a beer." The second chimes in, "I'll take the same, but half." The third says "Same, but give me a quarter." As the fourth mathematician starts to order the Bartender holds up his hand, turns around, pours two full beers, walks back to the counter and says, "You guys really need to learn your limits."
Haha, I laughed aloud
Good one
What if it was lite beer 🍺 - would that be like 2.5-3 beers 🍻
This guy
lol isn't this an infinite geometric series of (1/2)^n
42:55 The terrible pun comes from "népérien" sounding almost like "nepérien", which is homophonic to "ne paie rien" meaning "doesn't pay anything".
So e has to pay the beer because logarithm (in base e) "doesn't pay anything".
logarithme Népérien is a tribute to John Neper(or Napier), a Scot that invented the logarithm, as far as I know.
@@philippepons8924 yes. In the UK, natural logarithms are sometimes called Napierien Logarithms.
You really make me feel beauty in numbers. Never posted a comment on CZcams before, but I really need to thank you for all this.
Nice. I also commented only on four videos in my life and this video really deserves it. Comparing when I first learned and still learning math in university and it makes me want to sleep while this video I accidentally clicked was so interesting I watched till the end and learned things that university (at least the one I am in) never finds interest to learn
Math is absolutely beautiful, once you got into it. I think the main reason why so many people seem to outright hate maths is because they were never taught it's beauty, or were taught without passion.
@@lucaxtshotting2378 educational CZcams videos are as much learning as anything else
@@lucaxtshotting2378 depends from what you call studying and learning. I'd say these videos are the inspiration, what drives to discover again something we already learnt and maybe overlooked a bit. And, well, if that isn't the mathematical spirit, I'd be damned. But then of course, you also pointed out an important issue with your comment... Looking at things on the Internet doesn't make you an expert in the field, it merely helps to scratch the surface... That's a mistake we shouldn't make. Otherwise, what's the point in being a doctor, a plumber, an artist or even a philosopher, when you can read everything online, right? This is actually a pivotal theme of the world today. Let's judge what we find on the Internet and remember that books exist for a reason! Stay away from fake news!
I just noticed that Grant SOMETIMES crosses his 7's.
Look, I can get on board with crossing them or not crossing them, but crossing only SOME of them? What madness is this?
I do this, too...
For me a 7 without a cross in handwriting looks too much like a 1. Same for Z without a dash in the middle, without the dash it looks like a 2.
@@gregoryfenn1462 I see what you did there with the cross and dash!
@@gregoryfenn1462 That is exactly what started the convention
@@quadrannilator Oh thanks, umm I didn't mean to do anything lol
As a mathematics major who hasn’t used his degree in years, thank you! I love these! Reminds me of how much I used to love math in school.
Not using your degree?
At home I have electronic scales which show your weight in kilograms, and I often see the number 69.3 kg and I am always like "yay, the natural log of 2 once again!"
I remember when we learned about prime numbers in primary school, we had a homework assignment to figure out all the prime numbers from 1 to 100. I wrote a program in BASIC to do the assignment for me. In my mind, it told me that I not only understood the concept, but also understood it enough to explain to a machine how to understand it. I made the mistake of telling this to my dumbass teacher and she failed my assignment, on grounds of "cheating". This was back in the early/mid-90s. I wonder if teachers these days would still have the same attitude.
Your teacher was an idiot.
Probably, school loves to make students operate like machines and just do what they're told and nothing else.
@@everythingiseconomics9742 In hindsight, I suppose the power of IT wasn't so obvious to the "adults" back then either.
I never realised how differentiation had so many effects on logarithms and on series! That definition of exp(x)=sum(1,k, x^k/k! ) series is really paying dividends now.
Sorry if thats random af but youtube does have
openly racist people (non-subtle and therefore easy-to-find) AND a reportbutton.
Same for p0rn: Also easy to find via the searchbar.
Cant you... spare some few minutes and help flagging a bit?
It's amazing how you're able to explain things in such a clear and fascinating way!!
Lockdown math viewers: I understand logarithms well and could comfortably teach someone else to understand them well too.
Also Lockdown math viewers: MY FAVORITE NUMBER IS 69.
Take a like and make it an even 69😂
So I see you never matured beyond the seventh grade. Just once I would like to see a comment section where the deviant refrain from making inane comments! (for the ignorant, the word is inane not insane, look it up).
Knowledge shouldn't stop you from being childish
I love the approach you use to teach maths, you go right to the abstract concepts
1:04:39 - 1:04:52 Agree with me or not. IT WAS A WONDERFUL LINE! It motivated me somehow. Thanx Grant
Been watching for a while now.
I think I was lucky at school as you sound very like the maths teachers I had.
Concise, listenable, and showing off a love of how mathematics work and the surprises contained within it.
Thank you
Grant I have to say that I never noticed just how beautiful the music in your videos is. It really does a great job of rounding out the "math is beautiful" aesthetic of the channel. And I love that until it was there on it's own (in the wait screen) I never even consciously knew it was there? But I looked at the Spotify album and I LOVE how immediately it pulls me into elegant-math-visual world.
I don't buy this French stuff. Clearly, "ln" stands for "latural nog".
I can confirm that in French, "ln" stands for "logarithme népérien" and it is seldomly named "logarithme naturel". This is in reference to "Jean Neper" (or "John Napier" in English), the famous scotisch mathematician.
log natural?
I get the joke, but I've always thought of the "n" as a subscript. Even though it's not traditionally written that way, I just assumed lazy humans prevailed and it just became ln. Same as RL (R subscript L) meaning Load Resistance in EE. Also, isn't English the only common western language that puts adjectives before nouns? Obviously not my area of expertise!
@@djd829 German, Russian and Scandinavian languages have adjectives typically before nouns, Spanish, Italian French have them typically afterwards
I literally spent three minutes looking at "latural nog" trying to figure out what's wrong, lol
26:40 - 27:00 was the strongest acting I have ever seen in a math video. Or really almost any video. Talent.
26:31 Grant may be the first person ever to make me type a sum into my calculator and make me laugh upon getting the result
Wdym
The way I'm re-learning these things is incredible, I'm not a complainer, but I wish I had a calculus professor like you back in college
geezus yes
Thank you, Grant and the team. YOu guys blew my mind as always.
your last bit about 'making connections between patterns you've seen in different contexts' really resonated. I was reminded of Planck's struggle to understand the famous curve of energies given off by a black body. I learned that curve by making assumptions of little oscillators and calculating a distribution. In fact, he played around with curves and came up with his formula through familiarity. Another example was quantum mechanics.For example, the quantum description of light polarization and using the constant , i, is totally counterintuitive but it worked.
I thought everyone chose 69 because it is the largest number whose factorial can be expressed in a regular calculator (not mobile calculator)
Anyone who has played with a calculator would know that
damn there are a lot interesting facts of the number 69
Damn i thought it was because if you square 69 which is 4761 and if you cubed 69 which is 328509 you'll see that they cover all decimal digit from 0 to 9
Where in the world can you find a 100-digit calculator? (apparently it is so common that one can call it 'regular', yet I have never seen it before)
@@nickpro8116 regular calculator goes upto 10^99 which is equivalent to 100 digits. I think you thought of a calculator which shows 100 digits, but I meant a calculator which can handle calculations upto 100 digits but shows it in scientific notation
@@anuragjuyal7614 ah okay
I remember when I first started learning exponentials and logarithms at university, they tore down the traditional expectations of what these things were. To start with, there was the question of what a number raised to the power of x meant if that number happened to be irrational e.g 2 raised to the power of sqrt(3). Powers made sense when the numbers were rational but you had to go around that for irrational numbers - and it was typically assumed that we could approximate irrational numbers as rational and hence you could get a reasonable enough value for that exponent. This is to say exponentiation was continuous over the reals. But how such an assumption came to be was often overlooked. So instead of starting with exponents, the lecturer started by defining ln(a) as the area under the y=1/x from 1 to a. Since this area function is continuous, then we can start playing with some interesting properties of such a function and show that it satisfies our logarithm laws, where it is eventually revealed this function has an inverse which is none other than exp(x).
This channel can keep you hooked on math in a way I had never guessed it was even possible
I know you'll never see this, but I learned something today - thank you.
Great that you included integral of 1/x+1 in the end. That was a nice way to end todays lesson and give useful information as well
I'm so glad you did the French joke on exponential and logarithm, I wanted to make it in the last lecture, because of the introduction poll, and I was like "well nobody will understand it anyway"
27:45 omfg, I'm ded. Well known natural constant, oh yes yes, of course.
"Well known constant of nature" - this guy is amazing. His sense of humor is absolute great!
@@mh-lw1oe It's one thing to anticipate trolling, it's another to have a plans to steer trolling back to sincere discussion without having to even acknowledge you're being trolled. It's masterful, really.
At 23:00, was that a prime, mate?
(hides behind math textbook)
amazing :D
This deserves more upvotes
also the timestamp should be updated it earlier now
@@squibble311 to what time?
Omg, Grant has just got super advanced with this lecture. I just noticed the GORILLA! Super fun and horrifying as well :) Just to note that we also heard weird voices coming from the behind in one of the previous lectures :)
But how many times did the players in white throw the ball?
Short answer: we have the option of normalizing to something, and _e_ is the pick that makes the most complications cancel each other out.
You are my inspiration. You connect and teach complex with subtle things of nature. Man, I understood every bit of calculus from you. What a great teacher you have been to me without pay!
this is so well presented haha - nice job channeling a passion for the curious into a passion for math!
Pythonista here :p
the python code near the beginning doesn't need the has_factor, because a for loop can also have an else statement!
the else statement will go after the for loop, but if you break in the for loop, the else statement is skipped over!
for p in range(2, int(np.sqrt(x))+1):
if x % p == 0:
break
else:
result.append(x)
What a witchcraft I learned today!
This is one of the nicest features of python. I really miss it in other languages.
@@JoJoModding nicest features? it's conceptually confusing and unnecessary. When does this ever actually matter other than to confuse someone reading the code?
Holy crap. I've been a frequent Python user for over a decade and I never knew that. Thank you!
Now do the structured programming version
I love his cheeky smile when he talks passionately about the beauty of maths.
Please watch my vedio too
Watched so many very instructive videos of 3blue1brown. Thank you very much for all!
A big fan of your work. Most complex yet simply elegant.
Every time I watch a 3blue1brow video I wish he was my maths teacher. And then I realize he is, via these videos, and I'm happy.
27:09 I love democracy
This is perhaps one of your most beautiful lectures
These classes are amazing. Please keep it up.
Someone should grant (pun intended) this guy the Abel prize for outstanding zeal in talking and teaching math. :)
Towards the end of the video where he shows the connections between the integral [1/(x+1)]dx, ln(2) and the summation of an infinite geometrical series and how they are related; I'd also like to think that there are similarities in the patterns and connections here that you also see within the connections between geometrical polynomials and the sine and cosine wave functions that are found within FFTs: Fast Fourier Transforms. And this is why I find math to be so expressive! You can see it on paper but you can most definitely hear it in the awe of music! I played the trumpet for over 10 years when I was in school so I understand music, musical notation, etc. as I can read and play sheet music for that instrument. And since sine and cosine are wave functions and we use them to graph and map objects in nature that have waveforms such as energy, sound, light and more... we can express what we hear in music not only in a score or in terms of sheet music, but also in beautifully written summations of integrals and derivatives of sine and cosine wave functions expressed as geometrical polynomials such that you would see in different types of series such as Taylor Series, Laplace Transforms, Lorenze, Eulers, and FFTs. If you ever stop to read and break down the file structure of MIDI files, you'll begin to understand the power of mathematics to express the world - environment around us. Physics and Math are embedded within Energy, Mass, and Matter through motion and they are constructed on the simple principles of mathematics. For me, it is this duality that drives the COSMOS as they are the self-perpetuating engine! This is coming from the Mathematic - Physics - Chemistry perspective of the physical world.
Now as for the Spiritual side of things, I like to make the connection between everything above from basic addition (1+1) = 2 which generates the unit circle located at the point (1,0) on the XY-plane, and once you have the unit circle, you have every other function such as multiplication, powers their inverses, logarithms, constants such as PI and e, the trig functions and all of their properties and so much more... And since I've already stated that we use the cosine and sine functions to map the behavior of things that act, operate, or have the properties of being rotational, oscillatory, repeating, recurring, sinusoidal, periodic, etc... I like to make the connection with all of mathematics, physics, chemistry even concepts in biology and general engineering and that of Genesis Chapter 1 verses 1-5 but most specifically verses 1-3.
The more I study physics, chemistry, biology, and the complexity of nature and the physical world around us there is 0 doubt in my mind, conscience, heart, and soul that there is Intelligent Design Written all over everything we observe, witness and experience. To this degree, I make the statement and claim that everything we experience and witness is a product primarily of Sound and Light. These are two of the 5 most basic senses that we possess. The other 3 are touch, smell, and taste. Our vision comes from light, our hearing from sound, but as for our sense of touch, it is a combination of both. We feel the vibrations of things moving, friction, change in energy and temperature, textures between different states of matter, and so on. Now as for smell and taste they are related as they come from the product of chemical reactions, the aromas produced in the form of heat. We know through physics from Newton that F = ma and through Einstein that E = mC^2. We know that matter and energy can not be created nor destroyed... Sound and Light are Energy and their waveforms are governed by the properties of the Trigonometric Functions which are defined by Triangles and Circles. We can rewrite m as m = F/a and we can substitute that into Einstein's equation to get E = (F/a)*C^2 we can do the reverse and state that m = E/C^2 and put that into Newton's equation F = (E/C^2)*a if we look at E and F we can see that there is something that is common between both of them... E = (F*C^2)/(a) and F = (E*a)/(C^2) the have in common the acceleration of an object and the area of light in a given time frame. Don't think of C^2 as being velocity or the speed at which light travels in these equations as that would be misleading. Think of C^2 as C being the distanced traveled within that time frame (typically in physics the standard time for velocity and acceleration is m/s and m/s^2. So you have to as yourself what is the distance that light travels in 1 second. Take that distance and square it. This gives you an area. This area and the amount of acceleration an object have determines the amount of energy that is within that object and the forces that are being applied to it. If you take a look at the scale or octaves of music and you look at the color spectrum of light as octaves you will see that there are parallels between them, yes they have a different frequency range, but there is a mapping between specific colors and specific sounds or notes if you care to examine them close enough. There are no mistakes in these concepts.
So when I go back and read Genesis and make the connections. It's quite simple to see that when God said, "Let there be light" and it was so... It was his voice that shook the heavens and it was the sound of his voice that brought light into existence and from there the Cosmos was born. I believe that Sound and Light drive everything to move and that we are a part of their products! So when God said, Let there be Light... He painted the Universe with his Song and Poetry because Music is Colorful! Don't think of this as me trying to preach as that's not what this is about. I'm only stating the connections that I see and what I believe in. I just share what I know and understand and from there, it's up to the reader to do what they want and what they will with it. I believe in and support Free Will, so I say, take the information and make up your own damned mind about it! Yet, I will still argue and debate my position based on my knowledge, and the ability too understanding, ability to see patterns and to make the connections between them. If we consider the fact that the very first geometrical shape that has a defined area and three interior angles that add up to 180 degrees or PI radians from 3 intersecting line segments that can be expressed in the form of y = mx + b where we would have three different lines: y1 = m1x1 + b1, y2 = m2x2 + b2, and y3 = m3x3 + b3 and understand that from this Triangle we can define the sine wave that is used to map sound and light waves is no coincidence, just consider the Trinity as in the Father, Son, and Holy Spirit with the understanding that they are 3 different parts but are all one and the same such as the properties of a triangle with 3 different vertices and 3 different angles yet giving you a solid geometrical shape with a defined area! At the same time, knowing that the sound of his voice drives everything into motion including light is the E in Einstein's equation, and the Area of light is bounded by a 45-degree right triangle that is reflected over its hypotenuse. This is why the Pythagorean Theorem and the Equation of a Circle are basically one and the same and this is also why the Trig functions have an Identity - Property that involves the Pythagorean Theorem. The complex numbers are more real than you can imagine including concepts such as infinity because of the ability to create which is no different than the ability to count, compute, or enumerate... When you can take the expression (1+1) and generate 2. That is being Creative! According to his words, we are created in his image therefore we are creative beings! This is why we are able to understand and to do discover and invent the concepts you find within mathematics, physics, and chemistry.
We are creative, and because of that, we can make something that once didn't exist to now exist and we see this all the time throughout the course of history within mathematics, physics, chemistry, biology, and engineering. We possess and inherit all of his traits! This is why I love numbers, mathematics, physics, and equations, they do not lie, and when you study them enough and begin to truly understand all of their connections is when you will be able to open your eyes and see what I see, that there is Intelligent Design Written all over Creation! I don't believe that we are here from some spontaneous random chance explosion! I believe we are here due to an intentional perfectly planned explosion! Did the Big bang happen? Yes, it did! Is it the reason why everything is here? Directly, no. Indirectly, yes. Directly it is not responsible for everything. Indirectly it is what propelled everything into existence. You see, there are cause and effect. The big bang was the effect and not the cause! The cause of it all is from Sound a voice Shouting: "Let there be Light!".
I know most people in this comment section will come without reading it fully.
@@sjegannath6295 Probably, but if that's their prerogative... that's on them... I can't force anyone to read it nor would I want to. It's just out there so that people can read it if they want to.
Wow you are a lot younger than your voice would suggest. And great voice it is. Almost hypnotic. I have totally enjoyed this channel for years and and am amazed at how well the graphic content shows whats going on. Top notch.
You have such an awesome joy and love for math. Thank you for your videos. You've rekindled my love for math
I love all the facts that Grant has come up with in these videos about 69 and why people chose it :D That's amazing content right there!
31:42
Yeah... that's 100% the reason..
Edit: New timestamp: 26:40
hahaha
underrated comment right here
More immature people who get their jollies making seventh grade jokes.
This has been really helpful. Reading "The Road to Reality" and this makes a lot of that less intimidating.
What a brilliant math teacher... Keep up the awesome work.... You are definition for math inspiration
You should also explain the first series considering primes and why it result to log(pi2/6)
You will have to do little Work, but Here's How it go:
First you write the Series of zeta(2)= PI^2 /6 as an infinite Product of Primes (Using the Euler Product Formula).
Applying Log will distribute over the products, converting it into an infinite Sum. Now each log term itself can be written as an infinite series using Taylor Expansion of Log, Giving You an Infinite Matrix terms to be added.
Since The Series converges Absolutely, You can choose to write it in any persuasive order as he did.
I was doing my programming homework and came across a function that did something related to logarithms, especially the so called natural logarithm, so I Googled it, and I didn't understand a single word of it. And then I check my CZcams notifications, and my favourite math channel on CZcams just posted a "video" explaining exactly what I wanted. What a miracle.
Suggestion: A lockdown math video on conic cross-sections and quadrics surfaces would be a pretty cool video to see with your animation skills! :)
You always inspire me to keep learning!
Next month I'm going to get my master's degree in nanotech engineering and I still learn new stuff watching videos aimed at high schoolers. Not sure if that makes you an excellent teacher or me a big fraud. Maybe both
Where are you from? I am from India, i am also very interested to study nanotech , now i am in studying btech Ece 2nd year. Can you pls help me out ?? , what are topics you read / practised in master's ,materials used , which university should i look to pls help me out🙏
an excellent teacher. definitley.
That makes him a great teacher and you a great learner.
Please watch my vedio too
@@roundphysics ok, I well
Amazon explanation. Natural log is deeply rooted in calculus, which I wood say is the best branch of math. We should never 3b1b for Grant-ed.
I'm so grateful for this videos. My deepest respect to Mr. Sanderson.
Thank you for helping me realize the beauty of mathematics.
I want to request you to make videos on real analysis when you've got time , i believe this sector of math requires more of the visual and intuitive understanding even to walk ton few steps . I know you could unravel it better. Thanks for your awesome classes online .
Question: what is the cardinality of the set "5 favorite math pieces by Grant"?
Tip: bigger than five
Thank you for being the best math teacher I’ve ever had
Exactly how I learned exp(x) function in high school.
I had a really good teacher, I just didn't know it at the time.
2 years into a maths degree I went back and found him and thanked him.
The first question I realized that the density would be logarithm of some value. As the density of primes at lower values is very dense And sparse the larger the number as many other factors may go into it. I thought it was the log base 2 I good to know it's a natural log function.
Such an incredible video
Why?
It’s just so amazing that how he’s bringing maths into so many people. Changing how they think about it
I just want to thank you! I found my love for math again. All because of YT logarithm and your channel. I know you probably won’t read this, but still want to SHOUT it out to the world!
These streams areso great! I love them
44:12 it was a time when we were forced to joke about 69.
I love the way he looks and sounds. It's like "I assume that you people said it's 0.69 because it is the correct answer -AND NOT BECAUSE IT'S A SEX NUMBER- "
44:19
Same thing here
26:40
@@technoultimategaming2999 I wonder if Grant even knows what 69'ing even is.
This is the measured straightfacedness of someone who has had to tutor giggling undergrads and/or high schoolers before, and heard the same joke over and over, and knows that the only way is to leave it unacknowledged and patiently wait for it to get boring
@@carultch He wouldn't say "I assume everyone guessed that because it's the correct answer" if he didn't know.
Only the emotionally immature even thought about it. Grow up, and if that is not posible, keep your childish comments to yourself, the grownups are talking!
I've been watching this series religiously because I'm pi-ous and I have a primal urge to go on tangents.
"PRIMal"
Cos the failure to do so is a sin.
what kind of program do you use to record your video please help me?
In high school, we introduced the natural logarithm as the integral of 1/x starting at 1. That seems pretty natural.
We derived all rules for calculating with logs and exponentials from it, using that the derivative of ln(ax) is 1/x, too.
Only when you include complex number, the introduction through Sum(x^n/n!) as exp(x) becomes better.
10:1 odds say theres more than 5 in his list of top 5 pieces of math
I love these videos and haven't missed one yet. Due to my timezone, however, I can't catch them live. Could you please enable the live chat replay feature so I can see what other audience members are thinking/chatting about? It would make me feel more like a part of the lesson. When you revealed that that first infinite series approached ln(pi/6) my mind exploded and I wished I could see others' explode as well in the chat :)
He has chat disabled while live. It is only enabled before the show begins. That chat is a lot of spam.
@@ArrowRaider IkR , I do a good thing of not viewing chats ,just boosts my confidence down for some reason even tho not intended to me.
His voice is so soothing. It's hard to believe it wasn't just an act for videos
This is great to watch considering I just finished Sequence and Series on my Cal II class today. Aced the test btw!
fun fact: euler is the only mathematician that has 2 constants named after them (e and γ)
e stands for exponent not euler
@@kishorekumarsathishkumar1562 e stands for eggcelent
I was going to say “bernoulli technically has a whole bunch of constants named after him” because of the bernoulli numbers, but then I remembered the euler numbers are also a (closely related) thing
Maybe it's natural because it doesn't take any trenbolone or sustanon, after all Euler was not that buff.
In my freshman calculus class, the path to the natural exponential seemed odd, but it didn't have any of the logic problems you described. We first defined the natural log in terms of an integral, and defined the natural exponential as its inverse using the inverse function theorem. The derivative of e^x was obtained that way too.
3Blue1Brown is one of my best subscriptions. I am an engineer and enjoying a lot those informative vidoes.
I was hoping for more stuff regarding the prime power sequences
IkR ,but more up next video!!
You never explain what was going on with the sums where you picked out the prime power terms. It looks like it has to do with zeta functions and euler products.
Yep!
Taking the log of Euler Prime Product of zeta(2) and expanding each log term with it's own Taylor Expansion will give you the Result...
I'm in an engineering course and I think that throughout school and college, one of the most frustrating things is to know why e is so important, and how people even got to it, and how it is even computed. They just tell you it's e=2.7... and that the derivative of e^x is itself. No one explains why ln() is so important. Because of this lesson, which I wasn't even paying full attention, just realised how every exponential function really is just e^rx and why it is even written as exp(). I think it really makes it so much clearer to answer why we do these things instead of just doing them.
You let me fall in love with math even more
You gotta love his mathematical explanation of the infantile choice of 69 as the favorite number :D
When 69 came out, I really hoped he said "WTF ? What The Function ? Which satisfies e^(rx) = 69^x."
Partial sum of harmonic series is not **equal** to ln(N) + constant. And, also, you can't say that error doesn't exceed this constant,because for N = 1 error is 1. Instead, you could say "approximately ln(N) + gamma", but then it raises a question: what gamma we choose? If you pick gamma by Euler, then you have bigger maximum error than if you pick other constant. So, the right way of saying what you meant is that with more N you go, the sum is closer and closer to ln(N)+gamma. Or, use limit expression.
Thanks for what you are doing sir...i am one who got lost in maths...you are Gods blessing. I am going step by step. 🙏
23:00.. that gorilla probably has more iq than me
I think it was Euler in a costume of gorilla. I think he came to check if Grant was teaching his proofs well.
Wondering whether that was Grant's version of euleroid (mascot of flammable math which is fiery meteor with famous mathematician head as the meteor rock)
cant see it now
Gorilla is also at 18:03
I think I can outhink the banana.
Hmm, CZcams is consistently losing the ends of these.
"i may not call that my favorite piece of math because I use that term a lot"...
I really like this guy and how he sees everything.
You are like the greatest math teacher ever.