Euler's Identity (Complex Numbers)

[@gk10002000]

What people have to understand is how brilliant these guys were. They had no internet, few if any textbooks. They had to reason things from first principles, so much original. Just stunning

[@andraskovacs517]

They _wrote_ the textbooks & had them published; they published papers in mathematical journals, of which they had several; and some of them maintained extensive connections through personal letters.

[@ernst9100]

@@andraskovacs517 Thinkers existed in centuries gone by....they invented whole new concepts to explain reality of the times

[@boukharroubamediane119]

and above all, they did not have an electronic calculator! everything was done manually !! 😳 wow!!

[@chandradeepraut9306]

This is what I think ,we could also have been smart if we had no tv (in my childhood upto 6th) as I spent my childhood in watching tv and no critical thinking of sciences ,so I end up just learning what others had done but I started in 10th class inventing techniques to solve maths but it was too late

[@boukharroubamediane119]

Chandradeep Raut 👍

[@neilduran2335]

It is over 20 years since I studied the maths of Euler but this is by far the best explanation I have ever seen. I wish I had seen this video back then. Students of today have it a lot easily than years ago, when you were expected to just get it!

[@MarkNewmanEducation]

This is why I made the video. I was also just expected to know it and it frustrated me that I didn't. This video is part of a whole series on the Fourier Transform which I made for the same reason: howthefouriertransformworks.com/

[@AmikaofMan]

@@MarkNewmanEducation Thank you for the link and the video here for Eulers. Definitely makes more sense this way than the way my physics teacher tried to explain to us back in the day.

[@mp0011]

Now I am waiting for Euler's Supremacy and Euler's Ultimatum...

[@MarkNewmanEducation]

+Niespotykanie Spokojny Rowerzysta haha... Apparently, I might have been wrong in stating that it was Euler who gave e the name Euler's number. It seems to be that the name was coined later although exactly when and by who, I don't know. Euler probably used the symbol e simply to denote "exponent".

[@us-Bahn]

They are coming! But they wouldn’t be if Euler had never been Bourne.

[@petrophilip2279]

LOL, Good one Ian McCutcheon

[@ruffifuffler8711]

Happy you didn't move stuff across the = symbol without a triple deep incisive prayer.

[@georgesealy4706]

Excellent!

[@elmer6123]

For a long time 0 didn't exist, and some people who stupidly claimed that nothing existed had their heads bobbed. Now imagine imaginary numbers. That was like claiming the earth wasn't flat.

[@MarkNewmanEducation]

It was Descartes that called them "imaginary numbers". An unfortunate name. Perhaps he might have done better to call them lateral numbers or something that intimidated the idea of their working in more than one dimension. That might have made them easier to understand.

[@purplrshadowyay]

​@@MarkNewmanEducation A good name could have been surreal numbers (S) or just the other name it has, complex numbers (C)

[@bayestraat]

You should also highlight how euler's identity is nicely shown with multiplication of complex numbers as vectors around a circle plot on the imaginary plane. And how to maintain symmetrical values working out the power spectrum density in FFT.

[@MarkNewmanEducation]

Thanks for the idea. Will look into it.

[@petrophilip2279]

I have seen and used the constant "e" in the study of calculus, complex numbers, infinite series, natural logarithms etc, but no one explained what the number is. This is the simplest explanation I have seen. It takes a special kind of a skill to correctly explain a complex concept in simple terms. Thanks Mark Newman.

[@NoferTrunions]

Explanation by example or picture are the best.

[@NoferTrunions]

@@oldlonecoder5843 Unfortunately, mathematicians are the worst at explaining from example or pictures - they do their explaining by showing how to manipulate equations. Physicists on the other hand are the opposite - they use sketches and examples regularly. A meaningful humorous comment on mathematicians is that once they prove a solution exists, they lose interest and move on to the next problem...

[@parmachine470]

e is also involved in some of the most beautiful comprehensible markings on a chalkboard.

[@Chad-qk1ig]

@@NoferTrunions depends what kind of mathematicians you're talking about. Historically mathematicians heavily utilised geometry.

[@isaacrajagopal391]

It is not just beautiful "in mathematical terms," it is just BEAUTIFUL. Period.

[@MarkNewmanEducation]

Very true.

[@J0nasBern]

To me what is beautiful is that you have a number with infinite and random digits that is related to exponencial growth/decay, then you raise it to the power of a number that we find impossible to solve and so we call it imaginary, and to another number with infinite and random digits that is related to circles and it's geometry, and then you add a single unit, probably the most basic number that we know, to all of this only to get what we call "nothing"

[@jeanpaulniko]

My first ever comment in 10+ years of watching KZbin. Mark, you nailed it! This video has me feeling ecstatic. You have shown me the connection between sin, cos, i, e, and π as presented in Euler's famous identity. This reveals the deep foundation that underlies all of classical math and ties everything together. Now I have seen the light! Thank you so much.

[@MarkNewmanEducation]

Amazing. So happy to have helped and thank you for making my video the first video you commented on in a long time. Any suggestions for future videos you would like to see would be gratefully received.

[@bronzekoala9141]

This is the best mathematica axplanation I've found so far on KZbin for anything.

[@MorganBW53]

I watched this video awhile back and did not comment. It occurred to me that this presentation helped me to connect a couple of dots which enhanced my understanding. I actually had to spend a bit of time to find this video again but I felt the need to say thank you for your time and for the explanation.

[@MarkNewmanEducation]

+Benny Morgan my pleasure. Really happy to have helped you.

[@dianedong1062]

A lot of people seem to think that math is boring, but for me personally, studying mathematics has been like discovering a hidden cave full of beautiful treasures.

[@likeitornot6821]

Likewise

[@MarkNewmanEducation]

I think that the problem is the way that maths is taught, and that it is treated as a totally abstract concept with no relation to the real world when in fact it describes the real world with precise beauty.

[@jaredhouston4223]

@@MarkNewmanEducation You hit the nail on the head

[@jacquesjutras1635]

This gentleman Mark is a very good teacher he is a master.

[@MarkNewmanEducation]

You're most kind.

[@Ch4dBR]

This is the best explation about Euler's identity! Thanks.

[@nanzhang9837]

I don't believe Euler named the number after his own name. From what I know Euler was a very modest man, he instead named the number e because it was the next available letter that was not already taken. Listen to the podcast of 'In Our Times' discussing this number.

[@Soubhik12345.]

This video is so beautifully made ❤ Absolutely love it.

[@BlackNSB]

That sure as hell is beautiful--especially because, as a student, I didn't understand why this formula was so special. Great video.

[@MarkNewmanEducation]

I'll let you into a secret. As a student I didn't understand it either. I just had to accept it. It wasn't until years later when I had to work with it that I found out what the link was when I had to research it for my work. Glad you enjoyed the video.

[@screenflicker1]

I agree that the Euler's identity is beautiful but so was this vid. Hard work went into this!

[@MarkNewmanEducation]

+screenflicker1 thank you. I really enjoyed every moment of it.

[@audience2]

It also links exponents, zero, addition, equality, the identity element under multiplication, and when expanded, trigonometry, division, factorials, and infinite series.

[@DejiAdegbite]

And also indirectly, logarithm.

[@mingzih]

this explanation is insane, even a grade 9 student can understand.

[@ernstboyd8202]

you can consider e^ix to be (e^i)^x. then imagine e^I , (e^i)^2 , (e^i)^3 ... as a special case of a spiral on the complex plane that stays on the unit circle and advances 1 radian (57 degrees) each time similar to (1+i)^1 which is 2^.5 long and pointing at 45degrees. then (1+i)^2 = 2 units long at 90degrees = 2i which is 2*( cos(90)+I sin(90)) and (1+i)^3 is (2^.5)^3 long at 135degrees etc.

[@markproulx1472]

“I used to think math was no fun ‘Cause I didn’t know how it was done But Euler’s my hero ‘Cause I now know that zero Equals e to the j pi plus 1” - Paul J. Nihan

[@lytnin88]

You must be an electrical engineer since you used j instead of i to represent the square root of negative one.

[@manamsetty2664]

You done something to me in 8 minutes which many people could not do. Thank you

[@Frieza.exe.z]

8:10 "The brilliant thing about mathematicians is that . . . when they are on their way to some wonderful mathematical discovery, they don't let a little thing like "Numbers NOT EXISTING" stop them." Is it safe to say HERO ?

[@martlock1]

So beautiful that the simple identity e^(pi*i)+1=0 can link together the most important mathematical concepts (0, 1, i, e, pi) using the most fundamental mathematical operations (equality, addition, multiplication, exponentiation)!

[@BRORIGIN]

Equality is not an operation

[@hazemsaeed6372]

you have no idea how much i loved this video... beautifully explained...

[@Iwannaknow-di7jf]

One of the best math videos I have ever watched....thank you for ur efforts to deepen our love and understanding...respect from Africa, Sudan Hamza

[@thomashan4963]

I just want to say, there's so much efforts in making this video and I appreciate it. From animations to sound effects to historical facts and figures .. this is so much works.

[@JH-ho3qu]

I saw various equations named Euler's method or formula, I was so confused about what Euler's formula is. This is the best video I found to clear up my confusion. Thank you very much!

[@mtolympus-worstplayerever7924]

I have watched over a dozen videos on Euler"s identity, and this is the most clear and straightforward.

[@TheKellydelight]

Beautiful explanation i have never seen such a nice presentation skills . God bless

[@MarkNewmanEducation]

+Unique Family thank you

[@stmellion1]

brilliant, thank you Sir... i was "taught" this badly over 30 years ago... i get it perfectly now...

[@ApenstaartjeYT]

Sometimes I wonder if the internet made us numb.. Back in the day you were kind of 'forced' to think. Just look at this absolute beauty.

[@carlosalexandreFAT]

The association of the main numbers in the field of mathematics with each other, reflects numerical sequences that correspond to the dimensions of the Earth, the Moon, and the Sun in the unit of measurement in meters, which is: 1' (second) / 299792458 m/s (speed of light in a vacuum). Ramanujan number: 1,729 Earth's equatorial radius: 6,378 km. Golden number: 1.61803... • (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18 Moon's diameter: 3,474 km. Ramanujan number: 1,729 Speed of light: 299,792,458 m/s Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km. • (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371 Earth's average radius: 6,371 km. The Cubit The cubit = Pi - phi^2 = 0.5236 Lunar distance: 384,400 km. (0.5236 x (10^6) - 384,400) x 10 = 1,392,000 Sun´s diameter: 1,392,000 km. Higgs Boson: 125.35 (GeV) Phi: 1.61803... (125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97 Circumference of the Moon: 10,916 km. Golden number: 1.618 Golden Angle: 137.5 Earth's equatorial radius: 6,378 Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2. (((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62 Earth’s equatorial diameter: 12,756 km. The Euler Number is approximately: 2.71828... Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Golden number: 1.618ɸ (2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23 Earth’s equatorial diameter: 12,756 km. Planck’s constant: 6.63 × 10-34 m2 kg. Circumference of the Moon: 10,916. Gold equation: 1,618 ɸ (((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3)= 12,756.82 Earth’s equatorial diameter: 12,756 km. Planck's temperature: 1.41679 x 10^32 Kelvin. Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Speed of Sound: 340.29 m/s (1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81 Moon's diameter:: 3,474 km. Cosmic microwave background radiation 2.725 kelvins ,160.4 GHz, Pi: 3.14 Earth's polar radius: 6,357 km. ((2,725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000 The diameter of the Sun: 1,392,000 km. Numbers 3, 6 & 9 - Nikola Tesla One Parsec = 206265 AU = 3.26 light-years = 3.086 × 10^13 km. The Numbers: 3, 6 and 9 ((3^6) x 9) - (3.086 x (10^3)) -1 = 3,474 The Moon's diameter: 3,474 km. Now we will use the diameter of the Moon. Moon's diameter: 3,474 km. (3.474 + 369 + 1) x (10^2) = 384,400 The term L.D (Lunar Distance) refers to the average distance between the Earth and the Moon, which is 384,400 km. Moon's diameter: 3,474 km. ((3+6+9) x 3 x 6 x 9) - 9 - 3 + 3,474 = 6,378 Earth's equatorial radius: 6,378 km. Orion: The Connection between Heaven and Earth eBook

[@MadScientyst]

Thank u for this post friend....I intend to check these relationships out!

[@carlosalexandreFAT]

@@MadScientyst 👍👍

[@howardebenstein3204]

Apparently a higher authority uses the metric system.

[@GrandTeuton]

I think this is the finest maths video I've seen on KZbin - and I have sought out many. Well done!

[@lylecosmopolite]

Euler was perhaps the most productive mathematician of all time. The number e is named in his honour. It is to Euler that we owe much of our basic understanding of infinite series and complex analysis. But Euler had a weird flaw -- his proofs always fell well short of the standard laid down by Cauchy, Riemann and Weierstrass, and often were wrong even by the relaxed standards of the 18th century. But no result ever published by Euler was ever shown subsequently to be wrong. Everything he claimed to be a theorem in fact was, even though his proofs were never rigorous and were often downright wrong. Euler was perhaps the most spectacular example of history of mathematical intuition. Euler derived his eponymous identity using the infinite series for e^x, and his proof was largely correct.

[@sacredsanctuary420]

this is the best explanation I've seen so far :)))

[@MarkNewmanEducation]

That's very kind of you to say so.

[@ayoubmounadi2142]

Sacred Sanctuary i share the same opinion with you 😃

[@wickrenner488]

Yes, a very good explanation. I finally understand Euler's Identity, huge revelation, thanks.

[@sophialopez8024]

Maybe that's because you limit yourself to 'religious' sources..

[@michaelzhu6952]

Agree!

[@artha3283]

AMMAZING!!👍👍 !! this is how a story is told and a lesson is learnt 👌

[@afaqsiddique4940]

You rocked man.....we need teachers like you ...... Lots of love from Pakistan ❤

[@anymaths]

watch my maths videos.

[@giggetto71]

thanks Mark. Beautiful explanation. I would only add at the end, along with 1: e, 2: pi, 3: sin, cos, 4: i a 5th concept: the concept of 0 which is another great math concept.

[@MarkNewmanEducation]

Oh yes, of course. Thanks.

[@SassanRohani]

The most beautiful formula in Mathematics explained in the most beautiful way in this video. Thank You!

[@esuchil]

Other than having that ugly ass Pi in it, and needing to subtract 1 for no reason, it's pretty meh.

[@MarkNewmanEducation]

You're welcome.

[@MarkNewmanEducation]

...but PI is SO useful. It crops up everywhere!! I use it a lot in electronics and filter design.

[@The_Ass_]

Euler was so obsessed with complex numbers, he based his identity off them

[@samanthabanya1349]

This is the best explanation of this that i have seen

[@fikralafiq0361]

man, this is the easiest video to understand above all videos. thank you!!!

[@MarkNewmanEducation]

Amazing. Glad it helped. Check out my channel for more videos like this.

[@Mzt1_]

Aside the brilliant minds behind the formula, your presentation is also "beautiful" and very structured. No wonder, it is almost 2 in the morning and am wide awake!

[@ertansaygi]

Best explanation i've ever yet encountered. Thanks, man. Also, because of this video i understood "Thermodynamic absolute temperature"...It's because the universe gets cooler and cooler for long periods of time but it can only get cooler upto euler's 2.718...In Kelvin.

[@lumin4te644]

Holy ****, how did I not known this. Absolute zero is like -273 C (more or less) or 0 K, the temprature where molecular movement is no longer found to create 'heat'. This acctually gave me a moment of realisation. If I had to guesse I´l probably see this in one of my classes in the future, cause I just started my first year of engineering science.

[@ENBOmniGaming]

There is a great feeling when you understand a new math concept - I finally understand e and its relation to sin/cos after this video. Excellent work please keep it up !

[@MarkNewmanEducation]

I SO know what you mean. For years this stuff was incomprehensible to me. Then I got this project at work (I'm an electronics engineer) which was all to do with complex impedences and all things "imaginary" that I HAD to understand in order to get the job done and finally, after much research, I got that Eureka moment that I am so happy to have helped you arrive at as well, where everything just fell into place. Then I really wanted to make sure I understood it properly and the best way to do that is to try and teach it. I am now busy working on more stuff as we speak which I shall be putting into an online video course about the Fourier Transform. I have published my research notes for the course at: www.themobilestudio.net/the-fourier-transform-part-1. I'm posting progress reports on the course on a Facebook page facebook.com/TheFourierTransform/ which includes snippets from some of the videos that will be in the course. I'm just putting the finishing touches to the video on "phase" and will post an extract from it on the facebook page (and probably here on my channel too) during the next week or so, so please stay tuned. Thanks so much for your comment.

[@Bobby-fj8mk]

@@MarkNewmanEducation - good post - i might be imaginary but it gives you real results when you play with complex numbers & impedance's in electronics. It's all to to do with time - relating one voltage to another by a phase difference. When you do Fourier analysis you don't only get the amplitudes of the various harmonics you get the phase differences coming out of the equations. It's about time domain & frequency domain. We couldn't calculate anything much with using i. It is very powerful mathematics.

[@pwnmeisterage]

EEs describe phase impedance as _Z=x+jy_ ... I guess easier (visually relates to Euler's identity) when graphing with polar coordinates but why isn't _i_ good enough?

[@Forever._.curious..]

Offc 🌸 it's beautiful the two fundamental constants e and π comes in a equation along with an imaginary number

[@tarunsharma209]

This video really made me understand how beautiful Euler's identity is

[@lockdowntrendings1207]

I don't say this to every explainer or professor or technologist but I think it's suits you well. "You are real intelligent"

[@bryanbowen4193]

Euler's number isn't just beautiful. It literally describes reality to perfection.

[@ucctgg]

What is reality?

[@MeButOnTheInternet]

This is beautiful. I've never seen it explained so clearly

[@tamirerez2547]

A question: What is the volume (in centimeter cube) of a pizza with radius of "Z" cetimeters, and high of "A" centimeters? Answer: PI*Z*Z*A centimeters cube!!

[@whatever5575]

The beauty of the formula is that it says so much in so little space and in a simple and elegant way. That's what good literature or well written instructions ought to be. Say it simply.

[@hsenagrahdeers]

This right here, ladies and gentlemen, is what high-quality educational content looks like! I can't thank you enough!

[@MarkNewmanEducation]

You're most welcome.

[@jegadesh5632]

The true heroes are living in the world of youtube.

[@techno2371]

"They don't let a little thing like numbers not existing stop them" This is why math is amazing, the limit is how far you are willing to push it!

[@MarkNewmanEducation]

Very true!

[@lillysumathi7825]

So in simple terms, the value of the function of e raised to ix at pi rads is -1. That's mind blowing

[@vikasvishwakarma5263]

e^(i*pi) means you have rotated the complex number 0+i to 180 degrees. Because in polar form it is written as cos(pi) + isin(pi) and it is -1 :)

[@mdabusaqibalamansari9979]

This is the best video on the euilers identity

[@yarabamba]

The best explanation of Eulers identity.

[@tenshi.mp3]

This is one of the coolest math videos I’ve ever seen, thank you

[@MarkNewmanEducation]

Wow, thanks! It's part of an online course which I am producing on the Fourier Transform. You can access the whole course at howthefouriertransformworks.com/. So far the course is made up of a mixture of video lectures and blog posts. I am currently working on turning the remaining blog posts into videos. I've just released Video 7 - From Fourier Series to Fourier Transform Part 1 to my Patreons patreon.com/MarkNewman and I am hard at work on video 8

[@SuperKnowledgeSponge]

I think this is the only guy that ever clearly explained Euler's identity on youtube that I could actually understand. wow.

[@7rgrov198]

@@jacksmetana7932 then what did he spend 14 minutes doing?

[@perfectlimitless5944]

most beautiful explanation ever, i've learned more here than college ever, wish all lectures were like this

[@MarkNewmanEducation]

Wow, thank you!

[@2002budokan]

Yes, they are beautiful, and you explain them beatiful.

[@snavarro7713]

The formula IS a wonderful solution. The 4 concepts are combined all together and zero is appearing. That is Amazing, that is elegant, that is math.

[@agytjax]

Great Video. You might as well add that this is the only equation that connects all the important mathematical constants - e, pi, i, unity (1) and zero.

[@veerk3494]

this explanation much better than other videos that try to explain euler's identity by rotations

[@MissingPleiad]

One of the best videos on youtube

[@nellavaleri]

What an elegant explanation

[@MarkNewmanEducation]

Thank you

[@dollascience7808]

This is the best explanation I ever had except that one explanation when I was in 4th standard.

[@yasiramir3292]

I cant help but praise . Wonderful

[@deeneyugn4824]

Q: How much money you got in your account. A: Oh! Just e^ipi+1

[@goonzkelly4978]

hahaha😂😂😂

[@Nachruer]

Простое и наглядное объяснение. Спасибо!

[@MarkNewmanEducation]

Добро пожаловать!

[@03348826]

The most beautiful Eular equation described by a Beautiful way . Thank you so much . I am doing PhD in Quantum Physics.

[@mrisholukamba1696]

Heyy how old are uu and i really love physics am 18 yo

[@affanthephoenix492]

I am 18 Yo tooo

[@VenThusiaist]

this comment section is weird

[@NOOBCRASTINATOR69]

Seriously tho!.. lizard is a talk to elephant walking Mars on rock help from a giraffe.. right??😂

[@Tennisbord]

Lot of religious nuts in this comment section

[@clunapizarrog48]

Interesting comment... love it

[@CorekBleedingHollow]

If that wasn’t mind blowing enough.... brace yourself.... those people in the 1600s and 1700s did that math without calculators......... INCREDIBLE! That had to have been full time work and overtime!

[@mechailreydon3784]

Okay now my mind i properly blown.

[@parmachine470]

we had slide rules in the early seventees then we got calcs with a memory button. HP started selling a little better in75. HP 40? Never saw a calculator with a solve button or do calculus and differential equations any where near then. Let alone Put a man in space or The Manhattan project. Calculators are bran new

[@f3796]

e^x and cos(x) + sin(x) share similar pattern, but different signs. e^ix turns the signs same with that of cos(x) + sin(x), but i is now newly added, so have to multiply i to sin(x) to make them equal. We have 1. e^ix = cos(x) + (i) sin(x) Known that cos(pi) = -1. As there is imaginary number, we have to make sin(x) to 0, known that sin(pi)=0 So 2. e^ipi = cos(pi) + (i) sin(pi) = -1 3. e^ipi + 1 = 0 That’s amazing!

[@SpacePoolNoodle]

Thank you! I was frustrated because most videos did not show why e^ix = cos(x) + i sin(x), this made it very clear.

[@bluegtturbo]

As a maths teacher I have to say this is one of the best and clearest explanations I've ever seen! Well done!!

[@TheNabob57]

good thing you can handle math, your English is atrocious. I depise the hip hopping, ghetto-fying of MATH to maths.

[@bluegtturbo]

@@TheNabob57 How dare you! I am English (as in I live in England - there's a clue right there. Maths is short for mathematics (implicitly plural hence the 's' at the end)). Unlike you lot of ex-colonials, we know how to use the English language correctly. That's why we use the correct construction 'If + past tense' rather than 'If +would' as in 'If I knew' as opposed to the stupid American construction 'If I would have known'...

[@CommanderSp00ky]

It baffles me that even in a community related to mathematics and science idiots and douchebags are thriving to ruin everything.

[@dinoscythe6335]

It all makes so much more sense now. Thanks :D

[@choimdachoim9491]

Simple, lucid explanation that went clear over my head. I usually get mental pictures when I comprehend mathematical stuff...today nothing, not one thing. Oh well, haven't died yet, might still be time. Maybe as I die I'll sit up in bed shouting "I get it!"... Leave them all wondering. Maybe I'll do that just to mess with people...yeah...I like that. "I JUST FIGURED OUT THE MYSTERY OF LIFE! IT'S..." Klunk.

[@Snow-qt7pn]

This was the most beautiful explanation of the most beautiful identity.

[@larrylorenzen2449]

e appears when you try to differentiate the log(x) function. Everything works out as long as you use e as your base. That’s why e is called the natural base. Nature insists that you use it.

[@shinigamisteve5607]

I remember when my Calculus II teacher taught me this. It blew my mind

[@Atmosnofear]

Beautifully explained, thank you.

[@CoranceLChandler]

Beautiful! Inexpressibly and inexhaustible beautiful! Astoundingly and undeniably wondrous! Didn't understand a single, solitary syllable he said but I want to. Time to get back to learning math

[@Lawliet82]

This is basically a mix of all we learn in highschool but in one single calculation, absolutely madness bro

[@DennisRiungu]

God bless you for explaining this so beautifully!

[@unkup0w]

You. Are. Incredible.

[@harshadasonawane7445]

It was the best description I have ever heard That's why I love maths

[@MarkNewmanEducation]

+Kiran Sonawane thanks. I've always found maths SO hard. Once I understand something, it feels like a huge achievement. I find the best way to understand a subject is to try and teach it.

[@harshadasonawane7445]

That's really a big achievement Mark That's so awesome

[@harshadasonawane7445]

That's really a big achievement Mark Sir It's Awesome

[@沈國祥-s6b]

Kiran Sonawane ADSA , AND VGFYTIPO . 2E4DSR5466G TFGCGFYTT , TTTFGCXZASZDWCS GYFHCBGFHGHUJHIL常常吃車程遲遲唱出

[@zbuddha]

Numbers don't "exist", they are all created by human imagination and creativity. As our creativity expands, so does the set of numbers. The number i is just as much of a number as the number 2. It is used in many applications in our lives, such as the transmission of electricity. Many mathematicians argue that the terms "real" and "imaginary" are misleading and unfortunate.

[@boguslawszostak1784]

how the numbers "exist" is a problem for philosophers

[@traininggrounds9450]

Hear O Israel Yahweh our God: Yahweh is ONE and you shall love the Lord your God will all your heart and with all your soul and with all your mind. Everything comes from God. The difference between real and imaginary is the visible and the invisible. You don't understand why the cake tastes good but you can still make one and enjoy it. You don't understand how the plants grow but you do know what you need to do to make them grow and give you food. We can USE these numbers without understanding them and without explaining what they mean. But DON'T think for a minute that they aren't real just because we can't see them. However, just because a concept of a number evaluates other relationships of numbers does not mean that bridge exists in the universe. We want to reflect what the universe is doing because that is how we know we are doing what is right. If our system takes shortcuts to get to answers that the universe does not actually take then we will not find out the intricacy of God's creation and give him glory. There is no reason to think we can't create shortcuts that "work" mathematically according to some logic but that don't actually have any correspondence with the real world. So the question is how do we tell what is real from what is unreal between what is visible to what is invisible?

[@bustybonnie6700]

WOW! I've watched so many videos & asked so many math guys to explain this concept to me & just never clicked, but something about this video, I finally get it now!

[@MarkNewmanEducation]

Glad it helped!

[@jimrogers2579]

Mark Newman -- wow! Sorry I missed the release of this video almost 4 years go. Beautiful! This is a stunning exposition of an often spoke wonder I had never grasped. Your explanation left me gobsmacked twice. Holy cats man -- great! I will share this unfolding to help my students understand - and point them to this video. Can't wait for more! Extremely well done.

[@abhiramreddy790]

If we had teachers like this mathematicians would be everywhere

[@MarkNewmanEducation]

Wow. Thanks.

[@jkstudyroom]

Just beautifully explained! Taylor series with imaginary numbers imagine that!

[@divyeshpandya8674]

This video gives a satisfactory proof of an extraordinary equation which seems to be very confusing but beautiful after understanding :-)

[@MarkNewmanEducation]

Thank you.

[@teazer999999]

Wonderful explanation. I could have used this a few decades ago during my first class in Quantum Physics, when the professor wrote out Euler's Formula without any explanation. I couldn't understand how raising a number to a constant could create cosine and sine functions.

[@eamonhannon1103]

Really excellent explanation . Thank you

[@MarkNewmanEducation]

Glad it was helpful!

[@--tsatsa--]

Great explanation. Cheers from Bulgaria :)

[@MarkNewmanEducation]

Glad it was helpful!