# The number e explained in depth for (smart) dummies

## 100 thoughts on “The number e explained in depth for (smart) dummies”

1. Me Yes says:

Here's something that's been bugging me: Why is (1+1/x)^x = e when x approaches infinity?shouldn't it be 1+1/inf)^inf at the limit, which would basically be 1? Because 1/x when x->inf is 0, and (1+0)^inf should be 1.so then why is it e and not 1?

2. Creative Name says:

Why am you replace all the numbers with 2?

3. mohan153doshi says:

Absolutely awesome explanation. Everything that we just took for granted regarding e has been explained so very well here. Thanks for this lovely video.

4. Oon Han says:

5. Reivivus says:

I wrote a C++ program to estimate e^x with that summation series.
This program estimates Euler's number (e) to 10 decimal places if no more than 10 summations are added up,
and if more estimates are used the precision is increased to 18.

I think I can confidently estimate e^(x) to 18 decimal places if I add up 1000 terms from n = 0 to n = 1000.

Here is the program:

#include <iostream>
#include <iomanip>
#include <cmath>
using namespace std;

long double factorials(int n);

int main()
{
int x, maximum, precision;
long double total = 0;

cout << "This program will estimate the e^xnusing the infinite series: x^n / (n!) from n = 0 to large number nn";
cout << "For any user defined value of x" << endl;

cout << "Please enter a value of x. Use 1 if you want to estimate Euler's number: ";
cin >> x;
cout << "Please enter the span of your search from n = 0 to some number: ";
cin >> maximum;

if(maximum >= 0 && maximum <= 10)
precision = 10;
else if(maximum > 10)
precision = 18;

for(int n = 0; n < maximum; n++)
{
total += pow(x,n) / (factorials(n));
}

cout << "The estimated total of e^(x) over " << maximum << " iterations is " << fixed << setprecision(precision) << total << endl;

return 0;
}

long double factorials(int n)
{
long double factorial = 1;
if(n == 0)
return factorial = 1;
else if(n == 1)
return factorial = 1;
else if(n > 1)
{
for(int i = 1; i <= n; i++)
factorial *= i;
}
return factorial;
}

// Running the program I estimate Euler's number to 20 decimal places by adding up 1000 terms.

This program will estimate the e^x
using the infinite series: x^n / (n!) from n = 0 to large number n
For any user defined value of x
Please enter a value of x. Use 1 if you want to estimate Euler's number: 1
Please enter the span of your search from n = 0 to some number: 1000
The estimated total of e^(x) over 1000 iterations is 2.718281828459045235

Process returned 0 (0x0) execution time : 5.487 s
Press any key to continue.

6. Dan Kelly says:

LOL smart dummies.

7. Dan Kelly says:

I like that episode when Homer has the pencil removed from his sinus and becomes a genius. I think the pencil was putting pressure on his brain. I think he started as a genius when he was a kid but a pencil got shoved up his nose then he became a dummy.

8. Joseph Dunphy says:

excellent graphics. Just suggest people go to full screen at the start of the video, so they can get the benefit of the graphics.

9. Ronak says:

The equation on your T-shirt is amazing!👌

10. KaChun Li says:

how to solve the integral on your T-shirt ? amazing too…. (from Hong Kong)

11. greg55666 says:

Hi Mr. O'Loger, what I think you really need is a video on i^i.  These explanations of e^ipi are all similar, and it's easy to get so you can completely internalize how to picture e^ipi without, I think, really really understanding it at all.  i^i exposes this weakness.  What is REALLY going on here?

12. traso says:

13. FreezePro says:

So is it possible to get that on a channelin german? I think i can hear some german accent (not bad) and I am german too. So is it possible to get those videos in german?

14. videofountain says:

Around this time in the video https://youtu.be/DoAbA6rXrwA?t=8m22s it is stated that the error is equal to something else. Was that a correct statement?

15. Chris G says:

Look at self roots, that is the nth root of n:1 root 1=1, 2 root 2 = 1.414…, 3 root 3 = 1.442…, 4 root 4 = 1.414…,  5 root 5 = 1.380…, 100 root 100 = 1.047… 3 has the biggest self root of any integer, but the number with the biggest of all is – yes – e, at 1.444667861…I found this by myself, which doesn't necessarily make it a new discovery of course. I'm only a smart dummy after all. But please tell me if you know of it anywhere else in the literature.(Another fact that's news to me is that the self root of 2 and that of 4 are equal. They seem to be the only two numbers sharing a self root. Right?)

16. How mathematicians create maths says:

Well if that isn't Mr. "making maths entertaining again!" 🙂

The way you explain the math of e feels like eating the spinich of the famous Popey the Sea-Man!

You create the desire to also want to go deep down into other (hopefully still hidden) secrets of the quenn of sciences! It feels very motivating because you empower/reactivate the inborn intuition of people, where Einstein told us: "Thge only thing that really matters is intuition!" You make math feel come alive again and making it usefull for millions of people! Thank you for that valuable contribution to society!

I'm actually a colleague of yours but not so much into software. If you tell me which software to use, then, once I become successfull with it, I'll mention you as the man who gave me the oportunity and connect with you, so we can both help even more people to inspire their lives with maths! 🙂

17. David I. LEVINE says:

Love this. A detail: Around 8:44 you say “equal to 7 factorial,” when you meant “less than 7!”

18. Rishabh Dhiman says:

I prefer the other proof of cis(x)=cos(x)+i sin(x)=e^ix

Let f(x)=cis(x), note that f(0)=1
f'(x) = -sin(x)+i cos(x) = if(x)
i = f'(x)/f(x)
Integrating both sides and using standard u-substition,
We get,
ix + c = log(f(x)) => f(x) = e^(ix+c) and since we know that f(0) = 1, we get c = 0. Therefore, e^ix = cis(x)

19. Steeve says:

@ 4:13, do you mean polynomial coefficients? binomial would be just 2 terms

Amazing video. Exactly what I had been looking for: "How 'e' ties together various branches of arithmetic".

21. Neil Osborne says:

Fantastic video as usual! Keep up the good work. For some reason, I find your videos more explanatory/fun than those of Numberphile (though I like them too!)

22. Vince Ciricola says:

Excellent! A great presentation. I look forward to viewing your other lectures. Also, The Simpsons can be a very educational show… if one pays attention; thanks for referencing the show. I’ve recorded it for years [decades] just so I can ‘rewind’ or freeze frame an interesting/educational reference.

23. Lela R. says:

Thank you so much!
Your explanations are fantastic! I'm falling back in love with math!

24. 王星 says:

看着对数函数导数的推导，很漂亮

25. Mateusz Dziewierz says:

E=271801÷99990

26. Jacob Drake says:

e & i are variables not constants

27. Ion Murgu says:

A big Mathematician, can't to remain in a logical slavery of Conventional, I hope at the last you understand what it mean. #Math #Mathematics #Science #News #ScienceNews #EarthProudDay Fermat's Last theorem is FUNDAMENTAL in Math and Science and after 380 years we can say in Culture.Fermat-Murgu Impossible Equations Sent Fermat's Last Theorem in FUNDAMENTAL By two Methods – One need a Postulate and I hope soon a Youngest Mathematician will be coming soon with , is about Fermat-Murgu n Media for 3 Integers associated to Fermat Equations – but second method is Definitive, and Second Grade Impossible Equations are absolute Conditionals for General Cases of n as Power and not only for 3 , 4, 5 and so on. See it at :" www.climaticdisorder.com "

28. Taiken64 says:

So… why do math teachers say that pi can't be written as a fraction?
With this identity, it clearly can be written as pi = ln(-1)/i …

29. David Wilkie says:

"e" is the elemental rate of change, the fundamental constituent of all rates, it's the dynamic existence of specific uncertainty, (where is the source/orign Singularity positioning, in infinite/eternal possibility?), containing the rational-numerical relative rates of exponential-logarithmic change, ..that combined, is the Universal differentially-integrated wave-package @ 1-0D singularity. Ie existence is here-now in actuality, and everywhen-there in the sum of all history probability, so if e-Pi-i is the singularity-source of continuous containment connection, then e^Pi-i is the "empty" return to Origin by reflection, because +1 -1= f(0) remainder 1/1 = 1, all history probability difference, in infinite closed-continuous possibility.
It's the symmetrical phase-state of QM-Time.

In terms of math-philosophy, "e" is "numberness".., virtual-mathematical connection, and the base self-defining quantity, .dt-quanta, of the eternally-infinite spectrum-quality, QM-TIME.., by instantaneous reflection-connection.
(The "dumb" amateur explanation for smart people)

Admire the "Mathematical Magic".., the continuing science of existence beyond rational comprehension…, n!->holistic completeness, as n->infinite.

Covering the "sizeless" Singularity vanishing point location with infinite probability positioning points in Superposition = Superspin.
This is the self-defining, (because e is the derivative of itself, existing in temporal superposition), observable mechanism, of QM-Time modulation in Quantum Field Operation, = Eternal Iteration = Temporal Substantiation of infinite probability in possibility = +/- f(1) connection.., and is equivalent to the concepts proposed by Professor John Wheeler's "One Electron" theory in these assembled mathematical components.

30. Héctor Rodríguez says:

Not only this is super neatly explained, it also includes The Simpsons and Katy Perry. Best video ever!

31. Vincent Cantin says:

I would go to 1 million and 1 digit, make a quick test to know if I should continue to 1 million and 2 digit, quick test, and so on, as long as my quick "carry test" proves that there is a doubt about the 1 million's digit value (and maybe some of its predecessors)'s value.

32. Pixel Pusher says:

33. mehmet2247 says:

You are a genius teacher

34. Reality Versus Fiction says:

IRREFUTABLE SIMPLE/ELEMENTARY BLACK AND WHITE ARITHMETIC (SHARE)

THE LENGTH OF A CIRCLES EDGE
PRELIMINARY REFERENCES CONCERNING Pi
THE AREA OF A CIRCLE SUMERIA 1000 BC
THE AREA OF A CIRCLE – ARCHIMEDES TRIANGLE 287 – 212 BC
TWELVE STEPS TOWARD THE PRECISE VOLUME AND THE SURFACE AREA OF A SPHERE

THE LENGTH OF A CIRCLES EDGE

Using a 120-centimetre length diameter line

1. Multiply the 120-centimeters diameter line by 3
2. The Circles length is 360-centimeters
3. Every Circle has 360-degrees The Circles length is 360-centimeters, and each degree is 1-centimeter in length.
4. Multiply the 120 centimetres diameter line by 4, and the square of diameter line is 480-centimeters, and the circle's length is three-quarters of this length.

PRELIMINARY REFERENCES CONCERNING Pi

Reference Encyclopedia Britannica
Quote: Archimedes (c. 250 BC) took a major step forward by devising a method to obtain pi to any desired accuracy, given enough patience. By inscribing and circumscribing regular polygons about a circle to obtain upper and lower bounds, he obtained 223/71 < π < 22/7, or an average value of about 3.1418.

• MLA Style: "pi." Encyclopædia Britannica. Encyclopaedia Britannica 2008 Ultimate Reference Suite. Chicago: Encyclopædia Britannica, 2008.
• APA Style: pi. (2008). Enc

Reference Oxford English Dictionary
Circumference
1. The boundary which encloses a circle. 2. The distance around something.
Note the two words encloses and around

Boundary
1. A line marking the limits of an area.
Note the word line

Pi
The numerical value of the circumference of a circle to its diameter (approximately 3.14i59)
Note the word approximately

In Sum

It is clear from these definitions the original Archimedean formula (which was later given the name Pi and decimalized) of 22/7 as an improper fraction or 3 whole units of length plus one-seventh remaining, relates purely to the length of a line that circumnavigates around the area of a circle, to form a. *approximate boundary that separates the area of a circle from its surrounding area.

Therefore it follows that regardless of all beliefs and claims to the contrary, all of the calculation’s concerning circles and curvature’s that have been carried out using the formula Pi are approximates, and as such are mathematically inaccurate.

In nature rather than mankind’s linear geometry, shapes do not possess outlines or boundaries, a prime example of this is the circular shape of the Moon; the Moon does not have any form of outline but rather its light yellow colour is contrasted and silhouetted against the dark night sky.

This is as with the shapes of all things, the atoms of the surfaces of their physical forms merge with the atoms of their surrounding environment e.g. the Earth’s atmosphere. And as such the closest we can get to accurately calculating the area of a circle is by use of the length of a circles edge, rather than inaccurate linear approximates.

THE AREA OF A CIRCLE SUMERIA 1000 BC

Using a 120-centimetres Diameter line

Multiply the 120-centimeters diameter line by 3

The Circle is 360-centimeters long

Multiply the 360 centimeters diameter line by 360 = 129, 600 square-centimeters

Divide 129, 600 square-centimeters by 12 = 10, 800 square-centimeters to the circles area

THE AREA OF A CIRCLE – ARCHIMEDES TRIANGLE 287 – 212 BC

Proposition 1.
The area of any circle is equal to a right-angled triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.

The diameter line we are going to use to find the area of a circle according to Archimedes proposition 1, has a 120-centimetre length

The triangles base-line is 60-centimeters as per the radius of the circle

The height of the right angle is 360-centimeters as per the length of the circle

60-centimeters base-line x 360-centimeters circle length yields a rectangle with an area of 21, 600 square-centimetres, which when divided by 2, yields an area of 10, 800 square-centimetres to the area of each triangle and the 120 centimetres diameter line circle.

Using a 120-centimetre diameter line multiply it by itself to yield a square with an area of 14, 400 square-centimetres.

Divide the square into four quadrants and each quadrant will have an area of 3, 600 square centimetres

Divide one of the 3, 600 square centimetres quadrants by 4, and each of the four sub-quadrants will have an area of 900 square centimetres

Multiply the area of three of the 900 square centimetre sub-quadrants (2.700 sq cm) by 4 = 10, 800 square centimetres to the area of the circle and three-quarters of the area to 14, 400 sq cm square

The 60-centimetre radius of the circle’s diameter, gives 3, 600 square-centimetres of the area to one quadrant of the overall square.

SUMERIAN METHOD 1000 BC; 10,800 square-centimetres to the circle

ARCHIMEDES TRIANGLE 212 BC; 10,800 square-centimetres to the circle

THREE TIMES THE RADIUS SQUARED 2017 AD; 10,800 square-centimetres to the circle

FOUR IDENTICAL RESULTS, cannot be coincidental, and all four identical results are equally and self-evidently correct and incontestable.

TWELVE STEPS TOWARD THE PRECISE VOLUME AND THE SURFACE AREA OF A SPHERE

CUBE TO ITS CYLINDER

Calculating the surface area and volume of a 6-centimetre diameter sphere, obtained from a 6-centimetre cube.

1. Measure the height of the cube to obtain a length of diameter line = 6 cm

2. Multiply 6 cm diameter line by 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of the perimeter to the square face = Length 24 cm, Square area 36 sq cm.

3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.

4. Divide the cubic capacity by 4, to obtain one-quarter of the cubic capacity of the cube = 54 cubic cm.

5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.

6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.

7. Divide the cubes surface area by 4, to obtain one-quarter of the cubes surface area = 54 square cm.

8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.

CYLINDER TO ITS SPHERE

9. Divide the Cylinders cubic capacity by 4, to obtain one-quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.

10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.

11. Divide the Cylinders surface are by 4, to obtain one-quarter of the surface area of the Cylinder = 40 & a half square cm.

12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere.

www.fromthecircletothesphere.net

35. Kasia Zając says:

e is rational.
e=271801/99990

36. Daniel Șuteu says:

The number of required terms for computing e to 10^6 decimal places is 205022.

If we let f(n) to be the number of required terms for computing e to n decimal places, for 10^7 decimal places (and beyond) we have:

f(10^7) = 1723507
f(10^8) = 14842906
f(10^9) = 130202808
f(10^10) = 1158787577
f(10^11) = 10433891463
f(10^12) = 94851898540
f(10^13) = 869200494599
f(10^14) = 8019346203785
f(10^15) = 74419210652835

NOTE: f(n) can be computed very efficiently by using the binary search algorithm.

37. StarstormHUN says:

Ummmm, you need continuous convergence to be able to derive a function by deriving the sum of its component functions term by term, but I get why you didn't discuss that 😛

38. eddewit1992 says:

16:40 what is the significance?

39. Venkatesh babu says:

e is irrational log is rational phi is circle I is imaginary plus or minus one and a zero. Part 90% solutions to include most number type. Transcendental including would be better.

40. Kevin Thorton says:

Hey can you use your maths to create a huge magnetic suit that would allow me to walk on the streets and suck up cars onto the suit.

41. Stanley Nicholson says:

At about 14:39, when you are proving that the derivative of e is its self. You are using circular logic, because that series of "e" is derived from the fact that the derivative of "e" is "e".

42. Luis Saavedra says:

After 30 years since I studied calculus this has been the most BRILLIANT explanation of natural logarithms EVER ! FANTASTIC I WILL NEVER FORGET THIS 👍🏻GORGEOUS VIDEO

43. zim zom says:

i have a question about 5:30, how do they all go to 1? the last few would go to zero wouldnt they? because you'd have (x!)/(x^x) or close to it which goes to zero as x goes to infinity eg (1000!)/(1000^1000)= 4*10^-433

44. Lain311 says:

不好意思 可以拍一部關於墨卡拖級數的片子嗎？

45. Butters Mars says:

This is really beautiful!

46. Agus Widjaja says:

Hi, Thank you for your videos. I have a question regarding the end result of your equation at 17:59. Is it possible to input numerical values into that equation and get the left side of the equation equals the right side of the equation? How do you do that? Thank you.

47. tuckswa99 says:

Excellent, calculus in a nutshell!!

48. wolfguard says:

Say I take an error of 1/3!, therefore (1+1/1!+1/2!+1/3!). If I now estimate 8/3 as e: 8/3 – (1+1/1!+1/2!+1/3!) = 0 < 1/7!; Am I missing something here?

49. National Equality says:

Hi sir I am from India . I like your video ..this is very interesting and memorable……………

50. Rich Graziano says:

Excellent Presentation. I wish you would do more on Fourier series and Fourier analysis

51. Tunya says:

18:10 full circle kehehe…

52. Michal Nemecek says:

smart dummies?

53. Nwxxz Chen says:

integral(sin^4t/cos^8t-4(sin^5t/cos^7t)+4(sin^6t/cos^6t))dt = u^4(1+u^2)du – 4u^5du + 4tan^6(t)dt, set u = sin(t)/cos(t) and finally I get 22/7 – pi

54. vishwan raja says:

Thanks you very much. I have been banging my head for math theories like this. 😳🙏🙏💓💕💖💖💗. Please recommend some books to know about facts like this . I will definitely follow them. Mainly to know about caluclus , trig derivation and complex, probability , binomial stuff .

55. imago says:

Ok so now i know how. But I still don't know really why. This is very annoying.

This is the first of very many videos which made me understand how tho! Superb! Thanks!

56. Luke H. says:

So…. It's ｅrrational?

57. Emanuele Cerri says:

Fantastic, simply fantastic

58. Tha Geo says:

CORRECT ME IF IM WRONG PLEASE:
e^πi=-1
e^π(-1)=1
1/e^π=1
e^π=1
From what he says e^x =1+x+x^2/2!….
So for x=π
0=π+π^2/2!+….
Am i wrong?

59. aguaviva says:

simply awesome!

60. Murilo Vidal says:

Starting at 14:04. During any basic calculus courses, the fact that e^x is its own derivative is simply shoved down our throats. But after seeing the true explanation for it, I can now consider my mind thoroughly blown, and I can rest in peace.

61. Cédric Wiederkehr says:

Wow amazing video!

62. Meh ! says:

Wow. Such lucid explanations.

63. Zach Cioe says:

His laugh at 16:06

64. Mr Castrillon says:

What a wonderful video. Thanks

65. コンヅクトル says:

all is clear, thank you

66. Monsieur Godzylla says:

il y a les nombre cyclique aussi, ça fait 543210123456789X98765 et une histoire de lapin qui se reproduisent.

67. Michael Golub says:

Really beautiful!

68. Pastor Hass says:

69. agnichatian says:

Beutiful

70. Los El John Smith says:

Going through school, I noticed, that mathematical concepts stick in your head better, when you connect them to their actual, real life, application(s).

All this fraction, (etc…), juggling is useless, if one does not understand/know where, how, and for what purpose IN REAL LIFE this, or that, concept is used.

Dear Mathologer, perhaps, you could include an extra minute, or two, in all of your videos, with good examples of where, and how, the concepts you're talking about in that particular video apply.

Without that, all your wonderful work is for nothing. Enters into one ear, leaves through the other, and the understanding of the need of this, or that, concept for real world use never sets in.

71. Suani Avila says:

smart dummies

72. Suani Avila says:

can someone explain tp me why the xx is in the numerator instead of being multiplied by the exponent?

73. Goryllo says:

At 12:01 I laughed so hard… Never in my life had I heard a less excited "tadah!"

74. Mid night says:

Great

75. Azizi Zizi says:

76. Inquisitour M29 says:

You are doing a great job man.. Bless you!

77. Witold Domeyko says:

11:23 why has the ? to be a positive natural number (why cannot it be a negative one ?)

78. OrochiShaka says:

at 5:26, why do those fractions turn to 1? If the number on the top gets smaller than the bottom one, shouldnt it go to 0 instead of 1?

79. Bengt Bengt says:

When I saw your shirt, I immediately wanted to prove the reslut. It turned out to be a lot easier than I thought 🙂

80. Ylann Doucy says:

I think there could be a misunderstanding between log and ln : shall we not use ln in place of log in this video?
Of course ln(e) =1, and these are two differents functions.

81. Old Lion says:

Primarily, via Mathologer, I've learned a lot about the concept of infinite sets. I like the proofs they can be employed to solve. However, doing deeper research, I'm finding that some professional Mathematicians don't believe in infinite sets. Can you clarify this battleground?

82. sanoy samuel says:

Exponential series is summation integration- differentiation infinite series.
https://www.quora.com/anonymous/35be6a660d6e4f1fa58ebea77b05575f

83. IMO Reviews says:

Thanks for the knowledge, aligning the proof as you did [ 17:29 ], helped a lot. Thanks

84. Bert Blankenstein says:

I was able to follow this one without any difficulty. Thank you.

85. thomas guitard says:

very nice It's an approximation. that explain maybe why logarithm is approximation of harmonic number. Thank you very much I often have trouble understanding the exponential and logarithm now I understand better, thank you.

86. Vazgen Ghazaryan says:

Is there a way of interpretation of e^(i*pi)=-1 where one can explain it's meaning by simplest terms like debt and loss and whatever, avoiding all the trigonometry, infinite series and general calculus identities which kind are bound to get to the same point in case you start with Cantor's theorem and Continuum hypothesis as a beginning for all? Hence, in reality, all these equations are various ways of representation of the same thing, but this "thing" is an absolutely fantastic and totally incomprehensible unity where two totally different irrational (infinite) numbers somehow interact with each other and one totally imaginary symbol so to say – in a very non-trivial way – to produce the most basic number we are absolutely sure about. So, again, is there a way of interpretation of e^(i*pi)=-1 by very simple everyday terms (like debt and loss and gain and time etc.)?

87. Colin Holloway says:

Good. You mean me.

88. gore buster says:

(e × cbrt e × 5th root of e × 7th root of e and so on till infinity)/(sqrt e × 4th root of e × 6th root of eand so on till infinity ) = 2

89. Peter Marsico says:

I've watched quite a few of the Mathologer series and they are consistently well done, … and without a doubt one of the best at explaining complex mathematical concepts. Wish that these had been around when I was in school.

90. Smooch Pooch says:

Man, your german accent is off the roof XD. Great video by the way.

91. legendhero 45 says:

Thank you for the video! All of you friends are super awesome! Oh, moments in this video are sad.

92. Ζήνων Ελεάτης says:

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, . The natural logarithm of x is generally written as ln x , loge x, or sometimes, if the base e is implicit, simply log x. Just saying…

93. Yash 2223 says:

Lovely way to show the polynomial expansion.

94. Shay David says:

I love this dude.

95. KingRelic36 says:

If the million plus one-th digit is going to push the millionth digit up, we would want another zero to guarantee that the millionth digit remains the same.

96. K A Midhunkumar says:

2010 jee integral problem on your t-shirt, curious to know whether this integral has any specialities

97. McAllister Pulswaithe says:

He makes the same exaggerated hand gestures as that guy on Svengoolie.

98. Gautam Kumar Shukla says:

very good explanation,