The magical properties of "e" are endless. There is another surprising problem similar to the one above: Let N=n1+n2+n3+ .. + nk, where we are talking about the decomposition of N into arbitrary (integer) sums. The question is, how can N be decomposed into summables so that their product is maximal? The answer: the items to be added should be as close to "e" as possible, i.e. if possible, only 3's or maybe 2's should occur among the items to be added. For example, in the case of 10=3+3+3+1, the product= 27. The 1 does not make sense here, it is not an optimal solution. The optimum: 10= 3+3+2+2 The product=3*3*2*2 = 36 (In the range of real numbers, using exponentiation instead of multiplication, the maximum would be e^(10/e) = 39.5986..)
@s4br34 ай бұрын
I had this exact question in an oxford interview lol. I should have thought of the naturally exponential nature of e
@lyrimetacurl04 ай бұрын
(1 + (1/n))^n tends to e as n increases (was another one)
@hasanjakir3603 ай бұрын
Where can I get the proof?
@STEAMerBear4 ай бұрын
The slopes proof is much easier to teach. I recommend first teaching it with chalk or dry erase marker, then transitioning to GeoGebra where you can move A & B around so students can see how it works.
@MathVisualProofs4 ай бұрын
Yes. GeoGebra would be great for this.
@asparkdeity87174 ай бұрын
Wow, this is very similar/nearly identical to the comment I posted a while ago on e^pi vs pi^e; I recalled fixing the tangent line to log and rotating it clockwise/anticlockwise and considering the sign of the intercept, where you fixed the intercept and considered lines of different gradient (which is much easier haha); thanks for listening and making a great video that's so simple to follow! I also love how u kept the other cases open for viewers to try out, really engaging too. I copied and pasted my longer comment from before with the slightly different method for the sake of interest: with c = 0, m = 1/e gives one tangential intersection at x = e. Now, in the case c = 0, if we reduce m such that 0 < m < 1/e (visually “pivoting” the line clockwise about the origin), we will get two intersections points a,b where a < e < b. This means at a,b: the gradients of the straight line through ln(x) are equal: ==> (ln(a) - 0)/(a-0) = (ln(b) - 0)/(b-0) ==> a^b = b^a Now, same principle, but consider the case c > 0. Suppose two intersections with the straight line and ln(x) are labelled p and q, with q > p WLOG. Suppose p = e. Another lovely visual argument (pivoting the line clockwise about x = e by decreasing m) shows that we get two intersections for c > 0 with p = e and q > p, since we know there is only one intersection at e with c = 0. Again, we’ll run the same argument that the gradients at p,q must be equal: ==> (ln(p) - c)/(p-0) = (ln(q) - c)/(q-0) ==> qln(p) - pln(q) = qc - pc Now note in this case, c>0 and q>p ==> RHS > 0 ==> p^q > q^p -> Using another pivot argument and slightly adapting the one I did, try and show that p^q > q^p for any q > p >= e -> Now by considering the cases in which c < 0 yields two intersections, try and show that p^q < q^p for any 0 < p < q Notice there is a deadzone of values where it is impossible to determine which is bigger, in the case q >= e and p < e. A classic example of this 2^3 < 3^2 , 2^4 = 4^2 and 2^5 > 5^2.
@WhattheHectogon4 ай бұрын
0:57 doesn't lnx have a slope of 1/e at x=e?
@cupostuff99294 ай бұрын
That's what i was thinking
@MathVisualProofs4 ай бұрын
Yes. I misspoke. I meant the slope is 1/e and that’s why it passes through the origin. Thanks! I’ll note the mistake in the description.
@Kounomura4 ай бұрын
The function curve is magnified by a factor of e in the y direction so that the proportions are more visible. The function shown is actually y= e*ln(x). But this does not change the explained principles
@MathVisualProofs4 ай бұрын
I was able to kind of edit it on youtube and correct. Thank you for catching it!
@plamenpenchev2624 ай бұрын
Very nice video. Without doubt, visual presentation facilitates understanding but using only formulas also works.
@Achill1014 ай бұрын
Sorry, but I found the third proof the most instructive. Sure, it's more difficult to show x^(1/x) has only one maximum, but once we've got that, the cases for e=
@raulgalets3 ай бұрын
ABBA is the greatest of all time
@xjuhox3 ай бұрын
Let B = A+h, where h > 0. By the linear approximation ln(B) ~ ln(A) + h/A, we have B/A > ln(B)/ln(A) if A > e. That is, A^B > B^A if B > A > e 👀
@oida100004 ай бұрын
The first proof is the best but I prefer the 3rd over the second mathematically because integral.
@oniondesu96333 ай бұрын
don't think the second proof counts as a "visual proof" by any reasonable standard. first one is quite elegant and intuitive though
@MathVisualProofs3 ай бұрын
It is a published proof without words, so by some reasonable standard it is. I do agree it requires some decent background knowledge to make the jump without writing down the formulas, but it’s doable.
@Andy_Hendrix_98424 ай бұрын
ABBA 😁
@mikhailnikolaev49924 ай бұрын
ABBB💀
@Rippertear4 ай бұрын
@@mikhailnikolaev4992ABBC 😲
@Naej74 ай бұрын
ACDC
@StevenTorrey4 ай бұрын
I've always wondered whether there was a formula for solving that question, since it is a favored KZbin question.
@DOTvCROSS4 ай бұрын
Let A be a matrix (2x2)'s of trace(2) and zeros, let B be a matrix with trace(3) and zeros. |A|
@GenericInternetter3 ай бұрын
Doesn't really count as a "visual proof" when most of the proof is done using formulas.
@sonicwaveinfinitymiddwelle85553 ай бұрын
At first glance this question seems easy but once you look at it for a while it seems hard but when you actually try and solve it's actually pretty simple.
@deananderson77144 ай бұрын
To find the derivative in the third method you can also do this y=x^(1/x) => ln(y)=(1/x)ln(x) instead of taking both sides to the power of e, take the derivative of both sides (1/y)(dy/dx) = (-1/x^2)ln(x)+1/x^2 => (1/y)(dy/dx) = (1-ln(x))/x^2 => dy/dx = y(1-ln(x))/x^2 => dy/dx = (x^(1/x))(1-ln(x))/x^2 we can combine the powers of x to get dy/dx = (x^(1/x-2))(1-ln(x)) From here we can see this is only zero when 1-ln(x) is zero(as the other term is nonzero for all x) and we see that only happens when x=e (we have a point where the derivative doesn’t exist at x=0 but that point isn’t in the original function’s domain so it isn’t a critical point)
@FishSticker4 ай бұрын
Damn I really really like that first proof
@MathVisualProofs4 ай бұрын
😀😎
@rafiahamed29784 ай бұрын
🥰
@economicist20114 ай бұрын
As much as I love calculus, I love even more the ability to explain concepts like this one without it. Calculus is beautiful but it makes elementary topics like A^B B^A seem "gatekept" to people who haven't taken it. Good show!
@mathysvienne30322 ай бұрын
Can someone explain me how the fundamental theorem was used in the second visual proof ? I didn't get it
@beaumatthews64114 ай бұрын
2 < e! I was just teaching a kid about this yesterday
@Dalroc4 ай бұрын
The third is the least complicated.
@oinvestigard4 ай бұрын
ONGLY, if a is bigger than b oppocite oppocites that
@guessundheit64944 ай бұрын
Even with only knowledge of natural numbers, it's clear that if A>=3, there will never be a case where B^A is greater. And even if you know decimals, as long as A>=3 and B>A (e.g B=3.00000000...1), then A^B will be greater. But since 2 and 3 break that trend (2^3 < 3^2, 2^4 = 4^2), the point where it changes must be somewhere between. After fooling around with spreadsheets, it's interesting how exponentially large B becomes as A decreases (e.g. A=1.1, B=43.55774178) for A>B to remain greater.
@zsoltnagy56543 ай бұрын
The inverse function f^(-1) for f(x) = x^(1/x) is f^(-1)(x) = exp(-W(-ln(x))) for x∈[0, e^(1/e)]. So if 1 < A < e < B, then A^B > B^A, if and only if A^(1/A) > B^(1/B) and if and only if f^(-1)(A^(1/A)) > f^(-1)(B^(1/B)) as f^(-1) is a monotonically increasing function. So *if 1 < A < e < B, then A^B > B^A, if and only if A > exp(-W(-ln(B^(1/B)))) = exp(-W(-ln(B)/B)).* Otherwise it should be trivial, that *if A ≤ 1 < B, then A^B < B^A.* For example 2 < e < 3 and 2 < exp(-W(-ln(3)/3)) ≈ 2.48. So it's also, that 2^3 < 3^2 (which is also trivially true of course). So at the same time 2.5 < e < 3 and 2.5 > exp(-W(-ln(3)/3)) ≈ 2.48. So 2.5^3(≈15.63) > 3^2.5(≈15.59).
@sachavalette14373 ай бұрын
So instead of just seeing the result of a^b and b^a you must calculate 2 more complicated formulas with two receprical functions, that’s genius 💀💀💀
@zsoltnagy56543 ай бұрын
@@sachavalette1437 Well, alternatively you can also compare B to the following: A^B > B^A, _approximately_ if and only B > (e-1)²/exp(-W(-ln(A)/A))+1. With my previous example 3 < (e-1)²/(exp(-W(-ln(2)/2))-1)+1≈3.9525, such that 2^3 < 3^2.
@peterkron38614 ай бұрын
In the area proof, the critical step is applying the fundamental theorem of calculus. However, this handwaving “Applying the FToC we see that…” is totally unsatisfactory for what is offered as a “visual proof”.
@kennnnn3 ай бұрын
A^B B^A // log_A() both sides B A*log_A(B) this is simple comparison now 👍
@sundareshvenugopal65754 ай бұрын
If the sum of the numbers is a constant, their product is maximum only when the numbers are equal. To maximize the product N must be split into √N equal parts. This seemingly trivial fact, we choose to overlook, dismiss, or ignore, always has ginormous implications and consequences, not to mention applications, and uses. So it is with all things in life. The littlest of things always make the biggest of different. Every number always counts and all numbers always count.
@oinvestigard4 ай бұрын
if a isnt eaqual to b, b^a is the biggest.
@fabiansolisvalencia21583 ай бұрын
If, let's say, a = 3, b = 4, then, a^b > b^a. 3^4 > 4^3. 81 > 64. a ≠ b. a < b. ∀ (e ≤ a) ^ (a < b) ⇒ a^b > b^a. Read out loud, "For all values where e (clarification, e is Euler's number, a constant, which is weakly approximate [=~] to 3 and e < 3) is smaller or equal than a, and a is smaller than b, then a to the power of b is greater than b to the power of a, which I've already demonstrated (3^4 > 4^3). This holds true both ways. Given a = 1, b = 2, In which case, a^b < e, a^b < b^a. 1^2 < 2^1. 1 < 2. So, in conclusion, given that b is greater than a, and a is greater than 3, a^b is always greater than b^a. Unless we use negatives, in which case it gets... Complicated
@thenew3dworldfan4 ай бұрын
I have another question, if we change the domain from real numbers to Natural numbers, how do we solve is a^b
@agustinfranco03 ай бұрын
theres isnt that much "visual" here tbh
@sundareshvenugopal65754 ай бұрын
Most of them do not even seem close. If A is an n digit number then A^(B) is an n*B digit number, give or take a digit.
@pauselab55694 ай бұрын
It’s difficult to know whether something like e-1 to the e+1 is bigger than e+1 to the e-1 how do you get that instead of trying?
@Naej74 ай бұрын
And that’s why you need a long password, not a complex one
@Simpson178663 ай бұрын
There's an XKCD about that :D "correct horse battery staple"
@GreenMeansGOF4 ай бұрын
I have done the second challenge problem in the past. If 1
@BeaDSM4 ай бұрын
2^4 = 4^2
@_rd_50433 ай бұрын
2
@BeaDSM3 ай бұрын
@@_rd_5043 yeah exactly.
@dtikvxcdgjbv79754 ай бұрын
Obe must watch video at speed of 0.25 to catch all.
@nazrulhaque2604 ай бұрын
Thank you sir!!!❤
@yusufdenli93634 ай бұрын
Amazing... thanks.
@MathVisualProofs4 ай бұрын
👍😀
@MrEliseoD4 ай бұрын
What about in the case of 2^4 and 4^2? Kinda falls apart there…
@MathVisualProofs4 ай бұрын
Yes. That’s the ending challenge in the video. Figure out what to do when one is larger than e and one is smaller.
@mater59304 ай бұрын
Speaking as someone who only know elementary mathematics. The question in the title is simple but the explanation seems verbose. If you take a base 1 and raise it to the power of 1 billion, it will always be 1. But 1 billion to the power 1 is a billion time larger. My assumption is there are an infinite number of cases where the larger base is greater than a larger exponent. Similarly, there are an infinite number of cases where the opposite is true. The question cannot be answered. Unless we can determine infinity.
@JFxPress04 ай бұрын
Yes indeed, likely there are infinite solutions when the modal input is approaching limits ranges where A = [ 0, ±1, ∞, f(nⁿ) ] and B = [ 0, ±1, ∞, f(nⁿ) ] etc.
@eddtlpz4 ай бұрын
As you may have noticed, there's no way to determine the question in the title unless A and B are both less than e or greater than e, where the answer is, as shown in the video, the base closest to e is gonna be greater once raised to the power of the other number.
@beaverbuoy30113 ай бұрын
The question has been answered by breaking it into cases, as seen in the video
@shivpatel33113 ай бұрын
Just because 2 things are infinite in size does not mean there are equal amounts of both things. In math this concept is known as countable vs uncountable infinites. Fun fact there are more numbers between 0 and 1 than there are integers (even though both are infinite)
@JFxPress03 ай бұрын
Yes, yes indeed. Fun Fact: negative infinity is not equal to positive infinity -∞ ≠ +∞ Nor are certain continuums necessarily equal to certain "countable" ( limit approaching ) infinity groups or even unlimited infinity groups like integers... Fascinating how mathematics still has Undefined groups that are not even calculable according to modern theories. We need new terminology as addressed in statistics, stochastics, data science and AI to perhaps develop new improved concepts like 0 zero infinity or zero 0 continuums or null set or 1/0 x/0 undefined or undefined infinity or UI or AI or imaginary numbers or countable undefined or countable containable unidentified undefined function formulae ordination organization ( UFO) 👾 👽 ???!!!
@SHIVAPRASAD-hz4tj4 ай бұрын
Incredible that this was discovered 3 years ago!
@MathVisualProofs4 ай бұрын
The second proof only :)
@bilaalahmed89234 ай бұрын
which font do you use?
@MathVisualProofs4 ай бұрын
For what?
@bilaalahmed89234 ай бұрын
@@MathVisualProofs math. Your animations in your videos.
@MathVisualProofs4 ай бұрын
@@bilaalahmed8923 That is computer modern - the standard in LaTeX.
@lukatolstov55984 ай бұрын
3⁴>4³, BUT 2³
@MathVisualProofs4 ай бұрын
Yes. At the end I leave the most general question for you to consider. What can you say about A
@aldojerish13284 ай бұрын
Bro,if e
@kaaristotelancien30054 ай бұрын
why ma = 1/A * ln(A) ?
@MathVisualProofs4 ай бұрын
Compute the slope of a line passing through origin : it is the y-coordinate of another point divided by the x-coordinate (rise over run)
@Sans________________________964 ай бұрын
Fuck
@جوادعباسی-ر5د4 ай бұрын
So 2³ > 3² 😂
@vangrails4 ай бұрын
Yes, because 2 > e.
@vidal97473 ай бұрын
I just passed by to say that A and B are matrices. Use x and y like a reasonable human being.