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Thank you for these explanations. I really enjoyed the video, it was so easy to understand. Thanks.

Very simple - 3 people // you see 2 blue hats: you got the red one // you see 2 red hats: you got the blue one// you see 2 diferent colors NOW IT IS TIME TO GUESS BECAUSE YOU GOT OR RED OR BLUE = 50/50

If you multiply 0.999... by 10^n, where n is the number of decimal places( number of 9s), the result is 'wait for it' 1/e.

Great video! My only question really is, what formula are you using for area of a triangle in Level 5? I understand why the sides are x(1-z) and y(1-z) and why you integrate, but why is the area xy(1-z)^2, the product of the two sides? Wouldn’t that be the area of a rectangle with those dimensions?

I enjoyed this video very much. The fact that one can find e in Pascal's triangle really blew my mind. There are so much hidden gems in this mathematical object, I feel like I discover more of them every time, from unexpected grounds. Thank you!

The problem as stated in the "Ask Marilyn" column is ambigous.

1 is the multiplicative identity. No matter how many times you multiply a number by 1 you do not change that number. No matter how many times you multiply 1 times 1, it will equal 1 or 1 raised to any power is 1. One way to deal with infinite digit numbers is to see if a number of finite steps leads to a LIMIT. We know that in base 2, the limit of the partial sums of the series represented by ".11..." is 1. The actual limit it is based on is Lim n->°°, 1/10^n=0. This is, of course, all in base 2, then ".11..."^°° should be 1, again in base 2. Let's take finite steps. .11×.11=.1001 .11111×.11111=.1111000001 .1111111111×.1111111111= .11111111100000000001 This does not look like I am approaching 1 in base 2. Well maybe if I ignore the 0,s??? Do that with ".99..." in decimal digits! Not only do you have to ignore over half the digits of 0, the final 1, and a 8 in the middle. Can ".99..." represent the number 1, of course, as anything can. ".99..." is the closest representation less than 1 in decimal digits. The problem is infinity. It is incomplete,inconsistent and imprecise.

The Monty Hall problem may seem complex, but it's actually quite straightforward. With three doors, your initial chances of winning are 33.3%. When one door is removed, you’re left with only two doors, which gives you a straightforward 50/50 choice. Regardless of differing opinions, the math remains the same. The only real uncertainty comes from the fact that, if this were a television show, the outcome might be manipulated for entertainment purposes.

The root of the problem is that the math assumes 0 x 0 = 0 which is obvious nonsense. 0 x 0 is literally no zero or anything but zero. Let's say 0 x 0 = 10. For example, apples. Then 10/0 is a question of how many non-apples, like pears, are in 10 apples. Obviously zero. It works.

".99..." could be considered equal to 1 (Why?) In the same way that the length of a line could be considered 1 when we can never measure its length exactly.

You should have had an explanation for 3 year olds for me. I’m actually smart and don’t get it. Let me ask this at least, please answer if you see this. My mind puts it this way. If you have 6 cookies and divide them amongst 0 friends why is the answer not 6? You have 6 of them and distributed them to no one. You have 6. At least tell me that is not that crazy to think. Thank you.😼

e^0 = ? you prove 0! using the same unproved statement e^0, if e^0 = 1 and 0! = 1 is just manipulation this is not proof.

Dr Sean you’re the best.

Your five equations are a circular algorithm where equations 1 and 5 should be the same. Why not? If you modify the algorithm slightly and using the same "infinite" arithmetic. x=".99..." 10x="9.99..." 10x+x="9.99..."+".99..." 11x="10.99..." x=".99..." Why not 1? I say infinite arithmetic is incomplete, inconsistent and imprecise. The same goes for infinity itself.

some guy said it's a letter, imma clap that guy's cheeks

Best explanation I have ever seen sir. Thanks for resolving my queries.

What about: x=0.999... 10x=9.999... 10x-x = 9.999... - 0.999... 9x=9.000... x=1

Bob: Hey Alice, want to play two envelopes? Alice: OK. How's it go? Bob: I got a $10 bill and a $20 bill. I put them each into separate identical envelopes like so, seal them and now shuffle them so we don't know which is which. Pick one and you get to keep it. Alice: Cool. Uh..I'll have that one. Bob: What's even cooler is you can switch to the other envelope. Alice: But why? If I have the $10 then I gain by switching. But if I have the $20 I lose. But I don't know what I got, so how can I know what I get? Bob: Look at it this way. Let's say you have x in your envelope. Then that means the other envelope has half x or double x. So you stand to gain double what you stand to lose if you switch. Makes sense. Alice: So if I have the $10 then the other envelope has $20 or $5. But you never said nothing about $5, I never saw you put it in an envelope. Bob: Don't get awkward on me Alice. Alice: And if I have the $20 then the other envelope has $40 or $10. But you never put $40 in an envelope. So we're back to what I - and you - said before. It's just comes down to either $10 or $20 and we don't know which envelope they're in. Bob: I just thought it might help to think about it algebraically. Alice: OK let's think about it algebraically then. If I have x then the other envelope has half x or double x. But I don't know I have x. It could be that I have the half x or double x envelope, in which case switching for x stands to lose me double what I stand to win. Once again, I don't know what I have so I don't know what I get. Thanks but no thanks Bob.

Please look: viXra:1905.0008 submitted on 2019-05-01 20:40:00, An Interpretation of the Identity $ 0.999999...... =1$

I think Dr Sean nails it by pointing out there's only a 50% chance you have x. The other 50% chance must surely be that you have 2x or x/2. The confusion normally sets in as soon as a typical presenter starts off with something like "Let's say you have x in your envelope" as if by stipulating that they put it beyond serious question, and it's now just a matter of 2 possible amounts in the other envelope, 2x or x/2. Of course looked at like that it makes sense to switch. But we rarely get someone saying "Let's say you have 2x or x/2 in your envelope", though that would make it disadvantageous to switch to one with x. (The same applies as much when numerical values are used instead of x, like starting off with "Let's say you have $20 in your envelope".)

I'm a bit curious as to what you'd think about something like wheel algebra, which tries to define 1/0 (which ends up being unsigned ∞) and, by extension, 0/0 and all other indeterminate forms (which ends up being a new term called the nullity, represented by ⊥). the entire concept of something like this just baffles me, my mind can't quite wrap my head around these things, but it's very interesting to think about (probably no practical use for it though)

So glad the algorithm let me come and learn something from you again before I take statistics

I have a small doubt The flag explanation for 0! =1 says, that there is 1 way to arrange 0 flags that is no flag, but then for 1!, shouldn't there be 2 ways, flag is there, flag is not there And so on, with 2 flags, it could be flag A+flag B Just flag A Just flag B And no flag at all If we introduce the possibility of there being bo flags, then doesn't it mess up the calculations for others

Zero and infinity Zero and infinity should not be treated as numbers. To approach zero is to decrease a number indefinitely. An asymptote of no matter how small a number can never reach nothing, which is represented by zero. Zero is a placeholder, an idea. Infinity, on the other hand is also an abstract idea. It is supposed to represent more than ‘everything’. It is an absurdity. Approaching infinity is increasing a number indefinitely, but such an asymptote can not and will not reach infinity. Infinity cannot be equated to anything as there is nothing as large as infinity, nor is there anything a small as zero if zero is to be equated to anything. Zero is zero, 0 = 0. and infinity is infinity, ∞ = ∞, no other equation makes sense using either. A number is a result or a record of counting. Zero cannot be counted (I would like to see a Shepperd count zero sheep), not to mention someone counting till infinity. As long as something is a number, it cannot be a zero or infinity. They cannot, therefore, possibly be numbers.

E

".99..." is extremely close to the value of 1 in the decimal place system of numbers. "Prove me wrong" that there is no other "number space" IN BASE 10 closer but less than 1. (I am defining "number space" to be represented by a bound infinite number of base (decimal) place notation and other sets of adjacent numbers.) [All algebraic "proofs" that ".99..." is 1 have problems in the use of infinite " number spaces" in arithmetic. (Not enough space in this comment to have a "prove me wrong ".) ] Infinite numbers are incomplete, inconsistent and imprecise. "Prove me wrong". The sum of a infinite series of >0 terms can't be unique and precise as a single number. "Prove me wrong ". There can be no number in base placed notation adjacent to 1. The Archimedean property says this. Prove Archimedes wrong.

Then comes Kurt godel🤣

It’s just a number used by people in the oil industry. Get it? Ha ha ha

f(x) = ax + b, watching the slope of the linear function while "a" approaches 0 and considering the fact that the inverse of f can be seen as a plot with the vertical and horizontal axis swapped. So, when "a" reaches 0, f is a straight horizontal line, while it's inverse (f') is a straight vertical line. So, f'(y) only exists for y=b, and the result is "all values between minus infinity and plus infinity", hence f'(y) is not a function when a=0 (a function must have a single result for each parameter value), hence its result is "undefined".

level 6: lie groups

Great video.

This video was really interesting, you have a rare talent of making abstract ideas accessible. Thank you again. As a developer, I have to say that the definition via matrices is the one I like best. Very Beautiful.

He didn't explain what it is though? Just pointed out some places it pops up. No insight offered here...

I study linear systems, geometry, and rotations and to me, the compounding version of e is the best characterization. Understanding how e relates discrete and continuous actions when the actions encoded as products makes understanding what e to the power of a matrix or an imaginary number or quaternion means.

did we invent e or did we discover it...

amazing explanation! 🙌

My smart counterpart, thank you.

Great channel, smarter than your average ur-sine🎉

well done

Almost all of the confusion I think comes from the notation. Once you frame it as being simply a function, which we could write generally as f(x), then I think far fewer people would question why f(0) = 1 as lots of functions have this property. I think lots of people jump to 0! feeling like it should be zero is just based on it’s similarity to 0+0 or 0x0

How do you divide 6 cookies by 0 friends? You don't. You eat them all yourself! I think I'll be doing this 0 division more often!

This is one of the cleanest explanations of Bayes rule I have ever seen.

If you continue the diving rule for (-1)! you would get (-1)!=(0!)/0=1/0=inf When you use the binomial formula to get pascal's triangle you divide by inf outside the triangle so it's outside 0 (which makes sense when you apply the "add rule" with 0 and 1 to get 1 on the side of the triangle) And when you use (-1)! in Taylor's formula to get the "more left" terms you would get something divided by inf which is again 0 so you add a bunch of 0's _(Math from Ohio)_ 💀

I was taught that e is the limit of (1+1/n)^n as n grows beyond all bounds ("approaches infinity"). The compound interest example is a great way to introduce e. Before learning calculus, limits must be treated informally and intuitively. It took mathematicians two hundred years between the initial development of calculus and the formalization of what a limit is. With hindsight, the epsilon-delta definition can be derived from the requirement that the limit be unique if it exists, by defining the condition that leaves behind all other candidates for the limit.

Very cool. Examples related to information theory and communications are also very clear.

I love math. Thanks to RUclips algorithm. Finally, I found this precious learning channel.

I explained this to a Banker... He laughed and laughed. Ya...serf... We invented creative math.

This is gold! It should be taught in every college!

It would have been nice to include a little history about how zero was not considered a number by the ancient Greeks and others. Also, you touched on the pedagogical aspect with little kids, but there's a whole lot more there. I once asked my granddaughter (then 5 years old) which weighed more: zero rocks or zero feathers. Of course she said rocks. I don't think I was entirely successful in explaining why they weighed the same (nothing).

What a great explanation! Thank you for creating such content.