Infinitesimals and Non Standard Analysis

This video intuitively explains infinitesimals and the basics of Non-Standard Analysis. There are simplifications of advanced mathematics, just beware. Thanks to DM Ashura (Bill Shillito) for his awesome music and his constant support for my love of math and science. He is truly an inspiration. Check out some of his videos by looking up Bill Shillito on KZbin! All animations and voices were done by me for the Junior Breakthrough Challenge. It took me many hours to create this video so I hope you enjoy and thanks for watching! For more info about me and what I do visit www.nocodeneeded.com/shaun

[@KungFuJoeMarino]

Even on the Hyperreals, the fact of the matter is that 0.999...=1. These two numbers are not separated by a non-zero difference, infinitesimal or otherwise.

[@analysis-sb4ri]

The _decimal representation_ of the integer 1 (to be more abstract, don't interpret 1 as the symbol denoting it, which is 1, more so, interpret 1 as the unit length for the real number line) justifies the representation of 1 as 0.999... Instead of suggesting a non-zero difference, I would recommend the suggestion that those two numbers are, in fact, exactly equal as far as we are concerned with their properties, where the only difference between them is a non-significant one, that of which is their representation. If we were to imagine representing 1 with an infinite sequence of nested intervals (a common approach), all intervals of rational endpoints such that: _9*(1/10) + 9*(1/10)^2 + ... + 9*(1/10)^n < number < 9*(1/10) + 9*(1/10)^2 + ... + 9*(1/10)^n + (1/10)^n_ For _n_ approaching infinity as was the stipulation, by a geometric sum for _n_ approaching infinity, we see that we obtain 1, which is not a surprise - *_this is merely a tautology._* . _"These two numbers are not separated by a non-zero difference..."_ There is no separation between those two "numbers", as we have shown, those two numbers are exactly equal, though not identical, as represented differently. Noting that, we have operated on the basis of base 10, and can do so with any base that is greater than base 2, or base 2 itself.

[@didles123]

While the 0.999... =1, the hyperreal number (0.9, 0.99, 0.999, ...), is not equal to 1 which can be denoted in the hyperreals as (1, 1, 1, ...). When subtracting the first from the second the result is (0.1, 0.01, 0.001, ...), which is an non-zero infinitesimal. When people see 0.999... it is likely they're thinking of it as a hyperreal such as (0.9, 0.99, 0.999, ...).

[@okoyoso]

How do you interpret 0.999...? If you interpret it as the limit of the sequence 0.9, 0.99, 0.999, then it is indeed equal to 1. This is the standard definition. However, there are other ways of interpreting it. Check out the wikipedia article on 0.999...

[@janszwyngel4820]

limits are defined in the real numbers, they can be defined using hyperreals, but they still apply only to real numbers. therefore it doesn't make sense to consider the number 0.999... as the limit of a sequence (0.9, 0.99; 0.99, ...). It does however make sense to consider that sequence itself to be the number 0.999... in hyperreals. if you define 0.999... as the sum for n from 1 to an unlimited hyperinteger (because there is no single infinity in hyperreals) of 9*10^-n, the number that you get will be a sequence of real numbers, none of which equal 1 (you can get the sequence (0.9, 0.99, ...) with the unlimited hyperinteger (0, 1, 2, ...)), therefore within hyperreals 0.999 does not equal 1. the sequence that you ges is infinitely close to 1, which allows you to infer that 0.999... is equal to 1, but only in the reals, not in hyperreals.

[@KungFuJoeMarino]

@@janszwyngel4820 This isn't quite right. The symbol "0.999..." implies a rational number whose decimal expansion has a "9" at every place in the sequence. That is, 0.999... is equal to the sum of 9*10^(-n) for all standard integers n. If we were to alter this and look at the sum of 9*10^(-N) for all Hyperintegers N, we would arrive at another number entirely and one which is certainly nonstandard. The Hyperreals are an extension of the Reals, so anything which is true on the Reals MUST also be true on the Hyperreals. Since it is the case that 0.999...=1 on the Reals, it must therefore also be the case that 0.999...=1 on the Hyperreals.

[@luckyluckydog123]

very nice video, informative, entertaining, professionally done... well done!!! However, I think your introduction about 0.99999... /= 1.00000... is wrong and has nothing to do with non-standard infinitesimal. It is well know that the decimal representation of real numbers have this 0.9999... vs 1.0000... ambiguity. Nevertheless those two writings represent the same *real* number. The hyperreal number 1-epsilon is something else, and is not represented by 0.9999....

[@griffinwhite9929]

Lucky dog you are right. The video, in general, is a good primer but .999... = 1. There several very simple proofs for this fact. They are the same real number. Even in nonstandard analysis there aren't hyperreals between two points a and b when a = b.

[@AaronHollander314]

When people doubt me that 0.9999... is the same and equal to 1, I ask them for a number between the two. It's like saying 2^3 is the same as and equal to 8. Or 3.141592... is the same as or equal to Pi. They represent the same number.

[@william41017]

@@AaronHollander314 what about (1 + 0.999...)/2?

[@AaronHollander314]

@@william41017 ... also 1

[@tamptus3479]

the meaning of 0.999... is the sequence: ( 0.9, 0.99, 0.999, ...) Reals ars sequences / 0-sequences Nonstandart Reals ars sequences / Ultrafilter in both cases: x = 1 - 0.999 = ( 0.1, 0.01, 0.001, ...) x interpretet as real is 0 because ( 0.1, 0.01, 0.001, ...) is a 0-sequences x interpretet as Nonstandart Real is > 0 because the set of indexes for which x(i) > 0 is member of our Ultrafilter 0.333 is 1/3 as real 0.333 + (x/3) = 1/3 as Nonstandart Real

[@ahmedmahmud4238]

This whole analysis is flawed.

[@eonasjohn]

Thank you for the video.

[@helldad4689]

You did an awesome job with this. There are a couple of inaccuracies, which is inevitable. When you're trying to familiarize viewers with alien concept, you have to work in familiar analogies, but usually you want to ignore the parts where the analogy breaks down. Also, just a technical grip, but you should normalize your audio. This was recorded in multiple sessions, and when it switches to the video of you speaking, the volume drops. Still, great work!

[@schizoframia4874]

Thats some thicc ink 0:00

[@robfielding8566]

The easy way to understand it is to not think about math. Think about a machine that has to represent a number. If you add two digits that are from 0 to 9... then it's possible to get 18. You can temporarily store 18 there, or you can carry later by storing an 8 and adding 1 to the number to the left of it. They are the same number. To say that there's an infinite number of digits to the right, you are actually saying that there is NO pair of digits that doesn't carry. Stated that way, there is nothing infinite involved. Or you can say that there are an infinite number of zeroes to the right. And you can make a machine to demonstrate the idea. Ex: 5427 = 5*10^3 + 4*10^2 + 2*10^1 + 7*10^0 = 5*10^3 + 4*10^2 + 0*10^1 + 27*10^0. The numbers are the same. But one digit overflows. So, there are two different representations for the SAME number. This doesn't even involve infinity. If a carry ripples to the left, because nothing to the right DOES carry, then you get this same effect of adding an "epsilon" value.

[@phrainkdrums5704]

You have to define a real number before you separate 0.999... and 1. If 0.999... is a real number and you replace it with the variable x, you can do simple calculations to see that it’s the same value as 1. If infinitesimals existed, then 1,000 would be an infinitesimal % closer to ∞ than 10. But it’s 0% closer. Here’s why: if you have unlimited golf balls and I take away 10, you still have unlimited. If you have unlimited golf balls and I take away 1,000, you still have unlimited. Therefore, (∞ - 1,000) = ∞ = (∞ - 10). Therefore 1,000 is 0% closer to ∞ than 10, not an infinitesimal %. Infinity is not approachable. That’s why it’s not a limit. The limit of real numbers DNE.

[@wdfusroy8463]

A great video on a perennially fascinating topic! I myself have always thought the easiest way to see that what 0.99999 ... represents is NOT = 1 is simply to graph the various stages of the .099999 progression in such a manner that each time one goes to the next "9" one increases the scale of the graph by 20. Thus if the graph of 0.9 has a gap of, say, 1 cm, between itself and 1, the next graph will have a gap of 2 cms, since the gap between 0.09 and 1 is only one tenth the size of the gap between 0.9 and 1 plotted in the previous step, but, since we are now magnifying with a scale of 20X larger than in the first step, the gap's representation on paper must double in size. Thus, despite nothing having actually changed with the decimals represented themselves, the manner in which they are presented, here as a series, DOES make a big difference. Hence, in plotting the third decimal stage, representing the gap between .09 and 1, the multiplication of the graph scale 20 times yet again will mean that the point 0.09 will need to be placed 4 cms short of 1 etc. Now what this type of sequential graphing shows, at least to me, is that the more 9s we add to the end of the number 0.99999999999 ..., the FARTHER away from 1 its representation will APPEAR to be. The more decimals of the number that is so graphed, the LARGER the gap depicting the remaining difference between the last "9" and 1 will grow, indeed it will grow at the exact rate of 2^n for n = 0,1,2,3 ... This means that if we actually had graphs for an infinite number of additional "9"s the gap would grow to be infinitely large, and it would be comprised of exactly every number between the original 0.99999999999999 ... repeating forever, and 1, i.e. an infinitely long string of infinitesimals! Now when viewed THIS way the idea that 0.9999999999 ... will eventually actually reach 1, even if it takes an infinite number of steps to do so, clearly must be abandoned, since it is obvious that no matter how far out one goes, the gap in question will still exist and keep growing exponentially. It only "appears" to disappear when one insists upon viewing each stage in the process at the same scale, in which case the gap involved appears, misleadingly, to be decreasing to a single point, "1"not increasing.

[@Chris_5318]

Decimals are quite unsuited to representing hyperreals. If you redefine 0.999... to be 1 - 1/10^H, then you have the problem that there is no decimal for 10 * 0.999... - 9. Very few mathematicians use that , actually abysmal, redefinition. Albert Lightstone invented an extended decimal called Lightstone notation to represent hyperreals. The abysmal redefinition would be written as 0.999... ; ...999 (with a last 9) amfd the original would be written as 0.999... ; ...999... with no last 9. None of the digit in 0.999... represent an infinitesimal. The n-th 9 represents 9/10^n, where n is a natural number, and so is a rational number, not an infinitesimal. 0.999... is a constant series. Its value is not approaching anything. Its value is being approached by the sequence 0.9, 0.99, 0.999, .... Its value is the limit (and least upper bound) of that sequence. So is 1. Hence [the value of] 0.999... = 1.

[@colinjava8447]

Great video, I just don't like infinitesimals at the moment, but maybe one day I will understand them better. I'm happy with infinities, and power sets etc, but infinitesimals don't feel "real" in the same way. I see why they can't lie on the real number line as there is no place to put them so that they are smaller than any positive real number. Is this hyperreal number line logically consistent? I suppose we can view them like we view the number i, so its a type of number with special properties that real numbers don't have. I don't get this bit at the end, surely 1/epsilon = 2/epsilon as 2*infinity = infinity whichever infinity is used (aleph0 or c etc etc). It all seems a bit flaky to me, I think of the integral of a function not as an infinite number of rectangles, but as the limit of the process of using more and more rectangles to obtain the area under the curve.

[@TheRevAlokSingh]

They are logically consistent if and only if standard mathematics is (lookup "conservative extension of zfc"). So if you believe in the real line, you may as well believe in the hyperreal one.

[@vtrandal]

At 1:38 into the video you are misspelling "infinitesimal" as "infinitesimal" (and missing an entire syllable in your pronunciation of the word).

[@BillShillito]

Wishing you luck, Shaun! :) Great job on this!

[@ankitthakurankit4764]

2:48 no it is not curcuference as you are concidering the gaps left behind thus it is not that base =circumference if it does then you are simply assuming that a circle is made of triangles

[@AdlerMow]

1/3 doesn't equal 0.333... I looks so in base 10. In base 12, 1/3 equals 0.4. In any base there are recurring numbers, and for convenience we call the approximations the same as a fraction, like when dealing with tolerance and margin of error that limit how precise we choose to be. So an infinitesimal is not a number, but a process of iterative (non finite) division, and perfectly valid at that. In applied math, in engennering or statistics, however, we must "limit" the precision due to practical concerns. No problem with that, too. But 1/3 doesn't equals 0.333... for sure!

[@Chris_5318]

Rubbish. 0.333... = 1/3 exactly. In 1/3 = 0.4 (base 12) = 0.3BBB... (base 12) where B is the eleventh digit. If b is a natural number, then 1 = 0.bbb... (base b+1) Infinitesimals are numbers and they are not processes. The topic is pure math and it has nothing to do with engineering. Where on Earth did you get so many completely incorrect ideas from? I suggest that you look up the "geometric series" Wiki or any math textbook that deals with infinite sequences and series, ''cos you obviously have never seen one.

[@annaclarafenyo8185]

Abraham Robinson DID NOT USE ZORN'S LEMMA to construct the hyperreals! He simply added epsilon to any axiomatic description of the real numbers by adjoining the infinite list of axioms "epsilon>0" "epsilon < 1/2" "epsilon < 1/3" "epsilon

[@Chris_5318]

Thank you, that was very informative. It also helped me to resolve something that had been troubling me.

[@ronmaessen2021]

I don't get it. What I understand is that Robinson made a hyperreal number line that had infinitesimals on them and that he then said that the laws governing the real number line also govern the hyperreal number line. But what is explanation for how infinitesimals can be put on a line and what is the proof for hyperreal numbers being governed by the same laws as real numbers?

[@MeshremMath]

They are not infinity. They are omega and 2omega respectively. Greater than all real numbers, however they are not infinite. Just as epsilon is smaller than all real numbers, but still not 0.

[@Chris-5318]

In standard analysis, epsilon is a real > 0 and is not an infinitesimal.

[@Melki]

If infinitesimal exists, zero exists, then you could and imo should tell apart between unfathomably large number and limitlessness. Infinity can't refer to both

[@Springwight]

Oh man the dark side of KZbin math is very dark.

[@akrammohamed8374]

You're really talented you should do more videos bruh !!!!!!!

[@shaunregenbaum]

Akram Mohamed I am thinking about it! Thank you!

[@akrammohamed8374]

Shaun Regenbaum good luck 👍🏽

[@AbuSayed-er9vs]

Excellent!! Please make another video on this intuitive calculus understanding with the help of infinitesimal but not Cauchy-limit concept.And Please tell me where can I find more on this approach.

[@black_jack_meghav]

This is not a reliable source. Go for a nice book on nonstanard analysis like the one by abraham

[@rrr00bb1]

N = M - 9 M = 9 + (1/10)M You can solve for N=1. And you can also expand N to 0.999... for as many digits as you like. There is no notion of infinity used at all when you do it this way. It's straight-forward substitution into recursive formulas. This is the exact same phenomenon that causes S=-1 to expand like S = 1 + 1/2 +1/4 + 1/8 +... + 1/16 + 1/32 S. The trick is that n isn't infinite. The left and right are valid at EVERY step. -1=1+2(-1) = 1 + 2 + 4(-1) = 1 + 2 + 4 + 8(-1), etc. The pattern is S = Sum(S) + Tail(S). The magic here is that for a recursively defined series is that S is not the sum. Sum(S) = S - Tail(S). That's very non-intuitive! You subtract the tail from S to get the closed form as n expansions are done. S is the runaway recursion. Because of this, you can algebraically solve for 1 = 0.9999... while completely rejecting the concept of infinity. 1 = -9 + M = -9 + (9 + 1/10 M) = -9 + 9 + (1/10 M) = 1/10 M = 1/10 (9 + 1/10 M) = 0.9 + 0.01 M = 0.9 + 0.09 + 0.001 M = 0.9 + 0.09 + 0.009 + 0.0001 M Note the M in the tail. You cannot throw that away. That is the recursion that lets you do substitutions with no concept of infinity.

[@Chris_5318]

If anyone asked me to define what "balderdash" is, I'd refer them to your comment.

[@rrr00bb1]

it's about recursion. it's used a lot more in computer science than in typical mathematics. you can't iterate an infinite number of times, so computationally "+ ..." has more problems than recursion does. recursion is superior for explaining these. kzbin.info/www/bejne/mKjFgat9ptppqNU @@Chris_5318

[@Chris_5318]

@@rrr00bb1 Let me put it another way, 0.999... has infinitely many 9s, and you didn't so much as mention 0.999... in your recursion manipulations. All you can achieve is 1 = 0.9 + 0.1 = 0.99 + 0.01 = 0.999 + 0.001 = ... but you will never get to 0.999... + 0 as is required. For that, you have to take the limit. On which I note that body of the relevant epsilon-N definition of limit does not mention infinity. Far better than recursions, that don't really shed any light on the value of 0.999... is: 10 * 0.999... = 9.999... = 9 + 0.999... => 9 * 0.999... = 9 => 0.999... = 1 Whatever, it is possible to make the abysmal redefinition of 0.999... to be 1 - 1/10^H. I say it is abysmal because decimals can't really represent hyperreals. For example, there would be no decimal for 10 * 0.999... - 9 with that abysmal redefintion.

[@rrr00bb1]

@@Chris_5318 but, do the substitutons. literlly; with copy/paste s = 0.9 + 0.1 * s = 0.9 + 0.1 * (0.9 + 0.1 * s) = 0.9 + 0.1 * (0.9 + 0.1 * (0.9 + 0.1 * s)) note that you dont have to expand it an 'INFINITE' number of times, because 's' is pasted into the expression on the right at every step. it is not a coincidence that you can also solve for the number. this is why there is no notion of convergence. it's a notion of computation; like on a computer (s = 0.9 + 0.1 * s) * 10 (10 * s = 9 + s) - s (9 * s = 9) /9 (s = 9/9) 1 it is not a coincidence that it both simply solves for the number, and also expands into a sequence of digits. it's only a consequence of the definition of equals.

[@Chris_5318]

@@rrr00bb1 Once again you haven't mentioned 0.999... anywhere. You've just made a convoluted variation of the proof that I gave. Here's an improved version of what you did: 0.999... = 0.9 + 0.0999... = 0.9 + 0.1 * 0.999... (1 - 0.1) * 0.999... = 0.9 0.999... = 0.9 / (1 - 0.9) = 1 I don't understand why you are so keen to not mention 0.999... when you are trying to prove that 0.999... = 1, and why you bothered recursing after the first line. You only avoided mentioning infinity because you didn't mention 0.999..., the infinite decimal that you were thinking about. The only reason that these proofs work is because 0.999... is a convergent series. For instance, consider the divergent series ...999. Then 10 * ...999 = ...9990 = ...9990 + 9 - 9 = ...999 - 9 9 * ...999 = -9 ...999 = -1 and that is absurd. (Amazingly there are two ways of making that sensible, but you should treat that as merely luck.)

[@thomassaurus]

The teachers aren't wrong, just because an infinite decimal can be bigger than another infinite decimal, doesn't mean an infinity can be bigger than another infinity. Infinitesimals are technically quantifiable even though they are impossible to write. For example imagine the number 1.111111111... etc. We know that this number is larger than one but no matter how many ones you add to the number it will never be larger than two. However 2.22222222... is bigger than two and thus will always be bigger than 1.11111111... Infinity however cant be quantified. Imagine the number 1111111111... etc. Now imagine the number 2222222222... which number is bigger? If you add another 1 to the first number suddenly the first number is bigger than the second number. But when you add another 2 to the second number then suddenly that number is bigger again. If either of these numbers were longer than the other than that number would be bigger, but neither are longer because both are infinite, thus neither of the numbers are bigger than the other... that is infinity for you. I'm not sure if I'm using the word "quantifiable" right but I think it gets the point across anyway.

[@shaunregenbaum]

You are correct when using standard analysis, the normal way we look at things, but in non-standard analysis one infinity could be bigger than another infinity depending on the end behavior of that infinity which you must define. That is why this is called non-standard analysis as it is non-standard. Also infinity is standard analysis is not quantifiable or else you could compare them, but it can be countable, or have cardinality, however you want to say it.

[@thomassaurus]

Ok, looking into this more I'm starting to understand. But I think it's unfair to say that someone is wrong when they say one infinity can't be the larger than another without pointing out that you are using an alternate and apparently controversial way of thinking about infinity. And I realize you sort of did considering that the video is about non standard analysis but that is obvious to the regular viewer and it defiantly wasn't to me until just now. Besides this complaint though, great video! It helped me get an understanding on all this.

[@shaunregenbaum]

No problem glad you enjoyed! Non-standard analysis is used in the philosophy and explanation of calculus. If you have ever seen the dx in differential equations, you have seen an infinitesimal. Non-standard analysis was needed to prove infinitesimals, otherwise we would be stuck with limits and nothing else for calculus.

[@vuza752]

+thomassaurus there are actually different cardinalities of infinities, which is proven by Gödel's incompleteness theorems. For instance, they're are infinitely more reals than integers; the intuition being that you can fit an infinite amount of integers between two reals. - The distinction here in layman's terms is the distinction between countable and uncountable infinity.

[@thewolverine7516]

​@@vuza752I think you meant to say an infinite amount of reals between two integers.

[@zapazap]

Assuming the Axiom of Choice is one hell of an assumption.

[@antares2804]

Be chad assume Axiom of Choice everywhere

[@frankdominick2546]

Set 0.999 = x. Then multiply by 10. Subtract the original equation from the latter. Then x = 1. You would have to define a real number before saying that 0.999... is different from 1.

[@AbhishekSingh-zc8rw]

good one

[@fordmeisef9661]

Clear and fast video. Thank you. Infinity is a concept, not a number. Therefore, there is only one magnitude of infinity by definition.

[@Errenium]

eh, not precisely. there are multiple non-finite cardinals and ordinals representing distinct quantities or ideas

[@fordmeisef9661]

@@Errenium Thank you for correcting me. I have learned something. Can you explain complex infinity to the layman? Thank you.

[@Chris_5318]

All [pure] math is about concepts. It is abstract. What possessed you to say "therefore"?

[@analysis-sb4ri]

To be clear, in order to justify the hyperreals one would usually discard the Postulate of Eudoxus (or the Axiom of Archimedes) that ensures that the real number system is unique for which satisfies the standard sets of axioms that you might find from your typical book concerning itself with real numbers as the basis for real analysis. The hyperreal numbers specifically *_violate_* the Postulate of Eudoxus, which, from my perspective, unnecessarily so. Perhaps it was done because certain mathematicians always had a bias for the concept of the infinitesimal. Regardless, number systems are arbitrary in the sense that different mathematicians wish to satisfy different sets of axioms, and desires. As long as they are useful, then I see no problem with arbitrariness. Of course, the real number system for which standard mathematical analysts operate with is *_extremely useful._*

[@c.t.d.r.a.8295]

Niceness

[@megatrumper4512]

unlike my math teacher this guy speaks in English that all of us can understand

[@perpetualrabbit]

0.9999... = 1.000... Why? say X=0.9999.... Then 10X=9.9999.... So 9X = 10X - X = 9.999... - 0.99999 = 9. Therefore X=1 exactly, and 0.99999... and 1.00000... are just two different representations of the number 1. These representations are NOT an infinitesimal amount apart, but are exactly equal. Question: Internal Set Theory has a concept called Ilimited numbers, which are smaller than infinite yet bigger than any standard natural number, and infinitesimals can be viewed as their reciprocals, yielding numbers that are >0 yet smaller than any Real number. There has work been done on decimal representations of infinite numbers that are unending on the _left_ side, i.e. something like ...947690569751 (Or even both sides, like ...947690569751.0968613636... ). I wonder if those could taken as representations of those ilimited numbers, and the reciprocals being infinitesimals? If so, and if ALL ilimited number could be written that way it would mean there are as many of them as the reals, as in the same cardinality. If not, what IS the cardinality of the set of nonstandard numbers? And are they well-ordered? Can you compare two left-unending decimal representations? Which one is bigger? You cannot, right, because they never end on the left side so you can never compare them. Even when one such number would be all 9's and the other all 1's. A finite sequence of 1's is a bigger number than a finite sequence of 9's if it is one digit longer, so how would you compare two infinite sequences? Maybe by comparing their reciprocals somehow you could create an ordering?

[@analysis-sb4ri]

_"These representations are NOT an infinitesimal amount apart, but are exactly equal."_ Correct.

[@eldersprig]

Fluxions, fluxions you're the man! If you can't do it then no one can! Now read the chapter on limits in the appendix of the book.

[@peanut12345]

Euclid's Greek number line was Infinitesimal in 300bc. So Robinson steals for Ancient Greek Philosophers.

[@MisterrLi]

It was actually Edwin Hewitt that came up with the hyperreal numbers 1948, also he used the ultrapower construction of them attributed to Abraham Robinson in this video.

[@KFungMath]

very interesting video, may i know what software did use to make the video?

[@shaunregenbaum]

+K FUNG Everyday Math camtasia and some libraries from videoscribe in addition to blender

[@KFungMath]

+Shaun Regenbaum Thank you very much, look forward to watch your new video

[@shaunregenbaum]

+K FUNG Everyday Math I am tossing around new ideas, do you have any suggestions or topics that you would like to hear more about. It can be physics or math related, I actually know quite a lot more about quantum mechanics than advanced mathematical concepts.

[@magne14527]

Yeah... I get it but why do you claim the circumference to be 2.Pi.R when you already know that you're substracting an infinitesimal number when aproximating to a triangle? Thats why it fits the formula of the area of a circle. I know its a very normal thing to assume that infinitesimal numbers doesnt count, and I know it works most of the time, if not all the time. It just bothers me. Btw nice video. Keep it up.

[@shaunregenbaum]

+Magniverse That is the one problem that does arise from them, the "not intuitive" side. That is why i mentioned it, I wanted to show that no mathematical concept is foolproof. Also, this particular problem of turning infinite straight lines into a curve is a hot topic in mathematics, topology goes into it a lot more and tries to answer those questions, but truthfully there is no perfect answer, some things are just strange.

[@Creuilcreuil]

"Robi: Hey look Average Mathematician my shiny new math" "Average Mathematician: cool what it's all 'bout" "Robi: i brought back the concept of infinitesimal and it fix of a lot of things" "Average Mathematician: Naaa i prefer to say that 0.999... is equal to 1" "Robi: but... it doesn't make sense, 0.999 and 1 are different and i worked years on this...." "Average Mathematician: too bad, its gone now (you thought i'm gunna learn this new maths, which would mean i'm not a all knowing math god, i just say it's garbage and have to keep my prestige (mwahahhaha))" i swear it appear that each time through history it's the same thing again, again and again 1. make a in 2. people who don't want to loose their status on try to shut down 3. they succeed and this only manage to slow down the discoveries in 4. sometime pass and a new generation see the genius in 's 5. the cycle repeat

[@PichaelMoon]

entering again this year?

[@MathProofsable]

If .9 repeating is not equal to 1. Does that mean the decimal expansion of 1/3 is not equal to .3 repeating in nonstandard analysis?

[@frfrchopin]

The fact is that the decimal representation cannot just eliminate the remainders that will always be there, (10 - 3 * 3)^ω exist in the infinitesimals, but since no such thing exists in the real number line, such difference is just an equality.

[@berndhutschenreuther8342]

It means that the real part is equal according to the transfer principle. It does not mean that the hyperreal part is zero.

[@metatron4890]

Can you make more videos on this topic? Please?

[@unknownshell7673]

According to Richard Dedekinds definition of the real number system, any non-zero infinitesimals are non existent.

[@shaunregenbaum]

That is why we had to make a new set of numbers called the hyper-real numbers, which are outside the span of the real numbers.

[@unknownshell7673]

@@shaunregenbaum yeah I didn't watch the whole thing, sorry lol, hyper real system isn't widely used though.

[@dannyw8224]

Great stuff, makes alot of sense

[@shaunregenbaum]

+Danny W Thanks so much!

[@schizoframia4874]

Matheticians: infinitesimals work Also: dividing by zero is too hard

[@bjdthethreecoolmathfolk2528]

I love this this should have more views

[@RamanStudios]

Hats off to you...a very neat and good video

[@Ngtdggvkhij]

great!

[@shaunregenbaum]

+Chiarr Min Thanks!

[@tomhite3510]

Awesome video. Thanks.

[@milocarteret8770]

Eh, I don't agree with the idea that epsilon divided by two is a thing. I prefer to consider infinitesimals as also being indivisibles.

[@KungFuJoeMarino]

Hyperreal Infinitesimals are not indivisible. They can be divided potentially infinitely, just like any Real number. Indivisibles are an entirely different concept, and are not numbers.

[@googlr214]

I hope to attend your lectures, Doc!

[@AndrewMSmalley]

Excellent job.

[@tsukimineko]

finally found a way to explain some questions since learning Calculus. thanks a lot!

[@shaunregenbaum]

Your welcome!

[@willtaylor-melanson3014]

Incredible job on the video!

[@brunoamezcua3112]

incredibly good video, really well explained

[@3-to-the-power-of-422]

Great work dude keep it going

[@jandonnermayer6708]

"wierd" xD

[@Chris-5318]

First, "infinitesimal" is pronounced as "infin ee tes ee mal", not "infini tez mal". "Math e mAt ician" is not spelled "math mEt ician" You don't seem to understand the difference between a number and a numeral. Definition: The sum of a series is the limit of the sequence of its partial sums, if the limit exists. So the sum of 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... whose nth term is 0.999...9 (n 9s) = 1 - 1/10^n. Hence: 0.999... := lim n->oo 0.999...9 (n 9s0 = lim n->oo 1 - 1/10^n = 1. No infinitesimals are in sight. If 0.999... = 1 - ε, then what is 10 * 0.999... - 9as a decimal, and what is 1 - 0.999... as a decimal? In both cases - neither has a decimal representation. You claimed that infinitesimals are used in the majority of calculus - that is not true. Standard calculus is limit based. NSA is an advanced topic and many degreed mathematicians do not know it, yet alone use it. Your triangle-ised circle is NOT a circle, it is an apeirogon (in the limiting case). Circles do not have vertices. The area of your polygon was less than the area of the circle. Mathematicians will have a hard time agreeing with what you said because what you said is wrong. You didn't show the details for using NSA to do integral calculus. Do you even know them yourself? You should take this video down because it is unhelpful and misinforming.

[@oskarjankowski5709]

The way I feel about theoretical mathematics is that it's people making up new definitions and using over complicated methods and getting lost in other methods and laws to prove some very simple concepts. Honestly, I suppose it can be fun and all but far from smart or intelligent. Taking something most understand at face value and deciding that it needs a complicated mathematical proof. Obviously if someone enjoys theoretical mathematics, then that's cool, but I just feel like people look at these mathematical proofs and take them for the ultimate truth. I think what triggered me was the shot arrow example - please don't try to understand the real world with theoretical mathematics. Somethings just are, we don't need a proof for how that arrow travels. Honestly the arrow example just contradicts itself. We can divide time over and over into smaller increments but let's just focus on the real physical world. The arrow is moving from position A to position B and we can keep dividing the time scale from the time it leaves A to the point where it gets to B over and over but the arrow doesn't care about the concept of time that we have to measure such things. Yes the concept of time is very useful and we couldn't do a lot of important things without it, but it is just a concept. But interactions that happen in the real world between physically existing instances have no concept of time... They just are and they just happen. I guess my point here really is that theoretical maths can be fun on paper, like a strategy game or a puzzle, but it mostly has not real world application (hence the 'theoretical') so don't apply it in the real world. (I say mostly but really I have not clue if any of it does but I suppose if it does it stops being theoretical but feel free to correct me. Like I suppose a concept could be being developed in theoretical mathematics and then we realise it has a use in the real world. But I can say with absolute certainty that this one doesn't. Although, I would love for somebody to prove me wrong so fire away below.)

[@Lucky10279]

Oskar Jankowski You said it yourself - what starts out as pure theory may have important real world applications later, foreseen or unforeseen. I agree that sometimes mathematicians make things overly complicated and too often math isn't taught in an intuitive way at all. It can seem so abstract, when it actually represents things in the real world. And sometimes certain concepts can be helpful in problem solving, even if they don't appear to actually correspond to reality. Imaginary numbers, for example, were at first seen as a mere mathematical convenience invented to solve problems that require the square root of a negative number. It was later discovered that these numbers aren't any more imaginary than negative numbers are. I'm not an expert on any of this, but my main point is this: keep an open mind. What may seem like pointlessly overthinking simple things may have more to it than you realize and might have unforeseen benefits.

[@Lucky10279]

Oskar Jankowski Also, sometimes our intuition is very wrong, so using logic (of which I see math as a subset) to prove or disprove our intuitions can be very helpful. You might be interested in PBS Idea Channel's video on whether math is actually real.

[@TehJumpingJawa]

Presumably that means irrational numbers (such as pi) have an infinitesimal component to them? Thus it is an error to place them on the real number line.

[@shaunregenbaum]

+TehJumpingJawa Not Necessarily, Pi and other irrational numbers only contain real numbers, they do go on forever, but there is a major difference between irrational numbers and infinitesimal numbers. The more digits you list of pi the more accurate you get when using it or talking about it, but when using or referring to infinitesimals you do not get more accurate. This is one reason why irrational numbers are not hypereal, but there are dozens of other reasons.

[@TehJumpingJawa]

+Shaun Regenbaum My thought process to come to that conclusion was: The approximation of the circle's area is given by infinitely many real triangle segments. The actual area of the circle is infinitely many real triangle segments plus an infinitesimal for each.

[@shaunregenbaum]

+TehJumpingJawa Your intuition is correct and that is the best way to think about it, but that thinking only works in the hyperreal number line, not in the real number line by itself. In the real number line there is no difference between an infinitesimal and 0, therefore you ignore the space under the triangles, but when you enter the hyperreal number line you give substance to that space. To answer your question above a little better: when you are in the hyperreal number line irrational numbers are hyperreal, but when you revert back to the real number line by itself they are no longer hyperreal, just irrational. In all math today we can simply assume irrational numbers fit on the real number line as we can use them to a very accurate degree, but if you wish to think and use pi, the never ending number, it is no longer "fully" real and does have some infinitesimal component to it distinguishing it from any fraction or mathematical value we give it. Despite this, it is not a mistake to put it on the real number line as long as we accept that we are using an imprecise version of the real number which works splendidly in the real world and in any applicable math.

[@lucasrafaeldeandrade7516]

The problem is that: 1/3=0.3333 is just a approximation, is not correct