Woah. Im a 16 year old student from Germany and I actually understood everyword coming out of your mouth. This video is a great example on how real mathematics work, I love it!!!
Thanks for the great explanation, this is the first example I've seen of an infinitesimal being used in an intuitive rather than a hand-wavy-way (calculus always felt incomplete to me because of the lack of time spent on the infinitesimal). If you know more examples with infinitesimals being used in interesting (clear non-hand-wavy) ways it would be cool to see!
This is a good video and nice explanation. Just one issue, the definition of an infinitesimal ε you provided is incorrect w.r.t how it's standardly defined. You gave the definition for a *positive infinitesimal*. The broader definition of ε is so that -a < ε < a, for all 'a is in the reals'. If you want to write it your way, you'd want it to be 0 < |ε| < r for all positive reals. Either one of these is fine, because they include negative infinitesimals (which are also infinitely close to 0, just like positive infinitesimals). Other than that, good video.
What is the surface area of a sphere with radius infinitesimal? If there is a surface then must this surface inherently have infinite points on it? Our manifold, a minimal single sided closed surface Sin(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2) 0
Very nice. I am familiar with the Extended Real Numbers (Reals with +/- Infinity). It makes sense to also include an infinitesimal that is, as you say, smaller than everything else. I'm also learning about Dual Numbers with uses Infinitesimals for a very famous application, Autodifferention, for training (optimizing) a Neural Network. I am happy so many things are converging in my understanding of mathematics for (cringe) Machine Learning.
Great video! Battery current (both charging and powering a load) needs to be integrated correctly by both hardware circuitry and code using calculations that accurately reflect what is going on. Any discussion or approaches concerning current measurement total-izing would be appreciated. I've been wondering if a current measuring IC sensor voltage output could be simply integrated by a capacitor to determine the total amount of charge current going into a battery. So far, I get too much V across the cap!!
Ohh fun!! Sounds like an interesting problem! Whats the goal of measuring the current? You could brute force it if your circuit isn't too complex, and then there's always my fav tool: the multimeter! Generally speaking, the current in your circuit will equal whatever current source(s) are there (e.g. battery, power supply, etc.). Current is conserved (altho some gets lost as heat), so whatever is going into your circuit has to go somewhere! But yea, some context on the specific problem you're trying to solve would help narrow down a potential solution or approach!
@@JenFoxBot I am designing a 4 channel battery analyzer. The "easy" way is to use an IC that measures current (like the Allegro MicroSystems ACS70331EE). That measurement would be taken every, say, 100mSecs and totalized. That totalized number would then be converted to milliAmpHrs. So the "crazy" idea is coming up with some time of OpAmp/capacitor integrator that would give the same number, but without firmware. Then you could use a simple DVM, measure the volts out, and convert to mAHr. Lot's of used 18650 batteries are coming to market, (like from Battery Hookup) and it would be cool to come up with such a cheap device so people could safely re-purpose those batteries. PS:Wish you had of been my math teacher when I was in college. I barely made it through yr one calculus.
i ran into this video (which I think I have seen before), when I was contemplating an issue with limits. - an infinitesimal is a thing that squares to 0, but is greater than 0. ie: dx*dx=0, dx>0. - a number that is merely "small" still squares to greater than zero: smallx*smallx>0, smallx>0. that's why, this is just fine: dy/dx. dx>0, dx*dx=0. if y=x^2, then dy=2x dx. so dy^2=0 as well, because 2*2*x*x*dx*dx=0. i don't understand the total avoidance of using actual algebraic infinitesimals. when doing geometry, you end up dealing with things that square to -1, 0, and 1. in deep neural networks, it is common to use autodiff libraries, which feature infinitesimals to calculate giant derivatives with infinitesimals.
@@JenFoxBot actually, i intentionally mean that an infinitesimal called dx is infinitesimal BECAUSE it is a square root of zero: dx*dx=0, and it's not zero because dx>0 (for a positive infinitesimal). it's a super-important distinction from having something "small" but finite. in that case if small>0, then small*small>0; as it's a finite real number. 0 < dx < small A square root of zero that is not zero is the definition of an infinitesimal; and it's not a real number. It's no different from the objects that square to -1, which are not real numbers. And there are non-real objects that square to 1 as well. Geometric Algebra is all about objects that square to -1, 0, and 1; to produce all of geometry. You effectively add directions in space into the algebra; and it makes multiplication non-commutative. The thing that makes limits waffly is when you say "arbitrarily small". By that definition, you can't wrangle that definition as algebra. And that means that turning it into computer code is waffly and inconsistent. It's exactly why some people won't accept 0^0=1, because "arbitrarily small" isn't nailed down enough. If you are contemplating 0^small, then small*small>0 means that 0^small=0. But 0^dx where dx*dx=0 means that 0^dx=1. Limits are a way to waffle around not getting the algebra right.
unless you take the equals operator to mean "has the same real value", you can't say that "x + dx = x". the equals operator doesn't mean that. it means "can be substituted with, in both directions". y = (x+3)^2 d[y = (x+3)^2] dy = 2(x+3)d[x+3] = 2(x+3)(dx + d3] = 2(x+3)(dx + 0) = 2(x+3)dx it might be the case that only dx*dx -> 0. But not 0 -> dx*dx; which is weaker than "dx*dx=0". That might be a subtle point about infinitesimals. I am not sure if you can just factor zero into any square root of zero. ie: dy = 2(x+3)dx dy/dx = 2(x+3) dy = 2(x+3)dx + dx*dx //can we jadd dx*dx? dy/dx = 2(x+3) + dx But for sure, instead of dividing by zero when trying to set dx=0, you can divide individually by dx twice instead. // second derivative of x^3 // assume x is a line: d^2x=0 d[d[x^3]/dx]/dx = d[3 x^2]/dx = 3(2 x dx)/dx = 6x
@@rrr00bb1 1) complex numbers are real numbers in the sense that they exist in the physical world. 2) (dx)^2 = 0 is not the definition of an infinitesimal, it's a byproduct of the definition. infinitesimals are *different* than complex numbers (and all real numbers) in that they don't actually exist (not can they be measured) - they are a mathematical invention. it might be helpful to think of infinitesimals as limits instead of something infinitely small.
algebraic infinitesimals are used a lot in artificial intelligence libraries; as gradient descent is used to train the weights; billion variable calculus. in GA, you have a set of basis vectors, with kinds that square to -1,0,or 1. in that system, the things that square to -1 are usually not prime. "i" is usually meant to mean (right*up); which squares to -1.
The 'definition' does not really make sense, if we try to understand it using standard mathematical notation: the condition ε < ℝ should then mean that ε is less than every real number. But -1 is a real number, so ε must have the property 0 < ε < -1, which is impossible to make sense of. A proper definition would say that 0 < ε < x holds for every positive real number x.
Mathematical physics have always used infinitesimals as if they were actual numbers. While the matematitians didn't accept that until Robinson 1960 showed that they actually exists. But, of course, Robinsons way doesn't feel intuitive. But it still, I think, validates the physicists way of handle them. It would be nice if the mathematicians could go all the way to work with infinitesimals the intuitive way. But so far, I haven't seen that calculus book. Weierstrass technique was so boring! I like the videos proof of the area of the circle. But when I realized that metod many decades ago, I used that the sum of all the bases of the triangles is the circumference of the circle. So, I didn't even introduced the measurement epsilon for the basis on one triangle. Oddly enough, I haven't seen that proof in trigometry texts. It's so intuitive!
From my experience studying physics, what physicist call infinitesimals are actually small intervals, and not actually infinitesimal numbers. From there, they just do a bunch of approximations that they deem to be accurate enough, such as dy divided by dx being the partial derivative of y according to the variable identified with x, which is non-sensical since it implies that dx/dy = 1/(dy/dx). There's no calculus book written using infinitesimals (in the physics sense) because they make no sense whatsoever. "Proofs" that use them are wrong. If they want to use non standard analysis (which they don't use because most don't have anywhere near the knowledge needed to grasp it), they should be taught that, and this culture of teaching physicists and engineers standard analysis and then substituting what they learned in calculus classes for this "small enough interval" "calculus" should stop.
UA-cam won’t let me post more than a few lines until I get error 404. Tried several times but I’m tired of having to retype. Wanted to explain this from a different view.
Oh, you mean that as long as you don't understand and it is a big pile of dung, then it is beautiful? Tsk, tsk. Think! Think! It can be good for your mental health.
I wish mathematicians would embrace infinitesimals, like they do infinities. It's so strange they have a bias against one but not the other, and insist that we use the incoherent "limits" instead.
thank you can i said the infinitesimal number is smallest than any real number so it specific value but i can measure it and that mean the infinitesimal is the point which move our from algebra to calculus and the secret of continuity and limits
It's just as hard to measure an infinitesimal as it is to measure infinity! By definition it's not really 'measurable' in the same way as an integer or even a fraction.
Amazing video! Great explanation. I have watched many try to explain that but you are the first one that actually explained it perfect and I understood it.
@@JenFoxBot oh hello. Cos you were saying if i approximate a circle as a bunch of infinitesimal triangle widths, does that mean a curcle sblike a polygon? Cos a shape made of triangles is a polugon.
@@manicmath3557 ahh yea infinitesimals are supes weird! TBF we are getting into philosophical math territory -- is a circle on a screen a circle bc its drawn in pixels (tiny squares)? Technically, if we zoom in enough in a chalkboard we might be able to see individual grains.. is that still a circle? The brain breaking part of infinitesimals is that they don't *really* have any "size", just like an electron is technically a point particle with no volume (what does that even mean?!). This means that even tho we are using triangles to visualize and break our problem into tangible pieces, they aren't triangles in the traditional sense as its technically impossible to draw a triangle with an infinitesimal length. I hope that helps... and that I didn't just make things weirder hah
Infinity is not a number ma'am. It is a never-ending set in which has no limits. So does it makes sense to say infinitesimal is 0 that can be infinitely divided? And thus infinite?
If we're talking about math, then yes we can treat infinity (and infinitesimal) as numbers bc we can do calculations with them. They are diff kinds of numbers than "1" or "1346", just like -1, 1/3, and pi are diff kinds of #s than 1.
r is not a fixed number, it can be any positive real, including smaller than or bigger than an infinitesimal. the point of the infinitesimal definition is that, like infinity but in the opposite direction (i.e., infinitely small vs. infinitely large), there can always be a smaller number that is still greater than zero. in fact, there are infinite infinitesimals between two different infinitesimals! pretty wild. calculus has withstood thousands of proofs by millions of mathematicians. if there was a contradiction or something wrong w/ the definitions on which it relies, it is very, very likely that someone would have discovered that by now. that's not to say to trust the theory, i really believe in testing things, but we also have to be humble and respect the hard work of folks who came before us and folks who do this work professionally.
@@JenFoxBotWell, yeah, but the definition of the infinitesimal is it is the smallest positive real number. By taking two of them, you can show that there is a fixed real that is smaller than at least one of them. If r wasn't a fixed number, then dy or dx would have to be variables, which they aren't; they're infinitesimals. If dy
@@perseusgeorgiadis7821 just like with infinities, there is no "one" infinitesimal. there are infinite infinitesimals! you can always find a smaller infinitesimal.
@@JenFoxBot Isn't that statement contradictory, from the definition of the infinitesimal? If there's a smaller number than an infinitesimal, how was it an infinitesimal?
@@perseusgeorgiadis7821 nope, that is the definition of an infinitesimal. there's always a smaller number! just like infinity, there's always a bigger one! which is why we have to use math techniques to work with them, just as we do w/ infinities.
i'm not sure i understand what you mean by infinity as an error in math? doing calculations with infinities is crucial to calculus (e.g. limits) and in lots of areas in physics. i've done math with infinity more times than i can count! (although less than infinity 😜)
Woah. Im a 16 year old student from Germany and I actually understood everyword coming out of your mouth. This video is a great example on how real mathematics work, I love it!!!
That makes me so happy to hear!! Thank you so much for your kind comment, its words like yours that motivate me to keep making these kinds of videos 💖
Best explanation I've ever heard
A both practical and intuitive example, a good starting point for advanced stuffs!
Yay thank you !!
Thanks for the great explanation, this is the first example I've seen of an infinitesimal being used in an intuitive rather than a hand-wavy-way (calculus always felt incomplete to me because of the lack of time spent on the infinitesimal). If you know more examples with infinitesimals being used in interesting (clear non-hand-wavy) ways it would be cool to see!
LMAO. Intuitive = hand wavy
There are lots of hand wavy 'explanations' in the bogus formulation of mainstream calculus.
I actually learned today in spite of the beautiful teacher ; )
Krindž
❤ and subscribed ❤❤❤❤❤
This is a good video and nice explanation. Just one issue, the definition of an infinitesimal ε you provided is incorrect w.r.t how it's standardly defined. You gave the definition for a *positive infinitesimal*. The broader definition of ε is so that -a < ε < a, for all 'a is in the reals'. If you want to write it your way, you'd want it to be 0 < |ε| < r for all positive reals. Either one of these is fine, because they include negative infinitesimals (which are also infinitely close to 0, just like positive infinitesimals). Other than that, good video.
Amazing explanationn!! Thank you so muchh
omg omg litteraly u r the coolest teacher i have ever seen
thank you v much! unsure if this is referring to my rad outfit, my teaching, or all the things, but i appreciate and agree with your comment 🙃
@@JenFoxBot all in all is just amazing ❤️❤️
@@ayalouk3057 🥰🥰🥰 thank you!!
What is the surface area of a sphere with radius infinitesimal?
If there is a surface then must this surface inherently have infinite points on it?
Our manifold, a minimal single sided closed surface
Sin(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2) 0
Amazing..... thanks for the explanation....was really well worth it
Thanks for the kind words! Glad to hear it was helpful.
Congrats!!! This is awsome! You're a great teatcher, nice and also eye candy (lets admit it). This deserved much more views!
glad to hear the video was helpful! if you want to help promote the channel, share things you enjoy with your friends 😄
Very nice. I am familiar with the Extended Real Numbers (Reals with +/- Infinity). It makes sense to also include an infinitesimal that is, as you say, smaller than everything else. I'm also learning about Dual Numbers with uses Infinitesimals for a very famous application, Autodifferention, for training (optimizing) a Neural Network. I am happy so many things are converging in my understanding of mathematics for (cringe) Machine Learning.
That sounds fun! Super cool that you're studying mathematics and ML, I hope you enjoy learning it!
Amazing explanation mam ❤
Super cool stuff, thanks. 💐
Very good explanation
I am helpless I am unable to take off my eyes from the teacher because She has mesmerizing beauty 😍
Great video! Battery current (both charging and powering a load) needs to be integrated correctly by both hardware circuitry and code using calculations that accurately reflect what is going on.
Any discussion or approaches concerning current measurement total-izing would be appreciated. I've been wondering if a current measuring IC sensor voltage output could be simply integrated by a capacitor to determine the total amount of charge current going into a battery. So far, I get too much V across the cap!!
Ohh fun!! Sounds like an interesting problem! Whats the goal of measuring the current? You could brute force it if your circuit isn't too complex, and then there's always my fav tool: the multimeter! Generally speaking, the current in your circuit will equal whatever current source(s) are there (e.g. battery, power supply, etc.). Current is conserved (altho some gets lost as heat), so whatever is going into your circuit has to go somewhere!
But yea, some context on the specific problem you're trying to solve would help narrow down a potential solution or approach!
@@JenFoxBot I am designing a 4 channel battery analyzer. The "easy" way is to use an IC that measures current (like the Allegro MicroSystems ACS70331EE). That measurement would be taken every, say, 100mSecs and totalized. That totalized number would then be converted to milliAmpHrs.
So the "crazy" idea is coming up with some time of OpAmp/capacitor integrator that would give the same number, but without firmware. Then you could use a simple DVM, measure the volts out, and convert to mAHr.
Lot's of used 18650 batteries are coming to market, (like from Battery Hookup) and it would be cool to come up with such a cheap device so people could safely re-purpose those batteries.
PS:Wish you had of been my math teacher when I was in college. I barely made it through yr one calculus.
Good ...lv from Kashmir
Thnku
That was amazing!
Thank you !!
i ran into this video (which I think I have seen before), when I was contemplating an issue with limits.
- an infinitesimal is a thing that squares to 0, but is greater than 0. ie: dx*dx=0, dx>0.
- a number that is merely "small" still squares to greater than zero: smallx*smallx>0, smallx>0.
that's why, this is just fine: dy/dx. dx>0, dx*dx=0. if y=x^2, then dy=2x dx. so dy^2=0 as well, because 2*2*x*x*dx*dx=0. i don't understand the total avoidance of using actual algebraic infinitesimals. when doing geometry, you end up dealing with things that square to -1, 0, and 1. in deep neural networks, it is common to use autodiff libraries, which feature infinitesimals to calculate giant derivatives with infinitesimals.
ahh so it's not that dx^2 = 0, it's that dx + dx^2 ~ dx. in other words, we can 'discard' dx^2 in comparison to other things b/c it's sososo small.
@@JenFoxBot actually, i intentionally mean that an infinitesimal called dx is infinitesimal BECAUSE it is a square root of zero: dx*dx=0, and it's not zero because dx>0 (for a positive infinitesimal).
it's a super-important distinction from having something "small" but finite. in that case if small>0, then small*small>0; as it's a finite real number.
0 < dx < small
A square root of zero that is not zero is the definition of an infinitesimal; and it's not a real number. It's no different from the objects that square to -1, which are not real numbers. And there are non-real objects that square to 1 as well. Geometric Algebra is all about objects that square to -1, 0, and 1; to produce all of geometry. You effectively add directions in space into the algebra; and it makes multiplication non-commutative.
The thing that makes limits waffly is when you say "arbitrarily small". By that definition, you can't wrangle that definition as algebra. And that means that turning it into computer code is waffly and inconsistent.
It's exactly why some people won't accept 0^0=1, because "arbitrarily small" isn't nailed down enough. If you are contemplating 0^small, then small*small>0 means that 0^small=0. But 0^dx where dx*dx=0 means that 0^dx=1. Limits are a way to waffle around not getting the algebra right.
unless you take the equals operator to mean "has the same real value", you can't say that "x + dx = x". the equals operator doesn't mean that. it means "can be substituted with, in both directions".
y = (x+3)^2
d[y = (x+3)^2]
dy = 2(x+3)d[x+3]
= 2(x+3)(dx + d3]
= 2(x+3)(dx + 0)
= 2(x+3)dx
it might be the case that only dx*dx -> 0. But not 0 -> dx*dx; which is weaker than "dx*dx=0". That might be a subtle point about infinitesimals. I am not sure if you can just factor zero into any square root of zero.
ie:
dy = 2(x+3)dx
dy/dx = 2(x+3)
dy = 2(x+3)dx + dx*dx //can we jadd dx*dx?
dy/dx = 2(x+3) + dx
But for sure, instead of dividing by zero when trying to set dx=0, you can divide individually by dx twice instead.
// second derivative of x^3
// assume x is a line: d^2x=0
d[d[x^3]/dx]/dx
=
d[3 x^2]/dx
=
3(2 x dx)/dx
=
6x
@@rrr00bb1 1) complex numbers are real numbers in the sense that they exist in the physical world. 2) (dx)^2 = 0 is not the definition of an infinitesimal, it's a byproduct of the definition. infinitesimals are *different* than complex numbers (and all real numbers) in that they don't actually exist (not can they be measured) - they are a mathematical invention. it might be helpful to think of infinitesimals as limits instead of something infinitely small.
algebraic infinitesimals are used a lot in artificial intelligence libraries; as gradient descent is used to train the weights; billion variable calculus.
in GA, you have a set of basis vectors, with kinds that square to -1,0,or 1. in that system, the things that square to -1 are usually not prime. "i" is usually meant to mean (right*up); which squares to -1.
The 'definition' does not really make sense, if we try to understand it using standard mathematical notation: the condition ε < ℝ should then mean that ε is less than every real number. But -1 is a real number, so ε must have the property 0 < ε < -1, which is impossible to make sense of. A proper definition would say that 0 < ε < x holds for every positive real number x.
Its really helpful
very glad to hear! 😄
But isn't infinitesimal defined as 0 < ε < 1/n for all Natural Number n? And isn't what you described is either limit or non-standard calculus?
Mathematical physics have always used infinitesimals as if they were actual numbers. While the matematitians didn't accept that until Robinson 1960 showed that they actually exists.
But, of course, Robinsons way doesn't feel intuitive. But it still, I think, validates the physicists way of handle them.
It would be nice if the mathematicians could go all the way to work with infinitesimals the intuitive way.
But so far, I haven't seen that calculus book.
Weierstrass technique was so boring!
I like the videos proof of the area of the circle.
But when I realized that metod many decades ago, I used that the sum of all the bases of the triangles is the circumference of the circle. So, I didn't even introduced the measurement epsilon for the basis on one triangle.
Oddly enough, I haven't seen that proof in trigometry texts. It's so intuitive!
From my experience studying physics, what physicist call infinitesimals are actually small intervals, and not actually infinitesimal numbers. From there, they just do a bunch of approximations that they deem to be accurate enough, such as dy divided by dx being the partial derivative of y according to the variable identified with x, which is non-sensical since it implies that dx/dy = 1/(dy/dx). There's no calculus book written using infinitesimals (in the physics sense) because they make no sense whatsoever. "Proofs" that use them are wrong. If they want to use non standard analysis (which they don't use because most don't have anywhere near the knowledge needed to grasp it), they should be taught that, and this culture of teaching physicists and engineers standard analysis and then substituting what they learned in calculus classes for this "small enough interval" "calculus" should stop.
Hello teacher. Can you explain numerical methods heat transfer
Totes! I did a lot of that in grad school. I'll add to the list :)
Thank you💖😘
This is good jen from our planet
UA-cam won’t let me post more than a few lines until I get error 404. Tried several times but I’m tired of having to retype. Wanted to explain this from a different view.
Great explanation, and these type of videos showcase the beauty of maths :)
Oh, you mean that as long as you don't understand and it is a big pile of dung, then it is beautiful? Tsk, tsk. Think! Think! It can be good for your mental health.
thank you!! totally agree
😊
I wish mathematicians would embrace infinitesimals, like they do infinities. It's so strange they have a bias against one but not the other, and insist that we use the incoherent "limits" instead.
thank you
can i said the infinitesimal number is smallest than any real number
so it specific value but i can measure it
and that mean the infinitesimal is the point which move our from algebra to calculus
and the secret of continuity and limits
It's just as hard to measure an infinitesimal as it is to measure infinity! By definition it's not really 'measurable' in the same way as an integer or even a fraction.
Amazing video! Great explanation. I have watched many try to explain that but you are the first one that actually explained it perfect and I understood it.
so happy to hear it was helpful!
Liar. You understood nothing at all because everything she said was BS.
Wait then what is the difference between a limit and an infinitesimal
Here in India. It's 😁Tuesday
Ma'am, you are great, whew
But doesnt that suggest a circle is a polygon
How so?
@@JenFoxBot oh hello. Cos you were saying if i approximate a circle as a bunch of infinitesimal triangle widths, does that mean a curcle sblike a polygon? Cos a shape made of triangles is a polugon.
@@JenFoxBot btw the circle you draw looks unhumanly perfect somehow lol
@@manicmath3557 ohhh yes success!!! Circles are SO hard to draw, and jts like the more you think about it the less circular they are 😂
@@manicmath3557 ahh yea infinitesimals are supes weird! TBF we are getting into philosophical math territory -- is a circle on a screen a circle bc its drawn in pixels (tiny squares)? Technically, if we zoom in enough in a chalkboard we might be able to see individual grains.. is that still a circle?
The brain breaking part of infinitesimals is that they don't *really* have any "size", just like an electron is technically a point particle with no volume (what does that even mean?!). This means that even tho we are using triangles to visualize and break our problem into tangible pieces, they aren't triangles in the traditional sense as its technically impossible to draw a triangle with an infinitesimal length.
I hope that helps... and that I didn't just make things weirder hah
ghosts of departed quantities
Infinity is not a number ma'am. It is a never-ending set in which has no limits.
So does it makes sense to say infinitesimal is 0 that can be infinitely divided? And thus infinite?
If we're talking about math, then yes we can treat infinity (and infinitesimal) as numbers bc we can do calculations with them. They are diff kinds of numbers than "1" or "1346", just like -1, 1/3, and pi are diff kinds of #s than 1.
And no, an infinitesimal is not 0. With current # theory, 0 cannot be divided.
confused by the subtelity!
Could you be more specific on what you're confused by? Happy to try to give more details
I'm not satisfied with this definition of an infinitesimal. You claim that an infinitesimal is a number ε such that 0
r is not a fixed number, it can be any positive real, including smaller than or bigger than an infinitesimal. the point of the infinitesimal definition is that, like infinity but in the opposite direction (i.e., infinitely small vs. infinitely large), there can always be a smaller number that is still greater than zero. in fact, there are infinite infinitesimals between two different infinitesimals! pretty wild.
calculus has withstood thousands of proofs by millions of mathematicians. if there was a contradiction or something wrong w/ the definitions on which it relies, it is very, very likely that someone would have discovered that by now. that's not to say to trust the theory, i really believe in testing things, but we also have to be humble and respect the hard work of folks who came before us and folks who do this work professionally.
@@JenFoxBotWell, yeah, but the definition of the infinitesimal is it is the smallest positive real number. By taking two of them, you can show that there is a fixed real that is smaller than at least one of them. If r wasn't a fixed number, then dy or dx would have to be variables, which they aren't; they're infinitesimals. If dy
@@perseusgeorgiadis7821 just like with infinities, there is no "one" infinitesimal. there are infinite infinitesimals! you can always find a smaller infinitesimal.
@@JenFoxBot Isn't that statement contradictory, from the definition of the infinitesimal? If there's a smaller number than an infinitesimal, how was it an infinitesimal?
@@perseusgeorgiadis7821 nope, that is the definition of an infinitesimal. there's always a smaller number! just like infinity, there's always a bigger one! which is why we have to use math techniques to work with them, just as we do w/ infinities.
Infinity is regarded as a error in maths. But reality is infinity. Maths cannot represent nor reflect reality
i'm not sure i understand what you mean by infinity as an error in math? doing calculations with infinities is crucial to calculus (e.g. limits) and in lots of areas in physics. i've done math with infinity more times than i can count! (although less than infinity 😜)
not useful but not useless