Just wanted to add that continuity of addition and multiplication depends entirely on the topology used on both the domain and codomain sets, since in this case both are "similar", both operations are continuous, but in another case both can be not continuous.
Hi Dr Peyam, have a question for you. What is "halfway" between addition and multiplication? Can such an operator be defined? It seems we should be able to construct any number of paths between the two functions, but are there any that "make sense" to do? For instance, keeping associative and commutative properties. Natural next question is: what about between multiplication and exponentiation? And exponentiation/tetration? Hyperoperators? Where does the idea break down?
Well you can always just define such an operation. For example we could define something like: :: mad\f(x,y) = f(x+y)+(1-f)xy So if f is 0, its equivalent to addition and if f is 1 its equivalent to multiplication. Then if you want something halfway between the two, you just use f=0.5 and off you go. Of course being able to define such a thing doesn't mean that its actually a useful thing, and that's where artistry comes into mathematics. "Useful" things are created when someone needs to solve a problem that is difficult or impossible to describe using existing notation. If whatever you're working on requires an operation that's halfway between addition and multiplication then maybe that mad\f function _is_ worth defining. Of course practically speaking, that definition is too trivial to be useful in any real sense. What might be more useful is a definition where f=0 is a constant function, f=1 is addition, f=2 is multiplication, f=3 is exponentiation, f=4 is tetration, etc. I wouldn't be surprised if someone had already done that, though I don't know of it myself.
There is "typo" in the continuity of multiplication part. In scratch work you had bound of |x||y-y_0|+|y_0||x-x_0|, but later used a bound where you paired x's together and y's together.
@@drpeyam I thought that. So when you say "let delta be '*' .. the pick works w the proof known. Thanks for this great math series, super helpful and clear.
Very good video and mind blowing too !! Nice presentation. Sir we also have floor (pi) = 3 and ceiling (e) = 3 hence touching floor and ceiling both we get 3+3=6 very easily . Thanks with sincere regards. DrRahul Rohtak Haryana India
Sir If a Person Is Going for A competitive math Exam..What Should be the key Points So That The Math Subject Becomes Strong(The thinking Ability Just Like you)..Please Suggest Me.🥺🥺
π + e = 3 + 3
Fundamental theorem of engineering
It's a conclusion from the fundamental theorem, not a theorem itself. Please be rigorous 😁
That's funny!
I love that on math you can complicate a thing as arbitrarily as you want
Hahaha so true
You know what this video is missing? Category theory! 😛
Lets be honest. Pi=e=3
That makes "things" a lot easier.
FTE(Fundamental theorem of engineering)
"Close enough for government work."
Just wanted to add that continuity of addition and multiplication depends entirely on the topology used on both the domain and codomain sets, since in this case both are "similar", both operations are continuous, but in another case both can be not continuous.
Hi Dr Peyam, have a question for you.
What is "halfway" between addition and multiplication? Can such an operator be defined? It seems we should be able to construct any number of paths between the two functions, but are there any that "make sense" to do? For instance, keeping associative and commutative properties.
Natural next question is: what about between multiplication and exponentiation? And exponentiation/tetration? Hyperoperators? Where does the idea break down?
Well you can always just define such an operation. For example we could define something like:
:: mad\f(x,y) = f(x+y)+(1-f)xy
So if f is 0, its equivalent to addition and if f is 1 its equivalent to multiplication. Then if you want something halfway between the two, you just use f=0.5 and off you go.
Of course being able to define such a thing doesn't mean that its actually a useful thing, and that's where artistry comes into mathematics. "Useful" things are created when someone needs to solve a problem that is difficult or impossible to describe using existing notation. If whatever you're working on requires an operation that's halfway between addition and multiplication then maybe that mad\f function _is_ worth defining.
Of course practically speaking, that definition is too trivial to be useful in any real sense. What might be more useful is a definition where f=0 is a constant function, f=1 is addition, f=2 is multiplication, f=3 is exponentiation, f=4 is tetration, etc. I wouldn't be surprised if someone had already done that, though I don't know of it myself.
There is "typo" in the continuity of multiplication part. In scratch work you had bound of |x||y-y_0|+|y_0||x-x_0|, but later used a bound where you paired x's together and y's together.
I love advanced Calculus.
Stupid question: there's no rule of thumb way to choose a *good* delta 25:10, right?
Not really haha, it’s an art and takes practice
@@drpeyam I thought that. So when you say "let delta be '*' .. the pick works w the proof known. Thanks for this great math series, super helpful and clear.
It’s an important part for continuous , limits , derivatives and integrals how get to delta so it must more practices to learn that in addition to art
Yes it’s a great video so must save to learn when we want to study continuous
i am more concerned about : how tf is your comment 1 month ago when the video is posted just under 2 hour?
The Sugar Teacher... XD
visible truth 😁
Very good video and mind blowing too !! Nice presentation. Sir we also have floor (pi) = 3 and ceiling (e) = 3 hence touching floor and ceiling both we get 3+3=6 very easily . Thanks with sincere regards. DrRahul Rohtak Haryana India
Was wondering which video you showed the pair is continuous iff both components are continuous? Couldn't find it.
Continuity in Metric Spaces ua-cam.com/video/WTbcJYBLxAs/v-deo.html
@@drpeyam Oh I see. Thanks for help and your content.
It feels like using d_infinity as your distance would save a bit of trouble and conform better to how people think about dividing approximate numbers.
Could you explain limsup and liminf?
Already done
ua-cam.com/video/EvTpC5FlirE/v-deo.html
Thanks 😊😊
Would it be possible to angle your camera just a bit so we don't spend half the video looking at your back? Otherwise, awesome work!
I prefer 2*pi+e=9
Y0 - why not. Pi M is a king of the math puns 👍🏽
Third derivative of f(x) = sin(x*√3/2)/e^(x/2)
is there anyone who didn't know that addition is continuous?
😉😁
Most of us. There's a difference between "knowing" its continuous (intuitively) and _knowing_ its continuous (rigorously).
Est-ce que nous pouvons utiliser les structures algébrique pour prouver ça
Ah finally a man of culture, π.=e=3
Sir If a Person Is Going for A competitive math Exam..What Should be the key Points So That The Math Subject Becomes Strong(The thinking Ability Just Like you)..Please Suggest Me.🥺🥺
don't go easy
Just do lots of modelled examples
There’s more to math than competitive exams
@@drpeyam well said.
@@drpeyam yes weird maths olympiad
Great video
@ ~ 6:34 .. . .. "and again by the same Spiel" .....should be "and again by the same game" Spiel ist Deutsch!
That’s the point, it was intentional
@@drpeyam Ich dachte mir schon, aber ich wollte Ihre Aufmerksammkeit zu mir ziehen.
Der Schüler will von dem Meister gesehen werden.
Video Splash:
Engineers have entered the chat
pi^2 = g = 10 also :)