@scottmoerschbacher8664 thanks for watching! For the f(x)=3x example, i was using the standard topology on both and same thing for the tangent. Yes, continuity is dependent on the topologies, so a function could be continuous with respect to one topology and the same function could be not continuous with respect to another.

@@mathprofessor-11235 amazing, many thanks for the quick reply ✌️

@@dhwyll the idea that multiplying by a negative moves you to the opposite side of the number line is an interpretation that we can impose based on the result that multiplying a negative by a negative yields a positive (using the technique i demonstrated), however its not the reason for it.

@@dhwyll thanks for the comment and thanks for watching!

​@@mathprofessor-11235There are many ways to achieve the same result. Given that multiplication can be considered as a rotation and multiplication by a negative number would be a rotation of 180° (as multiplication by i is a rotation of 90° and thus multiplying by i² would be rotating by 90° twice and is equivalent to multiplying by -1), then it isn't really a question of the geometric representation being an imposition but rather just another manifestation of the same underlying principle. After all, we can also show it as follows: -a × -b = (a × -1) × -b = a × (-1 × -b) = a × -(-b) = a × b Thus, the product of negative numbers is equivalent to the product of the same numbers if they were positive. [And before complaints about justifying -(-b) = b, that can be shown from basic rules of associativity, commutativity, distribution, and the existence of the additive identity and inverse.) Since the goal is to show someone that two negatives multiply to a positive isn't just something we force onto things, having different ways to think about it is a good thing.

@@MathwithMing thank you and thanks for watching! By the way, i just subscribed back to you!

@@MathwithMing thank you! It was so much fun to write! And thank you for the sub!

@@Dippypirate Well, even though its a very basic result, it still requires a proof

@@mathprofessor-11235 exactly

@@maniyarala that is not correct. To divide by 2(1+2), the expression would need to be written as 6÷(2(1+2)).

Thanks!

@@SomeGuyFromUtah computer simulations verify the 1÷3, 2÷3 that I demonstrated

@@mathprofessor-11235 any idea how this concept affects the math on Deal or No Deal (I think there were 25 cases to start, and if you go clear to the end, you'd be in the same keep it or swap it scenario)

@@wingingglobe4095 lets try this variation. I give you 100 doors and say the car is behind one of them. You choose a door and then i open 98 doors but then ask if you would want to keep your choice or switch. Do you still think its a 50/50 shot?

You had a 2/3 chance of being wrong when you originally picked. You never stop having a 2/3 chance of being wrong. It is always, always, always, no matter what, more likely that the first door you chose was wrong. So you should switch.

Yes you idiot​@@mathprofessor-11235

The host might know, but the contestant doesnt. So no the chances are not 50/50. Its 1÷3 chance the choice is right and 2÷3 its wrong. After its revealed that door 2 is wrong, there is a 2÷3 chance that door 3 is correct....because there is still only a 1÷3 chance that door 1 is correct

@@jboixmangaming5325 thanks! I wasnt sure if i was going to be able to get it done in under 60 seconds. It took multiple takes lol

@ yep, and all that 60 seconds taught me something I didn’t know, so thank you

@@jboixmangaming5325 very welcome! Thanks for watching!!

@@mathprofessor-11235 you got it 👍🏽

Multiplying items is unphysical, there's no physical meaning that you can give to it. It seems reasonable until you conceive of them not as quantities of some abstract common unit but as separate items. Multiplying _that_ one dollar with _those_ two dollars makes as much sense as multiplying one mug and two pencils - it has no meaning. Adding items together on the other hand makes perfect sense - if oyu add your pcokets together you have 3 dollars and one mug plus two penicls is exactly that, one mug and two pencils. Multiplying quantities together also makes sense physically. 1 ft times 2 ft is 2 sq ft or 2 ft², which is a measure of a different quantity than length. Similarly, 1 dollar times 2 dollars could be 2 sq dollars or 2 dollar². What does that mean? Nothing, by itself. But you can give it good meanings - perhaps something gets cheaper as it gets more expensive. You could get a base price of 1 gallon gas per dollar to start with, but for every dollar your purchase gets more expensive, you get half a gallon more per dollar. Then you have a "cheapening" of 0.5 gallon/dollar². This can even be used concretely to compute how many gallons you get for some amount of money. In this instance, 1 dollar gets you 1.25 gallons, $2 gets you 3 gal, $3 gets you 5.25 gal, $10 gets you 35 gal and so on. Does it make any sense? Some, though it's not a very realistic situation. But it's clearly different from treating those dollars as separate objects which are matter and both dollars. In this case, the dollar is just a measurement of currency and thus we can have currency-squared measured in square dollars.

When using multiplication to count things, the thing you’re counting is usually multiplied by a counting number. The counting number isn’t a physical thing itself. By example: Let’s say you have x amount of dollars in your wallet. Suddenly, a wizard pops up and duplicates every dollar in your wallet. Congratulations! You have twice as many dollars as you had before. Now, here’s the thing. In order to find out how many dollars you have in your wallet, we multiply x by 2. “2 what?”, you say. Just the number 2, which represents 2 for each 1 of the thing you had to beginning. The 2 doesn’t mean 2 dollars. x is the number that represents the dollar amount. In this sense, “Multiplying a dollar by a dollar” doesn’t actually mean anything. How does one multiply something by the same thing? What could it even possibly mean?

@@andresfonsecaw multiplying money doesnt have any sort of meaning though.

Consider the following scenario: We begin with the number 2 and compute its square root, resulting in approximately 1.414213562373095. Let us denote this resultant value as “X.” Subsequently, we execute two independent operations on “X”: 1. Multiplying “X” by 2 yields 2.828427124746190, denoted as “Y.” 2. Raising “X” to the power of 3 also yields 2.828427124746190, which we also refer to as “Y.” Remarkably, both operations lead to the identical outcome, “Y.” Continuing this exploration, we proceed to take “Y,” divide it by 2, and then elevate the result to the power of 3. Once again, we obtain 2.828427124746190. This intriguing phenomenon highlights a curious mathematical property wherein certain operations on a number engender a cyclical or loop-like behavior. The equation encapsulates this condition, with representing one of its solutions. This is a loop = This is saying X to the power of 3 is equal to 2X which is equal to X + X This is an unatural equation, some would even consider mathematical fallacy.

This loop inherently describes the problem with multiplication - take now 1 dólar and multiply it against another dólar - why does 1 dólar magically disappears? What happens to the other dollar? It work with units as well

those comments can be very frustrating

@@Shashank_Shahi1989 you're welcome! I hope it's helpful!

@@zamplify the publisher, Kona Publishing, is selling it. Im just the author.

@@zamplify Im Nick Goins btw

@@zamplify konapublishing.com/category/71/

@@Hellon1832 i will for sure. Good luck in your topology course!

@@Hellon1832 thank you! Ive got more topology videos coming!

@ do you have somethign for differential equation? Looking forward

@@Hellon1832 the diff eqns playlist is pretty short at the moment. I will be adding more to it when I can

That's a great idea! I can definitely discuss that. What are you currently studying? Any topics you want to go through? Depending on the topics I can give some specific recommendations. Also, I have a video scheduled to post in about a week titled, "How to prepare for graduate school in mathematics", which might be useful to you as well. We appreciate you watching!

@mathprofessor-11235 Thank you for your feedback. Can you discuss the syllabus of Bsc and Msc mathematics and why / how should we study those subjects/ textbooks in self study mode in depth to understand fully. I m learning maths for hobby and AI in self study mode. Thank you

@@Shashank_Shahi1989 I can make a few videos out of that. I'll discuss bachelors degrees and graduate degrees, including what courses you would need/could take along with what you can do with such degrees. Also, I can talk about how to self study advanced math texts. It will probably take a few weeks since we have a list of videos already scheduled, but look for them to come out. Thanks again for the ideas!

@@mathprofessor-11235 Thank you. Sure , I will check your KZbin channel.

@@Shashank_Shahi1989 the video regarding reading advanced math texts will be posted in a few minutes. The other videos regarding undergrad and grad degrees will be recorded and posted soon. Thanks again!

Thanks for the feedback. I used the y equation with finding the partials for x and the same mistake for the partials for y. I'll update the video soon.

@@breannarawlins2401i dont think I noticed. But yeah its cool stuff!

@@williamvarenas2790 thank you!

@@breannarawlins2401 it is a good one. Be glad you took calc 2 where you did🤣🤣🤣

@@breannarawlins2401 i thought you'd like it. Maybe some of your uofm colleagues would now watch us! Lol

Lol! Well....thanks😂

@@EquipteHarry please share away!

We're trying! Still need to work out some issues with the lightboard, but I think it looks pretty good

Thanks! I need to mess with the settings to see if I can make the writing pop a little bit more

Thank you!

Appreciate it!

Thanks man! Appreciate your support!

Thanks for watching! Glad you liked it!

Our pleasure!

Thanks! Appreciate the kind words! I havent had much time to record any videos lately, but im trying to get done what I can.

Thanks! Appreciate it!

That wouldve made the simplification a bit cleaner. Again, appreciate the feedback.

The US

@@mathprofessor-11235 love from India🇮🇳🇮🇳🇮🇳