Indeterminate: the hidden power of 0 divided by 0

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You've all been indoctrinated into accepting that you cannot divide by zero. Find out about the beautiful mathematics that results when you do it anyway in calculus. Featuring some of the most notorious "forbidden" expressions like 0/0 and 1^∞ as well as Apple's Siri and Sir Isaac Newton.

In his book “Yearning for the impossible” one my favourite authors John Stillwell says “…mathematics is a story of close encounters with the impossible and all its great discoveries are close encounters with the impossible.” What we talk about in this video and quite a few other Mathologer videos are great examples of these sort of close encounters.

Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy :)

[NotQuiteFirst]

Great video, I rate it 0/0. Full marks!

[themeadowshadows]

this man sort of comes across as a bond villain but is friendly enough so that I think he would be the assistant to the bond villain and would end up somehow disarming the nukes of the villain as a sort of double agent. these are the things I thought about in college. and I wonder why my degree didn't work out.

[AlqGo]

4:09 "but as long as it's staying off 0..."

Nice! You'd be surprised how rare people explain this important piece of information when they explain derivatives

[doktoracula]

"Noone would know Isaac Newton. That would be really sad, right?" I bet Leibniz wouldn't agree.

[timharig]

The real reason that you are advised to avoid indeterminate forms is that your must invoke L'Hôpital's Law -- which you will not be able to pronounce to to everybody's satisfaction.

[wag-on]

The person who invented 0 gave nothing to mathematics

[alpheusmadsen]

My takeaway from this is that, because "0/0" is undefined or indeterminant, it can be anything -- and *thus* we have to look at it in context to see what value it makes sense to be (if sense can indeed be made). I've never thought of this that way, but it makes sense! And it makes sense not just in calculus, but linear algebra, too, where the determinant of a matrix being 0 means it has multiple possible values for an inverse as well. Heck, this even puts kernels of homeomorphisms in abstract algebra into context, as well, where you can describe the spaces of things that go to 0!

[antoniolewis]

I asked Siri what 0 divided by 0 is, and it broke my heart.
Siri why are you so cold!!

[Oinikis]

As an aeronautical engineering student I find it extremely satisfying to see stuff I am learning at the university.

[aiden_c]

You're a fantastic teacher. In less than a minute I went from not being sure why dividing by zero doesn't actually work to completely getting it.

[QuantumHistorian]

Most surprising thing in this video is a native German speaker holding Newton responsible for calculus rather than Leibniz!

[Math_oma]

Newton should be getting more credit because the term he used for derivatives/velocities was cooler: _fluxions_.

[michaelheimburger]

Just want to say that your explanation that both 1 and infinity are both functions just cleared up a lot of confusion about infinity for me and opened my mind to a totally new way of thinking about numbers. Thanks!

[pogonoah]

Before I started calculus I was determined that 0 divided by 0 was 0. When I was younger I had it explained to me that with x/y = z, z is the answer for how many times you need to subtract y from x to get to 0. And with 0/0... how many times do you need to subtract 0 from 0 to get to 0? Uh... 0 times, right? That's what I thought, but when we started doing limits I realized that it would create crazy jumps in otherwise continuous graphs, so I gave up on it.

[mrbangkockney]

Another good one Mathologer. The closing moments were the most important IMO: aspiring mathematicians and those interested should always remember that they're free to redefine expressions and loosen axioms depending on their area of work - much new material can be discovered in this way (e.g. NE Geometry).

Similar arguments made for defining what 0**0 should be, and different answers depending on who you ask of course. Perhaps even video worthy :)

Thanks again, have a good one.

[Mat_Rix]

For example I presented graph of the function f(x) =1/x ... without discontinuity point ! :) It can be presented for every x, and I'm also explaining why our traditional (wrong) graph has discontinuity at 0.
If you really want to understand it ... you need to read it and understand all presented examples.
Enjoy :)

[Cerealbox]

Man, I really wish someone showed me this video in high school. I kind of figured it out on my own, but it always really bothered me and made me feel like I didn't understand math. This is so simply expressed and explained. I suspect the reason why most people struggle with explaining why you can't divide by zero and related is because they don't actually know themselves. They just memorized that it causes paradoxes.

[jonathanfowler]

Zero has been my favourite number for as long as I can remember having one (sorry, 5). I love how it seems to act like a bridge between real and numbers like infinity. It's also a bit of a gem beneath our noses because the number seems so simple. I'm glad to see a mathologer video with plenty of zeros in it. Also, could you please tell me where I can get that shirt?

[avedic]

Whenever I find myself feeling too confident about my own look up a video about a mathematical topic.
And I am humbled almost immediately.
I think I'm a fairly intelligent person. But....there are expressions of intelligence that are as far beyond me as the things I'm capable of understanding are to a cat.

[alexhill]

I absolutely hated math growing up. It was the first class I ever got a B in in 5th grade. It then became the first class I ever got a C in 7th grade, and finally, the only class I ever got a D in Junior year of high school. I didn't study psychology in college because it required too much math. And yet in spite of that- or perhaps because of the void it left- I enjoy your work. Terrific way to make up for lost time and enjoy seeing patterns play out without the abstract jargon nor the pressure of testing.