Mathematics for aspiring mathematicians. Created by Dr. Katherine Stange at the University of Colorado, Boulder. Learn about me under the "Community" tab, or here: math.katestange.net/
Best explaination ever made for this topic by just one joing lines of a traingle you explained the smallest detail. Thank you very much for this it was so much helpful.
In terms of an orchid problem, I like Phi, rhe golden ratio instead if pi. Since it's "the most irrational number" meaning there is no good approximations. This approach or even the surreal numbers, allow for ording all the irrationals. so doesnt that allow a mapping from rationals to irrationals and showing that they have the same cardinality? Ruining the second diagonal argument? Instead of numebrs, you can also do this with powersets over sll rationals.
i truly wish i saw this video first years ago. how many times i have given up, confused on the notations, given its apparent inconsistency in use. now that you pointed out that it is a side note, it is so clear now. thank you. just wish i saw this years ago.
If you just have L/R options, would there be a significant difference between using fairy* subdivisions instead of binary ones? Also, the fractions you put on the grid don't quite make sense to me. (1,1) is further from the origin than (0,1) and (1,0), so it seems like it should represent 1/2 instead, since it looks smaller, with (1,0) representing 1, (1,2) and (2/1) representing 1/3 and 2/3, (1,3) and (3/1) representing 1/4 and 3/4, and so on. *[sic is what you get for being called Farey and not spelling it out :P]
RU-vid has a lot of trash on it - and then it has things like this. I think this is a serious contender for the best STEM-related video I've ever seen.
Here's how to calculate the numbers used in the sequence of a continued fraction. You take the number, - the biggest whole number you can find, then flip the remainder as a fractional addition, and then repeat. ie: 3.14 - 3 = .14, 1/.14= 7... 1/... = 15... 1/...= 1... 1/... = etc where 3,7,15,1 would be the 'address' instead of 3,1,4,...(base 10 address) Loved the video, and now that i understand the process to generate continued fractions, I agree with you, it's actually VERY easy to calculate the continued fraction. It only requires simple subtraction and division, where as calcuating base 10 of a number also requires simple subtraction and division. Now getting the actual 'value' back from a continued fraction, is a lot more involved (every address you go, requires another division and addition (and the addition involves multiplication), where as with any normal base address it only requires 1 simple multiplication per address, so indexing in is much easier than indexing out, which means numberical bases are here to stay, like you said, but its still quite clever and I loved thinking through the arguments and watching the visualization. Edit: I did some more thinking about this, and actually you need the base 10 system to write out the continued fraction in the first place. So this is an index, of an index. If we wrote all the numbers in base 2, the continued fraction would look way different. Doesn't change much, but I did overlook that, this isn't a 'better' address divorced of the base 10 address, it's an even more specific, base 10 address.
I am interested in understanding how things work rather than memorization, and in less than a minute of the video, I knew it was special. Content such is this is absolutely vital. Thanks.
At first, I didn't quite grasp why would the GCD remain same after we delete the smaller number from larger one (B-A). But it made sense this way: Hint: We are deleting pile A from pile B and then ask what's the new GCD of leftover pile B and the pile A? Well, just remember, the deletion is also made of new GCD as we just deleted pile A- hence the whole pile B and pile A have a new GCD ;) contradiction ! Explanation: GCD is basically the largest chunk of stones that will divide both piles in some number of parts, say- xa and xb. So, pile A has xa number of GCDs and pile B has xb number of GCDs (largest chunks common for both). => A = g . xa and B = g . xb (Imagine them as bigger balls that make up the pile) Now, we remove just one copy of pile A from B. This means: => B - A = g . xb - g . xa For a moment, let's assume, the common chunk size of A and B-A, could maybe get bigger after deletion- to say g' (read: g dash) => B - A = g' . x' and A = g' . xa' This means, the leftover of pile B is made of g' size chunks with count as x' and pile A is made of g' size chunks with count as xa'. But, here's the catch: the deleted pile A from pile B must also be made of g' size chunks with count as xa'. That means: => deleted pile A + left over pile B = the original pile B => g' . xa' + g' . x' = pile B => g' (xa' + x') = pile B So, the pile B is made of g' size chunks AND pile A is also made of g' size chunks! A common divisor for A and B! What's the largest common divisor for A and B? => The GCD(A, B) = g Hence, g' = g, the original GCD of A and B!
Came here after Leminno video about Kryptos. Nice video! And the puzzle was fun, althought at first I didn't know what to do with the fact that last row is incomplete. But when you think about it, it becomes more or less obvious.
7:25 once you dropped down into the origin my brain immediately made the connection between the inverse square law & what was being talked about previously
Hello Dr. Kate. Loved this video. Previously, I was convinced that on the interval [0,1) the number of reals was equivalent the number of whole numbers and that Cantor's diagonal proof was junk. Your video changed my mind about the former. :) Now the proposition that .999-repeating is identical to 1 has re-entered my life. As I currently understand this video, they aren't equal for .999-repeating never intersects 1.0 in the graph. (But, then again, thinking as I type this, It never passes 1-0 either.) For very-part-time, recreational, arm-chair mathematicians like myself, could you maybe, please, discuss .999-repeating and how it is or isn't the same as one? Thanks much if you do. (Pretty much the same if you take a pass.) In either case, thanks for the video.
Youre a wonderful teacher. I mean it. You made it very suggestive what the answer is so that I could come up with it myself. Brilliantly done and I bet you - now it is mine forever!
This also sort of explains why the golden ratio φ is like the “most irrational number” - its continued fraction ‘address’ consists of only 1s - so all the rational approximations are similarly bad! Since the coefficients are always natural numbers, 1 is the worst possible! Edit: fixed it’s instead of its.
🎯 Key Takeaways for quick navigation: 00:01 📏 *Introduction to real numbers and their representation.* 03:10 🧮 *Decimal expansion, limitations, and notable approximations.* 04:57 🎈 *Visualizing rational numbers, Reuleaux theorem, and their relationship.* 07:25 🌐 *Rational numbers explained through projective geometry.* 09:35 🛣️ *Fairy subdivision, its comparison to decimals, and pi's continued fraction.* 12:57 🥧 *Discovering Pi's true nature via its continued fraction expansion.* Made with HARPA AI
