Probably Tai-Dinae's best video yet, talking slowly yet fast enough to be appealing and sound enthusiastic. We all struggle starting anything new and it takes us a while to get the hang of things, not to mention Kelsey left big shoes to fill. But kudos Tai-Dinae, you're becoming really good. Onward and upward!

@Kalumbatsch

6 жыл бұрын

It's all based on feedback, equilibrium will be reached when there are as many posts asking her to slow down as there are posts asking her to speed up. For everyone who still isn't satisfied, they can try the buttons.

@SuperLkelley

6 жыл бұрын

Fully agree. Best one yet. Right pacing, right simplicity/detail ratio. Very clear throughout. Nicely done.

@MichaBerger

6 жыл бұрын

You can't make everyone happy. I would have preferred something faster moving.

@ARVash

6 жыл бұрын

That purpose of that phrase is not so you can air your grievances.

@paulthompson9668

6 жыл бұрын

Micha Berger I agree. I have to put her on 1.25x speed to get it to *feel* right for me. As for presenting the content, I think she did well. I would, however, like to see where all of this is going. For example, will they start to talk about the relationship between group theory and physics?

There was a lot of criticism in your last few videos, but this one was preety good, it didnt feel like there were any hand-wavy analogies, or statements left unexplained, so I'm glad the things people were complaining about were adressed, the only thing I thing I could say about this video is that, for people who never came into contact with modular arithimetic, talking about the mathematics of rotating regular polygons first would help motivate the video better, but talking about it later works too. Preety awesome video.

@martinkuffer5643

6 жыл бұрын

I completly agree!! It was really good

@someoneonyoutube8622

3 жыл бұрын

Although something that is unclear is how modular arithmetic handles values that cannot be evenly divided amongst the whole numbers, you know the irrationals, some of which are quite important to mathematics. Φ, e, π, √2, ... Just to name a few

@BenReillySpydr1962

2 жыл бұрын

@@someoneonyoutube8622 They'll explain it later

6:05 - I'm feeling the urge to tuck that yellow blade behind the red one.

@trefmanic

6 жыл бұрын

Phil P IKR, that asymmetry made this video quite less enjoyable.

@AashishVishwakarma

6 жыл бұрын

It was intended you know

@fakefirstnamefakelastname8305

5 жыл бұрын

Aashish Vishwakarma r/woooosh

@kaidwyer

4 жыл бұрын

They put the blade in front so that one blade would stick out as a pointer to a specific location on the modular pinwheel analogy Though I agree it’s an eyesore - like fingernails on a chalkboard but for your eyes

@blizzbee

4 жыл бұрын

lol

I'm sure somebody already mentioned this, but there IS something special about the number 5. It's a prime, and so Z/5Z is a field!!

This yt channel was a gift

By the title of the video I thought that it would talk about analysis (and in particular limits). I guess nobody expects the ring inquisition...

@killianoshaughnessy1174

6 жыл бұрын

no one ever expects the ring inquisition.

@ironicdivinemandatestan4262

6 жыл бұрын

we'll make you sit on the comfy torus

Group theory hype! Using multiple videos to build to the cool topics is definitely the way to go.

This is one of the most accessible introductions to what is essentially group theory I've come across. Kudos to everyone involved in writing the script for this episode, and to Tai-Danae Bradley who seems to be the main writer and presenter. Amazing usage of English to bring an otherwise tricky/difficult topic to a broader audience. Cheers! :-)

There's a typo at 4:34. It says "-9-6" when it should instead say "-9~6".

@jrhprs

3 жыл бұрын

same here i spotted it.

I just want to take a moment to say 'thank you' for managing to come back with such a video after massive amounts of criticism last time

Two hosts makes a great deal of sense for splitting the workload.

I feel like you get a lot of critics claiming you were misleading in the title without thinking about the meaning of it. Thank you for your explanation of this topic! I loved it!

This is one of the best vid from Tai-Danae. Thumb up. Good to explain throughly more easy pieces then build up towards the complex things.

This is really cool! I just discovered this channel and now I want to binge all the videos.

Nice video! Thank you for breaking down and explaining various terms that we will need to get into group theory. I am reasonably familiar with the mechanics (but not the theory) of mod arithmetic, so I found this to be a great intro with examples I understood. Thank you. (Since I really struggled with your last episode and was utterly lost at your follow-up, I particularly appreciated this one. Please, keep it up!)

@pbsinfiniteseries

6 жыл бұрын

Glad to hear it was helpful!

I'm disappointed that you didn't actually talk about division by zero, but rather the equivalence of the remainder of zero. Division by zero is an amazing topic.

@letstalkaboutmath2121

6 жыл бұрын

i'm little disappointed too. I heard of some kind of algebraic structure in which is possible to divide by 0, and i hoped was something like this. The topic is nice anyway, but is simply something that i already know

@terryendicott2939

6 жыл бұрын

In the integers mod 6 (for both addition and multiplication), both 3 and 2 are called zero divisors because 3*2 = 0. If you look at the integers mod 12 you get 2,3,4,6.8 and 9 as zero divisors.

@pbsinfiniteseries

6 жыл бұрын

Request for *actual* division by zero is noted!

@pbsinfiniteseries

6 жыл бұрын

Ah, so now the question is: For *which* n do 'the integers mod n' have zero divisors? And *why*? :)

@romajimamulo

6 жыл бұрын

Terry Endicott however, that's not division *of* zero, not division *by* zero. As for places you can divide by zero, they can't have additive inverses or distribution, because then you'd be saying that there exists a and b a×b=c+a×b and c is not zero. Showing of that down below c=0×b=(a+(-a))×b=a×b+(-a)×b c+(-(-a×b))=a×b c+a×b=a×b

Thanks for making this videos, I'm loving it

Best Tai-Danae video ever till now. Good job

Hey there! I like the new videos! You're laying down some basics to work yourself up to bigger subjects in the future. I like where this is going. It's different from many of Kelsey's videos which for me, as a total amateur, often existed a little bit in a vacuum, and often came across as "curiosities from the world of mathematics that Nuberphile didn't cover yet". Which was fun, don't get me wrong, but I love that you seem to be shifting gears and start to cover the "branches of mathematics", to really get an idea of what different fields are and what these tools even do exactly. Sort of similar to the shift in SpaceTime, but for me, from a different prespective. I don't know much about math, but I'm pretty well versed in pop-astronomy. So the "curiosities from the world of astronomy and phyics that Sixty Symbols and Deep Sky Videos didn't cover yet"-episodes were great, and I knew enough about the subject to really understand and appreciate what was going on. But when they started to cover relativity, black holes, cosmology, quantum mechanics, not just by showing off some bizzarre facts, but really explain from the ground up, over many episodes, why those things are that way and behave that way, that.. that's when shit got real good. And I kind of hope that's what's happening here. Covering these different ways of doing maths, getting a feel for what those tools even are exactly, what you can use them for, what they (can) represent, how they related to one another. I'm used to think in abstract terms as a programmer, but I never got into math the way I should have, and I hope this might change this years with a little help from you guys! And don't let the hive bring you down! People don't like change, and it's not just this new job, but this (kinda) new direction for the channel that will take some time for everyone to get used to. And you didn't do anything wrong that the dudes are doing all the time too ;) Have fun! I'm exited for many more episodes!

What an excellent presentation of the subject - very cool. Thanks

The description should credit Tai-Danae instead of Gabe for this episode.

@pbsinfiniteseries

6 жыл бұрын

Thanks for catching this! Fixed!

that was so beautifully explained

I love watching you guys! Great team! PBS is the best!

Thanks Tai for taking all the previous criticism into consideration. You were amazing this time. Please keep it up, good work. And good luck with you're PhD (if that's what Gabe meant).

Okay so I just found out Gabe is here now. Thought I wouldn't hear him again, today is a great day! And I love Tai-Danae's style as well, no matter what bullshit people are giving her in the comments, it seems. To all of you on Infinite Series, keep up the great work !

Really liked this one, specially the glimpse into some further abstractions that I hope to partially understand, if they are as carefully presented as this video.

Excellent video, very interesting and well presented!

Vortex Math. This stuff changed perspective on numbers forever

Great episode !

Great video, Tai.

This is gonna be a great video :D

Great video!

Group theory gets used a lot in Raman spectroscopy for inorganic chemistry. Very cool stuff.

Good video and explanation. The description of how the nℤ set behaves just like zero was very clear and sufficiently detailed. Just wanted to provide some words of encouragement to Tai-Danae.

YAAASSS GROUP THEORY !!.. Also category theory is cool too

This is really cool!

I had hoped this video would be about the Projectively Extended Real (Number) Line, which -- by making (-∞) and (+∞) the same thing -- allows for "proper" division by the number 0. I was rather disappointed. Still, despite the (potentially) slightly clickbait-ish video title, it's a decent explanation of Modular Arithmetic.

Interesting, thought provoking

I love the female host, she has a really calm and collected voice, and does a great presentation.

Very interesting interpretation of "division by zero"! I thought you were gonna take the Ring Theory approach and show that if you assume divisibility by 0, the ring collapses to the trivial ring.

Ooo, I want to learn about group theory! And graph theory, and number theory... So many theories I want to learn!

Great topic to dive into, I'm very excited in which direction you want to go forward! :) Just a minor nit-pick: Explaining quotient sets as a way to divide by zero might seem a little confusing. In algebra, you typically cannot divide by zero, not even by the equivalence class of zero in Z/5. What you are basically doing, is "making" the number 5 to be the same as zero, which may seem even more confusing at first I admit^^ but it really comes with a nice intuition of "gluing" the number line into a loop by simply declaring 5 to be the same as zero.

@pbsinfiniteseries

6 жыл бұрын

Great point! The division analogy was motivated by a desire to help folks make sense of the fractional notation “Z/nZ”, as well as the terminology behind "quotient" sets/groups/etc. (Once you first learn about these things, it’s quite natural to wonder, “Why the allusion to division?”) But the “gluing” analogy is absolutely the key. Incidentally, that’s exactly where Gabe will pick up next time! -Tai-Danae

@DucciVinci

6 жыл бұрын

Wow thank you, I didn't expect an answer that quickly *g* I just don't really see the analogy to "dividing by zero", but again, as long as your viewers understand what you mean that's perfectly legitimate. I have myself often sucked at explaining these kind of constructions to non-mathematicians^^ When we speak of "dividing out" things, in algebra we almost always mean forcing these things to become zero. It's a little bit like teleportation when you think about it^^. This is fantastic, but just to be clear to your other viewers (not Tai-Danae of course): In the normal integers, you cannot divide by zero. In Z/5Z, not only can't you divide by zero, but now you cannot divide by 5 either. Well, that didn't help, you might say. ;) But other divisions may suddenly become possible! When you multiply 2 numbers which both lie in the equivalence class [3], the result will always be in the equivalence class [4] (very easy to check). So in a sense, we can write [4] / [3] = [3] (because that is what division means), and note that the number 4 itself is not divisible by 3 to begin with! By declaring 5=0, we lost some things (division by 5, also the sense of size of numbers), but we also gained a lot (all divisions except divisions by the new zero make sense, which is not the case in the ordinary integers). Those are exactly the trade-offs mathematicians have to do all the time^^ and finding the right trade-off is an art.

@kindlin

6 жыл бұрын

I don't see the divide by zero myself, more like "divide by something that is, for all intents and purposes, zero while considering this specific type of modular arithmetic." It just doesn't have the same kind of ring to it. EDIT: Wtf? Arithmetic is spelt with an E? Who turned MathemAtics into ArithmEtic?

@DucciVinci

6 жыл бұрын

Declaring something to be, for all intents and purposes, zero is completely fine. I just wanted to point out that you cannot actually "divide" by these elements that you just declared zero. This construction is very important, but it is not technically division. It is called "dividing out" because in the case of finite groups, the sizes get divided: when you for example take Z/10Z (just the equivalence classes of digits 0 to 9) and you now want to further declare 5=0, you divide out a group consisting of only two elements (0 and 5) and you are left with Z/5Z. So you started with 10 elements, then divided out 2 elements and were left with 5. The sizes get actually divided, that's where the word "dividing out" comes from. Division by zero (or the things you just declared zero) still isn't allowed though. The most popular way to actually divide by zero is through differentiation, which is also possible in topological rings like the p-adic numbers, but let's not dive into that now^^

@kindlin

6 жыл бұрын

Well, in this case, it seems like 5/[0] is defined as... actually,. I got nothing. I understand how division works to create these sets, I don't know how to interpret the creations logically.

It's time for bed... I will probably be asleep by the end. Thanks for making my brain fall asleep (in a good way).

So clear.

Good video thanks:)

Abstract cry concept made delightfully interesting saying ..thanks is understatement..

google -> "This paper solves the problem of division by zero. Starting with a deep analysis of multiplication and division, it is presented that they are one in the same, as an operation. Together with the operation of selection, they are different forms of the transformation operation, that changes one pair of numbers into another pair of numbers. It is presented that the numbers are always in reference to some other numbers. Therefore, the only correct form of the number, is when it is a ratio between the certain value and the base measure, which this value is related to. It is clearly proven that the problem of division by zero is the result of unauthorized simplification, which was done by bringing all the rational numbers (which are represented by the ratio of the value to the measure) to the fractions, with a denominator of 1."

I'm glad to have gave back... Kind of gave up on understanding spacetime though. I like to play it in the background sometimes but the topics aren't easily accessible...not that they're supposed to be. But I remember gave did a really good job talking about relativity...had to watch the videos twenty times but it was understandable and it made it a lot of fun. I also enjoyed the ratio video and am gonna give tai a couple videos to win me over. The other girl left big shoes to fill but I like what I hear so far...kinda. I'll have to watch again I got kinda bored cuz it sounded like the way you divide by zero is that you don't...

ooo, is this leading toward Andrew Wiles's proof of Fermat's Last Theorem and the modularity theorem for elliptic curves?

Gives AWESOME headache 😅❤

@1:55, I was wondering, why do we go all the way up to four? Any integer would be off by 2 max right? Take 14 for example, 10 + 4, or 15 - 1, so 1 integer off is already sufficient. Similarly, 8 can be described as 5 + 3, or 10 - 2, where 2 off would be sufficient. Would do we continue?

@pbsinfiniteseries

6 жыл бұрын

I love this question! TimeMasterII asked something similar (also using 14 as an example), and I've just posted my answer there. Take a look! -Tai-Danae

also, mod math can be used to translate a 1d set into 2d. where n%m, (int) n/m is the x, y coordinate

These kind of videos that are year 1 level are what you should've started with. Work your way towards the more complex stuff first. Also, the title is a little misleading; equivalence class of integers congruent to 0 is not really something most people picture when thinking about division by 0.

@jasondoe2596

6 жыл бұрын

the title is *very* misleading

@pascalwuyts8998

6 жыл бұрын

Too much misleading divide by 0 is like not dividing but the whole idea of or math needs to be changed a bit. As the theory of numbers is incorrect

@tonysparks1682

4 жыл бұрын

Yo check out the comment I made. I actually divide numbers into the actual number zero and use algebra to the numbers back out again. Tell me what you think

I have never looked at the number 0 as i used to thx to this video.

This 5z thing is basically describing the ones digit of base 5. If you did 10z and took the number 100,456, you'd have [6], the ones digit of that number.

I think the most interesting thing about quotient sets and equivalence classes is how intertwined they are with the structure you're cutting up. As in, there usually is a very limited amount of ways you can partition a structure and still have your operators `work` there, like in Z/nZ. If you divided Z into classes not given by a set different nZ, the induced operators, the summing the x and y in [x] and [y], wouldn't work. :D It's basically why congruence is important.

I love that moment when I rewatch it and it clicks

indeed librepenseurs, this was fun to see! I take a more ross-littlewood "combinatorialist/representationalist" stance, from non-standard analysis... towards that ratio issue. it is more like the Phi-Rho-Sigma relation, or, as Conway-Coxeter describe... an irregular truncated manifold... which answer depend on your stance to Poincare-Perelman-Fermat eh... it is also related to the E8 structure's p-brane resistance. it is related to the Wall-Sun-Sun, Wolf-Wolstenholme et hyper-brane collision... wieferich, wierbach et some gateaux or siefert prime also. I am interested in it for its self-error-correcting potentials eh. it could work with Wilber, Jung et Galtung numbers, not to mention in fuzzy-logic / grey logic applications...

Great video, but very hard for me to REALLY understand (i.e. to chew thoroughly and not just watch). Can anyone recommend a book with lots of examples and even exercises? Thanks.

i miss this channel 😭

It's funny, i was playing around with the metallic ratios problem while bored in class. Found that sine ratio and thought "yikes". Started looking through every trig identity i could find that could maybe clear it up a little. Not much luck as you might expect.

Guys, I proved that for n>3 there is no regular polygon in which sigma_n is the ratio between a diagonal and a side (actually, it can be easily generalised to show that sigma_n is not the ratio between any two diagonals of this polygon). I sent you the proof but you didn't answer, and I'm really looking forward to getting a feedback. Hope you read this comment (and my proof haha)

Division by 0 without set theory... Take the slope-intercept form y = mx+b of a linear equation where m is the slope defined as rise / run and b is the y-intercept. For all following examples we will assume that b = 0. We will also start with m = 1. Before any examples are shown let's state the following identity properties of basic arithmetic. a+0 = a, a*1 = a, and a^1 = a. With this and our initial condition we can write y=mx+b as y = x. This simple equation without any arithmetic operations that applies some kind of transformation demonstrates identity, equality, and associativity. The expression y = x also has the following properties. It has perfect symmetry and is a point of rotation and or reflection about the origin. It is also a bisector of a right angle. Now we need to understand what this implies and in order to do so we need to understand what m or its slope is in more detail and how it directly relates to the trigonometric functions. Slope intuitively represents steepness or a gradient. It is a ratio of rates of change. We can find the slope of any linear equation by any two points on a given line by the following equation m = (y2-y1)/(x2-x1). We can simplify this as m = dy/dx stated as the change or difference in y with respect to the change or difference in x. How does this relate to the trigonometric functions you might ask? This is a linear equation and not trigonometry... Well it's quite simple. We need to look at the graph of y=x. Then we can choose any two arbitrary points P1 and P2 on that graph. We can then draw a vertical line starting at P1 upto the y coordinate of P2 and label this dy. We can then start at that new point and draw a line horizontally until it intersects the line at P2 and label this dx. Here we can label the angle that is generated by the horizontal line that is above the line y=x and the line of y = x itself. Through basic geometry, this angle is the same exact angle that is between the line y=x and the +x-axis. If we look at the two lines dy and dx we can see that we have created a Right Triangle and with respect to theta dy/dx = tan(t). Let's test this observation. We know that the slope of y = x is 1. We know that the x and y axis are orthogonal and perpendicular to each other as they create a right triangle that is either 90 degrees or PI/2 radians. The line y = x is a bisector to this angle giving us an angle of either 45 degrees or PI/4 radians. Now, what is tan(45) or tan(PI/4)? It is sure enough 1. Knowing this and through trig substitution from one of the trig identities we can also write m = dy/dx = tan(t) = sin(t)/cos(t). This then shows us that there is a direct correlation to (y2-y1) = dy = sin(t)​ and (x2-x1) = dx = cos(t). Both of these trig functions are continuous for all of their domains. They both have the same range and domain. Their limits exist for all values. They are sinusoidal, cyclic, repetitory, rotational, oscillatory, wave, and transcendental functions. There are only two difference between the sine and cosine is that they are exact linear transformations of each other. They are horizontally translated by 90 degrees or PI/4. This means that these two functions are orthogonal or perpendicular to each other. The other difference is that the sine is an odd function and the cosine is an even function. Both functions have a direct relationship and correlation to Right Triangles, the Unit Circle and the Pythagorean Theorem. Now let's use these to examine the tangent function a little closer. When we have zero slope where the plane is horizontal or parallel to the x-axis we have the following condition: m = sin(0) / cos(0) = 0/1 = 0. This makes sense and we can normally accept this. However when we have "vertical" slope everyone throws their hands up and claims it's impossible or "undefined" by example: sin(90)/cos(90) = 1/0. Division by 0. Also within Trigonometry and the graphing of the tangent function this is exactly where we see the vertical asymptote, and for the same reason we state that it is undefined. If we apply a technique that we learn in pre-calc and look at the limits we can see a connection. This will help shed some light into this phenomenon. The limits of sin(t) both right and left for all of its domain exist and are equal and it is also the same for cosine. However when we look at the tangent function we have a problem at 90 degrees or PI/2 and all PI*N iterations of them. Here both the right and left hand limits do exist but they are not equal as the limit towards the right tends to -infinity and the limit towards the left tends to +infinity. They do both exist, however, they are not the same so the tangent function unlike the sine and cosine is not continuous from what we have been taught. What if I was able to demonstrate that this may be a misconception. If we take the absolute value of the limit regardless of how far the limit is approaching either + or - infinity, the limits will then have the same exact magnitude or distance. The only difference between them is their direction or orientation. And the difference in that orientation is 180 degrees. Why 180 degrees? It's quite simple. 180 degrees or PI radians is the angle of a line with respect to or in regards of itself. Also 180 degrees is the summation of the three interior angle of any 2D Flat Planar Right Triangle. While considering the concepts of rotation about a point, symmetry and reflection and under the observation of this perspective, I'd suggest that the tangent function is still continuous to some degree. The problem here isn't that it is undefined as this is a fallacy. The problem here is that the tangent function is ambiguous at this value. If 0/1 is okay for a fraction than its reciprocal must be 1/0 and if they were to be orthogonal to each other then one or the other must be its additive or multiplicative negative. In other words, 0/1 and -0/1 are 90 degrees from each other and -0/1 and 1/0 are 90 degrees from each other. When we rotate the line y = x in a counter clockwise fashion from 45 degrees the change in y or sin(t) tends towards 1 as we approach 90 degrees and the change in x or cos(t) tends towards 0. So how does the tangent at 90 degrees tends towards both + and - infinity? Well the line y = x that is being rotated isn't just in the first quadrant, it is also in the third quadrant and as we rotate this to become a vertical line or parallel with the y-axis we then have no displacement in the x and constant displacement in the y both in the + and - negative direction. The reason division by 0 is ambiguous and not "undefined" is due to the fact that an arbitrary point that is a 0D object has an infinite amount of slopes running through it. You can take any point on its own such as the point (3,5) and from that you can draw an infinite amount of lines that all have different slopes that pass through that point from all variations of y = mx + b were this point is found on those given lines. Rotation, symmetry and reflection. All of this is embedded within the simplest of all equations and the most fundamental expressions within mathematics y = x. And this is the Identity Property that in my understanding also defines Associativity. This is how linear equations, quadratics, all other polynomials, trig functions, exponentials and logarithms, etc. are related. Without being able to equate or assign one thing or another through Associativity mathematics itself would not be possible. (continued...)

@skilz8098

2 жыл бұрын

(...continued) It's more than just the numbers themselves and their relationships and properties and its more than just the operators that can be applied to them. It's the ability to associate equivalence after a transformation has been applied. Your arithmetic operators are just that, a transformation! If 1 did not equal 1 then we could not add one onto itself to acquire 2. This gives us our very first arithmetic equation 1+1 = 2. This also satisfies the linear equation, but it's more than just that. It is also more than just basic arithmetic. 1 + 1 = 2 is also the unit circle located at the point (1,0). Each argument on both sides of the addition operator can be seen as a unit vector which is also an independent radius of the unit circle and their combined magnitude 2 is the diameter of that unit circle. The unit vector itself can be considered the unit circle only after you rotate it by 360 degrees at its tail. And this is why 0D points have an infinite amount of lines all with different slopes intersecting through it. Division by 0 is not undefined, tang(90) or (PI/2) is not undefined. I don't like this terminology as I prefer to declare them as being ambiguous! They are well defined but they do not have function like properties. A function has the property that it only has a single output for each of its individual inputs. When an equation or expression yields multiple outputs for a single input we do not consider this to be a function but rather an equation or an expression as it does not pass the vertical line test due to its ambiguity. Here's the equation of a circle (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center of the circle, (x,y) is a point that lies on its circumference and r is the length or magnitude of its radius. If we look at the equation of the unit circle located at the origin (0,0) we can simplify the previous equation to (x-0)^2 + (y-0)^2 = 1^2 = x^2 + y^2 = 1. If we solve for y we end up with y^2 = 1-x^2 = y = +/- sqrt(1-x^2) and as you can see we have both a + and - on the outside of the radical. Thus the equation of the circle is not a function because it doesn't pass the vertical line test, yet it is continuous! Also if we look at x^2 + y^2 = 1^2 doesn't this look familiar? It sure does A^2 + B^2 = C^2. The equation of the circle is a specialized form of the Pythagorean Theorem and the equation 1+1 = 2 gives us the unit circle located at the point (1,0). So division by 0 is in my interpretation tan(90). They are mutually the same exact thing. How do I come to this conclusion? It's quite simple, the slope of a line in itself is a fraction, ratio, percentage, etc. as it has both a numerator dy and a denominator dx and dy/dx is sin(t)/cos(t) which is also tan(t). All of your fractions 1/2, 2/3, 3/5, 7/8, 15/32, 113/97 etc. are results of the tangent function with a corresponding angle to those fractions in terms of them being a given slope. So what exact is vertical slope and how is it not undefined? Well when you walk on a flat ground such as a parking lol or the beach, you have 0 slope that is represented by sin(0)/cos(0) = 0/1 where you can move only in the horizontal plane. When you walk up hill you have a positive slope where the angle theta is in the range 0 So how do we distinguish the tangent between being + or - infinity? Here it comes down to which limit you are referring to. If you look at its graph, the left limit is +infinity and right limit is -infinity. Also if we look at just the first iteration of the tangent function between [-PI/2, PI/2] we can see that this graph is very close to or similar to that of y = x^3, not exactly the same but similar and has a unique property to it. It is symmetrical across the line y = -x. After this first bounds from [-90,90] or [-PI/2,PI/2] it just repeats as in the same concept above within modular arithmetic. Within physics, music and audio according to wave motions and sound, the sine and cosine functions are what gives us a visualization of sound in motion. And these can construct harmonies as well as dissonance or distortions. And from this we can see wave interference both constructive and destructive. Since both of them are 90 degrees phase shifts of each other and when they are in ratio form sin/cos they have a direct relationship to the tangent. In most cases we don't normally associate sound or music via tangent as we typically will use either the sine or cosine functions but can you guess when wave cancellation happens? I'll leave that for an exercise for you to think about. These are the connections and patterns of what I have observed and nothing here requires the concepts or notions of sets other than y=x with its Associative property and its symmetry implies a set. I believe that mathematics are not rooted from within set theory at all. I tend to believe that mathematics is rooted in Associativity, Symmetry, Reflectiveness, Equality, Assignment and Translations or Transformations. You can not have sets without numbers or quantities or items or objects and you can not have numbers or quantities without having the ability to count or enumerate or the ability to translate or to apply a transformation to some already existing value. From all the years of my studies and understanding, y = x is at the heart of all mathematics. Why? It's quite simple. There can exist no mathematical equation or function without first having an expression and y = x is the identity expression that relates on thing to another stating that they are equal or assigns one value to another thus providing you with an identity between two representations of the same thing. Just as recognizing that division by 0, tan(90), and or vertical slope produce basically the same exact results and are basically the same thing just a different representation or perspective of this conceptual idea. And as you have seen just from a few basic algebraic and trigonometric properties I was able to show you that you can generate every number and function from y=x. Think of y=x as f(x) = x. Here x is the same as x^1. Let's progress: f(x) = x^2, f(x) = x^3, etc... f(x) = x^n. Here we have all of the polynomials... And once you have exponents, you also have logarithms. All aspects of math are have roots in y=x. And all numbers can be either be integrated or derived from the unit vector or integer 1 by applying translations or transformations to it via addition, multiplication, exponentiation, and their inverses.

So coooolll

Excited to understand group theory better. Awesome channel.

Ah, I'm in Ring Theory, and this does feel like a good explanation. And, well, 5Z isn't zero when you divide it, it becomes zero after you're in your new ring. So, a little bit off

@Ennar

6 жыл бұрын

I liked the video, but yeah, you are right. I think it was just a clickbait.

Ms. Bradley your presentations are improving with each one. Keep working. Thanks.

Tai-Dinae, I love it when you spoke faster.

Tai-Danae did a super job. I hope she's doing well.

Awsome stuff. Quite dense, though.

@aaaa-hj9vv

6 жыл бұрын

Honestly this wasn't even close to one of their densest videos. I'd say the videos about homotopies, and sizeless sets were all much harder; everything in this video was covered in the first year of my math undergrad.

@TheYoshi463

6 жыл бұрын

We actually had all of this in our second week :D

@elischlossberg5275

6 жыл бұрын

aa aa same here. I've dealt with quotients nearly every day for the past semester 🙃

I think I finally understand wheel theory.

and now i wished this had come out when i was studying this stuff at uni, now i'm done i only NOW get this stuff having watched the video (I skipped this question in the exam...)

@Ennar

6 жыл бұрын

What did you study at your uni? I mean, this was just the definition. The action of Z/nZ on regular n-sided polygon is usually covered later when dihedral groups are discussed. I personally like it that it was introduced immediately after definition in the video.

I remember vihart once mentioned that there are certain types of calculus where dividing by zero does have an actual meaning and it is, in a way, equal to Infinity. Any idea what she was referring to?

@RandomBurfness

6 жыл бұрын

Complex analysis. Look up what Möbius transformations are.

Tryna watch this shit with no knowledge of what’s going on is crazy man😂 I have never learn any of this I just got into the concept of topology

Make a video on intuitionism/intuitionistic logic

Can anyone tell me how we change our arithmetic operations (i.e. addition, multiplication, etc) when deciding to work with different equivalence classes? I.e. 2 mod 5 + 6 mod 5 will give me [3], but that's assuming we want our answer to be in the mod 5 group. What if I want to represent the answer in the mod 3 group? Then it would be [2]...but that would be confusing. Can anyone elaborate?

Regardless to what everybody said ... Thank You.

I thought the suggestive notation for quotient groups was related to the fact that when both G and H are finite, the size of G/H is equal to the numical quotient of their sizes. I don't think the origin is related to modular arithmetic in the way the video describes.

@brendanmurphy7640

6 жыл бұрын

This is also good to note, but `G/H = G/~` where `x ~ y` iff `xH = yH`, so you are really just quotienting on an equivalence relation

@diribigal

6 жыл бұрын

Brendan Murphy that's true, but equivalence relations came later, and there's less of a connection to division then. See math.stackexchange.com/q/857539/26369 for some history of quotient groups and mathoverflow.net/q/135347/28209 for that of equivalence relations.

I am always surprised how complicated things can become. However, there is a limit to simplicity. And there is a simple, yet not public method, to break RSA encryption. It's so incredibly simple, that it's impossible not to see it. However, it has not been seen yet!

Maybe the metallic ratio proof is related in some way to Fermat's last theorem.

quite enjoyed the video. However, I must say that the pinwheels being asymmetric bothered me to no end. oh well.

Golden ratio and silver ratio is a link between addition and subtraction.

This show is profound

@PBS Infinite Series: Video Request: I recently had to build a concept map, essentially a graph where every node has one or more links to other nodes. To make my graph easy to follow, I tried to arrange the graph such that the lines between nodes overlapped as little as possible. That got me thinking: is there an optimal arrangement strategy to achieve this goal? I started by putting the nodes with the most links in the center, then switched to moving them toward the outer edges, but in both cases I don’t think I achieved the minimum number of overlapping links. Maybe this problem is trivial and I overlooked an obvious solution, but I thought this might make for a cool video idea.

What is a metallic ratio?

Can you please turn on captions for your videos?

I thought this video was about the l'hospital principle :)

Can a imaginary/complex number form a quotient set?

@RandomBurfness

6 жыл бұрын

Sure! If you consider, say, C/(Z + iZ), then you get the set of equivalence classes of complex numbers such that [z] is the set of all complex numbers w such that Re(z-w) and Im(z-w) are integers, where Re(z) is the real part of z and Im(z) is the imaginary part of z. Thinking about it in another way, the equivalence classes in C/(Z + iZ) consist of complex numbers whose real and imaginary parts only differ by an integer.

Having a logarithm of base 1 yields the same results for the exact same reasons as dividing by 0.

Might sound crazy and I have no idea bout this but imagine ya have infinite set of nothing a univers akin to a blank slate imagine having multiple of those now get rid of some would it be labeled as 0 to the power of (insert number here ) or somthing else ? I’m curious please tell meh the answer :)

I watch pretty much all of PBS Digital's shows. This one is the only one that I cannot understand 😵

Shits lit yo

@folf

6 жыл бұрын

weegeniss rival It's gonna be 2,000°F in a minute or two 🔥🔥🔥🔥

@user-iu1xg6jv6e

6 жыл бұрын

*Wolfy The Wolf* 2 hours already passed, and amazingly no body commented on you using F instead of C...

@folf

6 жыл бұрын

ɐɯɹɐʞ ɐıuɐɯ Haha

@folf

6 жыл бұрын

ɐɯɹɐʞ ɐıuɐɯ 4 hours now lol

@completeandunabridged.4606

6 жыл бұрын

Wolfy The Wolf y r u using farenheit not celsius u silly billy

This is essentially coded into the physics of work as well.

This is why 0 is my favorite numbah

So what she is trying to say is that the remainder of x/0 is 0 no ? Or if you have a construct wherein modular arithmetic is sufficiënt to save the day, you don't have to care about x/0 because it''s neatly boxed (under the rug) ?

Its relation and function

...notation should include the modulus because it's not assumably infinity, say [n:5] or n₅... ...odd, we use "/" 'solidus' or '⁄' "slash" for division not subtraction yet you've used it ≡ ℤ−5ℤ where ℤ is the countable infinity of integers and therefore ℤ−5ℤ is the generalized off-set... (i.e. n−5ℤ = n+5−5ℤ because 5ℤ = 5ℤ−5 as infinite set ℤ = ℤ−1 the same set of all integers)

Lady, your each fingers and their joints are *standalone entities!*

Talking speed is good but please slow down a little at the abstract parts

It’s all fun in the decimal system