GraphicMaths
GraphicMaths
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1.9 Limits pathological case Dirichlet function
If you enjoyed this video you might be interested in my book Calculus on Leanpub (leanpub.com/calculus-book) or Amazon (amazon.com/Calculus-Martin-McBride-ebook/dp/B0DLBH5Y5M)
In this video we will look at the Dirichlet function, an unusual function that takes the value 1 if x is rational, and 0 if x is irrational. We will see that the function is discontinuous everywhere and has no limit anywhere, making it an interesting pathological function for limits.
Take a look at the Calculus playlist for more videos on the same topic: ua-cam.com/play/PLblmfcIBkAwpeRJoXHduQ2X8c48kt7GLq.html
Please join my Substack newsletter to be notified about new videos and articles: graphicmaths.substack.com
00.00 - Introduction
00:30 - The Dirichlet function
01:04 - Rational and irrational numbers
04:01 - Definition of a limit
04:55 - Dirichlet function has no limit anywhere
See also:
ua-cam.com/video/-OTVYOWd3pg/v-deo.html
This video was created using generativepy, a free and open source Python library for creating mathematical art and animations, available from github.com/martinmcbride/generativepy
Переглядів: 68

Відео

1.8 Limits pathological case sin(1/x)
Переглядів 100День тому
If you enjoyed this video you might be interested in my book Calculus on Leanpub (leanpub.com/calculus-book) or Amazon (amazon.com/Calculus-Martin-McBride-ebook/dp/B0DLBH5Y5M) In this video we will look at the concept of limits in calculus. We will look at how limits work with various simple functions, together with the definition of a limit. We will also see several situations where a limit do...
1.5 Limits in calculus
Переглядів 12221 день тому
If you enjoyed this video you might be interested in my book Calculus: www.amazon.co.uk/Calculus-Martin-McBride-ebook/dp/B0DLBH5Y5M In this video we will look at the concept of limits in calculus. We will look at how limits work with various simple functions, together with the definition of a limit. We will also see several situations where a limit does not exist. Take a look at the Calculus pl...
Coin rotation paradox
Переглядів 1,1 тис.6 місяців тому
The coin rotation paradox If you enjoyed this video please sign up to my substack newsletter to be notified of new videos and articles: graphicmaths.substack.com Imagine we have 2 coins. The radius of the larger coin is 3 times the radius of the smaller coin. We roll the smaller coin all the way around the edge of the larger coin, without it slipping. The question is, how many times will the sm...
Circle theorems - the opposite angles in a cyclic quadrilateral add up to 180°
Переглядів 1,2 тис.Рік тому
Circle theorems - the opposite angles in a cyclic quadrilateral add up to 180°, with proof If you enjoyed this video please sign up to my substack newsletter to be notified of new videos and articles: graphicmaths.substack.com You can help to support this channel on Patreon: patreon.com/graphicmaths A cyclic quadrilateral is a four sided shape where all four vertices lie on the circumference of...
Why does complex number multiplication cause rotation?
Переглядів 869Рік тому
Why does complex number multiplication cause rotation? If you enjoyed this video please sign up to my substack newsletter to be notified of new videos and articles: graphicmaths.substack.com You can help to support this channel on Patreon: patreon.com/graphicmaths Complex multiplication of a and b has the effect of rotating b through angle that depends on a. In this video, we will investigate a...
Newton fractal
Переглядів 244Рік тому
Newton fractal See my book Calculus leanpub.com/calculus-book You can help to support this channel on Patreon: patreon.com/graphicmaths The Newton fractal is a fractal based on the Newton-Raphson method applied to functions in the complex domain. In this video we will look at the Newton-Raphson method itself, how to differentiate functions in the complex domain, and how to apply the Newton-Raph...
Why are sin, cos and tan called sin, cos and tan?
Переглядів 2,6 тис.Рік тому
Why are sin, cos and tan called sin, cos and tan? You can help to support future videos on Patreon: patreon.com/graphicmaths Where do the names of the trig functions come from? This video explains where the terms sine, tangent, and secant come from, and how they relate to the geometry of the unit circle. It also explains the origins of name cosine, cotangent and cosecant, and why the inverse fu...
Differentiation from first principles, x² example
Переглядів 122Рік тому
Differentiation from first principles, x² example See my book Calculus leanpub.com/calculus-book This video describes differentiation from first principles. It explains how differentiation from first principles works, including an example of how to differentiate the function x squared. It also includes a geometric proof of the derivative of x squared. Suitable for UK A Level, IB Diploma Program...
Differentiation - slope of curve
Переглядів 729Рік тому
Differentiation - slope of a curve. See my book Calculus leanpub.com/calculus-book This video is an introduction to differentiation. It looks at slopes of straight lines and how they relate tangents of curves. It also discusses methods of performing differentiation, from first principles or by using product, quotient and chain rules. Suitable for UK A Level, IB Diploma Programme, US grades 11 ....
What is 2 to the power pi?
Переглядів 854Рік тому
What is 2 to the power pi? Good and bad approximations. See my book Calculus leanpub.com/calculus-book This video explores the idea of raising a number to an irrational power. It takes the example of 2 to the power pi? The normal rules of exponents only apply to rational powers. It is not obvious how these rules should be applied to an irrational exponent. We look at the idea of using a rationa...
Similar triangles proof
Переглядів 843Рік тому
Similar triangles - proof of AA and SSS rules You can help to support future videos on Patreon: patreon.com/graphicmaths This video provides proofs of the AA and SSS rules for similar triangles. Two triangles are similar if they are the same shape but not necessarily the same size. Ths means that all three corresponding angles of the two triangles must be equal, and all three corresponding side...
Cosine rule
Переглядів 15 тис.Рік тому
The cosine rule You can help to support future videos on Patreon: patreon.com/graphicmaths This video explains the cosine rule. The cosine rule can be used to solve any triangle where either two sides and the enclosed angle are known, or where three sides are known. As a result of applying the cosine rule we will know three sides and one angle. The sine rule can then be applied to find the rema...
Similar triangles
Переглядів 1 тис.Рік тому
Similar triangles
Maclaurin expansion of exponential function
Переглядів 482Рік тому
Maclaurin expansion of exponential function
Maclaurin expansions
Переглядів 121Рік тому
Maclaurin expansions
Newton-Raphson method
Переглядів 793Рік тому
Newton-Raphson method
Similar shapes
Переглядів 701Рік тому
Similar shapes
Einstein's proof of Pythagoras theorem
Переглядів 6 тис.Рік тому
Einstein's proof of Pythagoras theorem
Circle theorems - angle at centre of circle is twice angle at circumference
Переглядів 1,2 тис.Рік тому
Circle theorems - angle at centre of circle is twice angle at circumference
Negative and fractional exponents
Переглядів 61Рік тому
Negative and fractional exponents
Rules of exponents
Переглядів 129Рік тому
Rules of exponents
Parabola focus and directrix
Переглядів 218Рік тому
Parabola focus and directrix
Circle theorem - two radii form an isosceles triangle
Переглядів 993Рік тому
Circle theorem - two radii form an isosceles triangle
Proof of Pythagoras' theorem
Переглядів 105Рік тому
Proof of Pythagoras' theorem
Pythagoras theorem
Переглядів 139Рік тому
Pythagoras theorem
Rotational symmetry of polygons
Переглядів 2,5 тис.Рік тому
Rotational symmetry of polygons
Straight-line graphs
Переглядів 124Рік тому
Straight-line graphs
Circle theorem - the angle in a semicircle is a right angle - with proof
Переглядів 2,8 тис.Рік тому
Circle theorem - the angle in a semicircle is a right angle - with proof
3D geometry - cubes
Переглядів 173Рік тому
3D geometry - cubes

КОМЕНТАРІ

  • @DrTWG
    @DrTWG 5 днів тому

    Thanks I'm trying to improve my maths knowledge - this was very clear . I don't know why it matters in real life !but you can't have everything !

  • @DonkeyLord113
    @DonkeyLord113 9 днів тому

    ngl this video made me understand cosine rule. I never understood it until now which was a BIG issue since I'm doing methods next year. mr graphicmaths7677, you're an absolute lad

  • @mailtoramu
    @mailtoramu Місяць тому

    Clearest video on differentiation I could find. Explains about slope from the basics and even has a part two with detailed explanation. Good one

  • @deankaraniya7422
    @deankaraniya7422 Місяць тому

    Your videos are great.

  • @bhkti821
    @bhkti821 Місяць тому

    Acha beta

    • @kkjj5127
      @kkjj5127 8 днів тому

      Tu yaha bhi aa gya

    • @bhkti821
      @bhkti821 8 днів тому

      @kkjj5127 ha beta

  • @BtMeszinet
    @BtMeszinet 2 місяці тому

    I can not see the PROOF in this video.

  • @arjitgupta9622
    @arjitgupta9622 3 місяці тому

    You are the first one to explain this correctly in the best way ❤thank you. Stay on the ways, god is with you 😊

    • @davido3026
      @davido3026 3 місяці тому

      God, written with G

  • @Pixel_GmD
    @Pixel_GmD 4 місяці тому

    Holy crap is that a geometry dash reference?

  • @chrisg3030
    @chrisg3030 4 місяці тому

    Lovely and clear. A couple of interesting variations: 1) the small circle does one less rotation when rolling round the inside edge of the big one. One consequence is that when the two circles are equal then the "inside" one can't roll, it does 1 - 1 = 0 rotations. 2) You can also have a small square rolling round a big one .

  • @VikingDjburnt-qz2uk
    @VikingDjburnt-qz2uk 4 місяці тому

    Nice lesson

  • @gangstaaa7837
    @gangstaaa7837 4 місяці тому

    thanks

  • @DorahMooketsi
    @DorahMooketsi 4 місяці тому

    Thank you thank God I got it right 🙏🙏🙏

  • @IsmailAkhrif
    @IsmailAkhrif 5 місяців тому

    Thank you teacher for your efforts to explain mathematics on its origins, I just have finished my bachelor degrees in mathematic science in Morocco, and in my free time I love to understand mathematic theoretically, as a result of this, I found your channel by chance.

  • @seth111yta1
    @seth111yta1 5 місяців тому

    @5:45 so you're saying the *true reason* complex number multiplication produces a rotation is because, at the end of the day, it is "arithmetically equivalent" to a matrix transformation? (great vid btw you should have more views)

  • @TinaWollenkowski
    @TinaWollenkowski 5 місяців тому

    What's the difference between a normal geometric kie and a right kite?

  • @TinaWollenkowski
    @TinaWollenkowski 5 місяців тому

    I saw how you changed the form of the trapezoid. One of them is the obtuse trapezoid which is one with a obtuse angle. The same way with the parallelogram by making it lean to the left or to the right.

  • @TinaWollenkowski
    @TinaWollenkowski 5 місяців тому

    You didn't mention the rhomboid which is another word for the parallelogram.

  • @Cal97g
    @Cal97g 5 місяців тому

    Thanks!

  • @wasgehtsiedasan8660
    @wasgehtsiedasan8660 6 місяців тому

    I don't get how you get from C=A+B to c^2=a^2+b^2.

    • @bilaalahmed8923
      @bilaalahmed8923 6 місяців тому

      he said C= c^2, A=a^2 and B=b^2 in the first place, assigning them to respective places results a^2+b^2=c^2

    • @BuleriaChk
      @BuleriaChk 5 місяців тому

      Neither does he. (Binomial Expansion for the case n=2) And thus proof of Fermat's Last Theorem for n > 1.

  • @BuleriaChk
    @BuleriaChk 7 місяців тому

    c=a+b c^2 = (a+b)^2 = [a^2 + b^2] + [2ab] (binomial expansion) c^2 <> a^2+b^2 The "proof" in the video is only valid in the imagination. (Pythagoras was also confused).

    • @nguyenn7729
      @nguyenn7729 7 місяців тому

      You are confusing capital C = A + B with lowercase c = a + b. Clearly someone else besides Pythagoras is confused

    • @BuleriaChk
      @BuleriaChk 7 місяців тому

      What does capital C have to do with it? C = A + B C^2 = (A+B)^2 = [A^2 + B^2] + [2AB] C^2 <> [A^2 + B^2] C = A + iB C* = A - iB CC* = A^2 + B^2 only in the imagination. Note that A = 4. B=5 CC* = 25 <> 49 49 = 7 ^2 = [49] = [25] + [24] = [CC*]+ 24

    • @nguyenn7729
      @nguyenn7729 7 місяців тому

      The Pythagorean theorem, a^2 +b^2 = c^2 refer to side length lower case a,b,c of a right triangle. Capital A,B,C in this context refer to the areas of squares with side lengths a,b,c respectively. The areas of such squares are clearly a^2, b^2, and c^2 respectively. Hence it suffices to prove capital A + B = C.

    • @nguyenn7729
      @nguyenn7729 7 місяців тому

      And you dont even understand imaginary numbers. If c = a + ib is a complex number, what you have described is the norm of c not c^2. c^2 = c*c in the normal understanding of powers.

    • @BuleriaChk
      @BuleriaChk 7 місяців тому

      @@nguyenn7729 The hypotenuse of the right triangle is sqr(cc*) (i.e., 5)

  • @zayan6924
    @zayan6924 8 місяців тому

    THANKK YOUUU 💪💪💪💪💪💪!

  • @levizst9458
    @levizst9458 9 місяців тому

    thank you!!!

  • @fun3721
    @fun3721 10 місяців тому

    Good stuff

  • @kiranjain3290
    @kiranjain3290 10 місяців тому

    Thank you for explanation

  • @theclipreaper
    @theclipreaper 11 місяців тому

    So did I get this straight: complex numbers do a specific matrix multiplication "in the background" and in a way that's more intuitive for us humans to understand?

  • @dishwasherdog
    @dishwasherdog 11 місяців тому

    damn u have a nice voice

  • @HassanDinbalidinbalihassan

    اسناد خیلی احمقی متوجه نشدی عدد پی غلط است

  • @HassanDinbalidinbalihassan

    استاد خیلی ابلهی متوجه نشدی عدد پی غلط است

  • @pelasgeuspelasgeus4634
    @pelasgeuspelasgeus4634 Рік тому

    I have a silly question: how do you know that imaginary axis is vertical to the real axis?

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      It's not a silly question at all. Possibly the simplest hand-waving explanation is that real number and imaginary numbers are different things, so every complex number is a unique combination of a real and imaginary number, and vice versa. A standard way to represent that sort of idea is to use orthogonal axes. Also, viewing complex numbers that way turns out to be very useful, and it doesn't lead to any contradictions, so it is very convenient to treat it as being true. It is a good way to understand Euler's formula, and multi-valued complex roots. There are probably deeper reasons too.

    • @pelasgeuspelasgeus4634
      @pelasgeuspelasgeus4634 Рік тому

      ​@@graphicmaths7677So, it's an assumption. And, why do you treat complex numbers, which are points, as vectors? I mean that multiplication is common in vector analysis.

    • @user-ug2ty8rh9f
      @user-ug2ty8rh9f 9 місяців тому

      @@pelasgeuspelasgeus4634 multiplying by i rotates by 90 degrees

  • @im_misfar940
    @im_misfar940 Рік тому

    Thanks alot❤

  • @sidahmedbensaad1838
    @sidahmedbensaad1838 Рік тому

    Thanks, can you give the example when newton Raphson can't converge.

  • @johnpolo7160
    @johnpolo7160 Рік тому

    I like the way to study some important ideas by comparison. Not so many people can explain it this way. You sir did it in a very nice way which is very important mainly for newcamers studying this subject

  • @ophelia_xmaryx8506
    @ophelia_xmaryx8506 Рік тому

    Very very helpful thank you

  • @dozsy_sy
    @dozsy_sy Рік тому

    Hi qah

  • @jakobthomsen1595
    @jakobthomsen1595 Рік тому

    Very nice!

  • @SzymekCSC
    @SzymekCSC Рік тому

    I don't understand the formula of that tangent.

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      There is an article covering the same topic on my website graphicmaths.com/pure/numerical-methods/newton-raphson-method/ It is the same derivation, but maybe it will be clearer in written form rather than video. The basic idea is to calculate the slope in two different ways. One is based on slope of the line AB, which is simply the change in y divided by the change in x. The other is that the slope at A is equal to the first derivative of the function at that point. Since these two ways of calculating the slope must yield the same result, we can equate the two values and solve to find the position of B. Hope that helps.

  • @trdway5887
    @trdway5887 Рік тому

    Good to explain ✅

  • @graphicmaths7677
    @graphicmaths7677 Рік тому

    Find more circle theorems in the playlist ua-cam.com/play/PLblmfcIBkAwpLUNFaQLwS65Lwu-oJCcG_.html

  • @miguelcerna7406
    @miguelcerna7406 Рік тому

    Quality content.

  • @graphicmaths7677
    @graphicmaths7677 Рік тому

    There is now an article about regular tessellations on the graphicmaths website graphicmaths.com/gcse/geometry/regular-tessellations/

  • @richardDara-ot9zp
    @richardDara-ot9zp Рік тому

    Second viewer great info Subbed 💪🏼

  • @xshadowfistx
    @xshadowfistx Рік тому

    I can tell if you watch this vid

  • @yiutungwong315
    @yiutungwong315 Рік тому

    π = 2 in Riemann Paradox And Sphere Geometry Mathematical Systems Incorporated ❤😂🎉😢😮😅😊 So 2 ^ π = π^π = 2^2 = π^2 = 4 = Tau.

  • @graphicmaths7677
    @graphicmaths7677 Рік тому

    The equations in the video can also be found at graphicmaths.com/gcse/trigonometry/cosine-rule/

  • @rowennacu5529
    @rowennacu5529 Рік тому

    What About The Penrose Dart Or Arrowhead, It Has An Reflex Angle. Also A Kite Is A Penrose Too.

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      This video is mainly aimed at UK GCSE level or equivalent, so it only covers certain shapes. But Penrose tiling is an interesting topic, I might cover it in a future video.

  • @saenzmath
    @saenzmath Рік тому

    Very good visual way of explaining.Thank you for sharing

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      Thanks!

    • @leif1075
      @leif1075 10 місяців тому

      ​@@graphicmaths7677How is it known for sure Einstein came upmwotjbthis and not someone else? Is there written proof or something? Thanka for sharing

  • @StephRenee812
    @StephRenee812 Рік тому

    I am trying to draw my own patterns for a quilt I want to make. MILLEFIORI quilt is an example. How to get, say, about 6 to 12 shapes and these all can be put together like a mandala. I've been trying to figure out how to hse my drawing compass for all shapes or even a hexi and draw shapes inside that go together like a kelodoscope. Idk.

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      I use generativepy (Python software) to create these videos. One idea for creating mandalas is to make some circular patterns of points, and use a Voronoi diagram to create regions. There is an article about it here: martinmcbride.org/post/2021/voronoi-diagrams-with-scipy/

  • @dunno418
    @dunno418 Рік тому

    Hey can you make visual proof of derivatives of trigonometric functions? You can see 3blue1brown video on it There seems to be confusion, as why some steps were taken and as such. Please make proofs for all trig functions!

  • @dunno418
    @dunno418 Рік тому

    Thanks!

    • @graphicmaths7677
      @graphicmaths7677 Рік тому

      You're welcome. I hope to add more circle theorem videos soon.

  • @lynnmay6436
    @lynnmay6436 Рік тому

    Love this guys videos . Need likes? -> *PromoSM*!!