Understanding the number e | BetterExplained

00:09:15
https://www.youtube.com/watch?v=yTfHn9Aj7UM

Ringkasan

TLDRVideoen præsenterer essensen af den matematiske konstant e, som er central for at beskrive kontinuerlig eksponentiel vækst. Gennem enkle eksempler forklares det, hvordan e er relevant i både naturlige processer og finansiel vækst. Indlægget fremhæver, at e repræsenterer 100% vækst og er unik ved at være en jævn, glat vækstform uden pludselige spring, som ofte ses i naturlige processer - for eksempel cellevækst - og økonomiske mekanismer som renter, hvor både hoved- og renteværdier kan tiltrækkes af yderligere renter. Formlen e^rt (e til kraften af rente gange tid) forklarer fleksibiliteten i at ændre vækstrater over tid. Denne konstante natur gør e til en uundgåelig del af mange matematiske formler givet dens universalitet i vækstbeskrivelse.

Takeaways

  • 📈 e er en universel konstant for eksponentiel vækst.
  • 🧬 Naturlige processer, som cellevækst, følger ofte en kontinuerlig vækstkurve.
  • 💸 I økonomi tillader e beskrivelse af renter, der vokser kontinuerligt.
  • 🔄 Kontinuerlig vækst betyder jævn, ikke-springende vækst.
  • 🧮 e bruges i formler takket være dens konsistens i at beskrive vækst.
  • 🎯 100% vækst kan teknisk beregnes som en fordoblingsrate.
  • 🔍 e kan justeres ved hjælp af forskellige vækstrater i formler.
  • 🌿 Vækst i naturen og økonomi deler lignende grundprincipper takket være e.
  • 📊 Vækst ved e er en uendelig række af små incrementer.
  • 🧾 Forståelse af e hjælper med at forstå logaritmiske mønstre i vækst.

Garis waktu

  • 00:00:00 - 00:09:15

    Khalid fortsætter med at beskrive, hvorfor e er universel og fleksibel. E udtrykker 100% kontinuerlig vækst og kan skrives som e^rt, hvor r er raten og t er tiden. Selvom r ofte er 100%, kan den tilpasses til fx 50%, hvilket ændrer vækstkurven passende. E kan også tilpasses for at matche hakket vækst, som 2^x, ved at sammenligne det med en glat kurve e^q. Han konkluderer, at e ikke er magisk, men simplificerer vækstforståelse ved færrest mulige antagelser om kontinuerlig, jævn vækst uden ujævnheder.

Peta Pikiran

Video Tanya Jawab

  • Hvad er e?

    E er en matematisk konstant, der repræsenterer det naturlige basis for eksponentiel vækst. Det er cirka 2,718.

  • Hvordan relaterer e til vækst?

    E repræsenterer kontinuerlig vækst, hvilket betyder, at den beskriver en proces, hvor mængden konstant vokser på en glat og ikke-jagged måde.

  • Hvorfor er e universel i formelbrug?

    E anvendes universelt i formler, fordi den pålideligt beskriver kontinuerlig vækst, et fælles fænomen i både naturlige og finansielle processer.

  • Hvordan anvendes e i finansiel vækst?

    I finansiel vækst beskriver e, hvordan penge vokser via renter, hvor også renterne selv kan forrente sig, hvilket repræsenterer en uendelig række af små væksttrin.

  • Hvad betyder 100% vækst i konteksten af e?

    100% vækst refererer til, at den oprindelige mængde fordobles efter en given periode, hvor væksten sker kontinuerligt over tid.

  • Hvordan kan e modelleres i andre vækstformer?

    E kan modellere forskellige vækstformer ved at justere vækstraten over tid, hvilket tillader variabilitet fra fx 50% til 100% vækst.

  • Hvad er betydningen af kontinuerlig vækst?

    Kontinuerlig vækst betyder, at væksten sker jævnt og uden pludselige spring i værdien, modsat en trappelignende vækstform.

  • Hvordan kan e anvendes i naturprocesser?

    I naturprocesser kan e illustrere glat vækst over tid, som hvordan celler deler sig og vokser gradvist.

Lihat lebih banyak ringkasan video

Dapatkan akses instan ke ringkasan video YouTube gratis yang didukung oleh AI!
Teks
en
Gulir Otomatis:
  • 00:00:00
    hey guys this is Khalid here I just want
  • 00:00:02
    to give you a quick overview about the
  • 00:00:04
    key insights about e this is essentially
  • 00:00:06
    what I wish it could have gone back in
  • 00:00:08
    time and told myself about e you know
  • 00:00:10
    way back in high school already here we
  • 00:00:12
    go okay e is essentially epitome of
  • 00:00:15
    universal growth what does that mean why
  • 00:00:18
    is it universal well we can start off
  • 00:00:20
    with regular growth if you consider
  • 00:00:22
    something that changes over time like a
  • 00:00:24
    simple progression is doubling right so
  • 00:00:26
    you are a time one you have certain
  • 00:00:28
    amount of time to you double a time
  • 00:00:30
    three you double again so you have four
  • 00:00:33
    and so on and you basically have some
  • 00:00:35
    kind of exponential curve and then we
  • 00:00:37
    can call this two to the X so that's so
  • 00:00:39
    amount of time you know as you go along
  • 00:00:42
    you basically get some amount of growth
  • 00:00:43
    happening now this might seem Universal
  • 00:00:47
    and it is for a few reasons the first is
  • 00:00:49
    that - that's 100% growth which
  • 00:00:53
    basically means that you're doubling by
  • 00:00:54
    the amount that you have so it's kind of
  • 00:00:56
    universal that sense right it's a unit
  • 00:00:58
    growth so if you want if you triple
  • 00:01:00
    that's kind of you know 200 percent
  • 00:01:02
    growth which is you know it's not
  • 00:01:04
    exactly is clean as 100 percent growth
  • 00:01:06
    so there is a cool kind of you know
  • 00:01:09
    symmetry here where you're growing by
  • 00:01:11
    the amount that you have so basically
  • 00:01:13
    this goes here and then you get a new
  • 00:01:15
    interest and then these go here and you
  • 00:01:18
    get new interest as well so 100 percent
  • 00:01:20
    is kind of a neat idea but there's a
  • 00:01:22
    problem you waited until the end of the
  • 00:01:25
    period to get all your interest
  • 00:01:27
    you had nothing you were just normal
  • 00:01:29
    then suddenly boom this guy popped in
  • 00:01:31
    out of nowhere and this actually isn't
  • 00:01:34
    that common right if you think about
  • 00:01:36
    natural processes things don't just
  • 00:01:38
    appear out of nowhere they grow slowly
  • 00:01:39
    over time for example take a look at a
  • 00:01:41
    cell if we start with same idea of
  • 00:01:46
    growth we might have a cell and over
  • 00:01:50
    time it has a little buddy and this
  • 00:01:54
    little buddy
  • 00:01:54
    is kind of emerging out of them right
  • 00:01:57
    and eventually it'll pop out so it
  • 00:02:01
    didn't pop out all of a sudden over time
  • 00:02:03
    it's sort of emerged and then each day
  • 00:02:05
    got a little bit more and finally it was
  • 00:02:07
    a new soul of its own so the idea is
  • 00:02:09
    that instead of getting all your growth
  • 00:02:11
    at the end which happens with
  • 00:02:13
    to to the X that we saw kind of like a
  • 00:02:15
    staircase right it sort of goes up like
  • 00:02:17
    that we're talking about kind of as more
  • 00:02:19
    smooth gradual growth so if you think
  • 00:02:23
    about this this is actually the more
  • 00:02:24
    like nature right in nature things grow
  • 00:02:26
    slowly over time they don't sort of
  • 00:02:28
    suddenly appear out of nowhere and have
  • 00:02:29
    a kind of a discontinuous jump so if you
  • 00:02:33
    really want to have a universal way to
  • 00:02:34
    talk about growth it really should be
  • 00:02:36
    more of a smooth change instead of kind
  • 00:02:38
    of a staircase like pattern because
  • 00:02:39
    really that never happens things you
  • 00:02:41
    know don't happen isn't aeneas lee it
  • 00:02:43
    happens over time now there is one
  • 00:02:45
    little side effect though when we're
  • 00:02:47
    growing this cell it couldn't actually
  • 00:02:50
    grow on its own we sort of waited until
  • 00:02:51
    we had a red cell and then that red cell
  • 00:02:54
    could start having its own interest like
  • 00:02:56
    a green cell coming out so we have a red
  • 00:02:58
    cell and then we have a green cell
  • 00:03:01
    emergent right so the red was kind of
  • 00:03:05
    stuck in the green started coming out as
  • 00:03:07
    well so this again isn't quite as smooth
  • 00:03:10
    as it could be because while we're
  • 00:03:12
    growing the red the red can't grow on
  • 00:03:14
    its own and one kind of neat thing is
  • 00:03:16
    that money actually it's a good example
  • 00:03:20
    of something which doesn't have this
  • 00:03:21
    problem so here's the idea if you're
  • 00:03:24
    looking at the way money grows we might
  • 00:03:26
    have one dollar and over time it's a
  • 00:03:28
    dollar but you know what it earns
  • 00:03:30
    interest so as time goes on your dollar
  • 00:03:33
    earns interest and eventually you have
  • 00:03:35
    the new dollar shiny new dollar all for
  • 00:03:38
    yourself but the cool thing is that with
  • 00:03:40
    money interest can earn interest so this
  • 00:03:43
    red amount that you earned well that can
  • 00:03:45
    own own its own interest here which kind
  • 00:03:48
    of shows up here and this green interest
  • 00:03:51
    well it can earn its own may Brown
  • 00:03:53
    interest that shows up here as well if
  • 00:03:55
    you look at the article there's kind of
  • 00:03:56
    a worked example but the idea is this
  • 00:03:58
    your regular amount come goes along like
  • 00:04:02
    that it earns some interest which is
  • 00:04:06
    this kind of red growth here and that
  • 00:04:08
    red growth well that earns its own
  • 00:04:10
    interest which is this kind of green
  • 00:04:11
    growth here and that green growth earns
  • 00:04:14
    its own interest which is this brown
  • 00:04:16
    growth and so on so there's actually an
  • 00:04:18
    infinite amount of growth happening and
  • 00:04:20
    it sort of looks like that so what ends
  • 00:04:22
    up happening is you get this curve which
  • 00:04:25
    has sort of an infinite number of
  • 00:04:27
    opponents the biggest components come
  • 00:04:30
    from these kind of giant chunks in the
  • 00:04:32
    beginning but you end up having smaller
  • 00:04:34
    and smaller groups emerging so it's a
  • 00:04:37
    bunch of little slices and eventually it
  • 00:04:39
    caps out here's the idea
  • 00:04:42
    E is that cap so if you add up all the
  • 00:04:46
    slices you get growth which is actually
  • 00:04:48
    e to the X so E is basically the idea of
  • 00:04:52
    taking growth adding its and Trust
  • 00:04:56
    adding bats interest adding bats
  • 00:04:59
    interest and so on and so on and each
  • 00:05:02
    time you do it it gets a little bit
  • 00:05:03
    narrower and narrower and the max speed
  • 00:05:05
    limits you hit that's e so the reason is
  • 00:05:08
    Universal is the following one it's 100%
  • 00:05:11
    growth so the idea is that your original
  • 00:05:14
    item it'll actually double itself at the
  • 00:05:16
    end of one period so you start at you
  • 00:05:18
    know let's call this time zero at the
  • 00:05:20
    end of one period your black amount has
  • 00:05:23
    actually doubled itself so that's why
  • 00:05:24
    it's how 2 percent growth but two it's
  • 00:05:27
    continuous and this is really the key
  • 00:05:30
    here it's not jumping suddenly as time
  • 00:05:34
    goes on each instant is earning red and
  • 00:05:36
    the eds
  • 00:05:37
    the Reds earning green and the Greens
  • 00:05:39
    earning brown and the Browns earning
  • 00:05:40
    gray and so on and so on so this
  • 00:05:42
    combination actually that gives you some
  • 00:05:45
    amount of growth at the end of the
  • 00:05:46
    period which is about 2.718 dot and that
  • 00:05:51
    number is and basically if you want to
  • 00:05:55
    find out how much growth you have at the
  • 00:05:57
    end of two periods or three periods you
  • 00:05:59
    just take e to some exponent so the
  • 00:06:01
    reason that he appears in every kind of
  • 00:06:03
    formula is because it's a very common
  • 00:06:05
    base to verse Allah to talk about growth
  • 00:06:08
    and you can actually change the rate so
  • 00:06:10
    this is actually okay now jumping into
  • 00:06:12
    the kind of flexibility B so E is
  • 00:06:14
    basically 100% continuous growth but
  • 00:06:17
    it's also flexible so E I actually
  • 00:06:21
    consider it e to the RT so this is e to
  • 00:06:26
    the rate times time
  • 00:06:27
    often times the rate is 100% so you
  • 00:06:29
    don't see it because 100 percent is 1.0
  • 00:06:31
    right so you don't actually see it it's
  • 00:06:33
    you know invisible but the idea is that
  • 00:06:35
    if you adjust the amount of interest
  • 00:06:39
    that you get at the end of a period so
  • 00:06:40
    right
  • 00:06:40
    100% let's say you only get 50% interest
  • 00:06:43
    so you'd have something like this you're
  • 00:06:45
    growing and at the end of one period
  • 00:06:48
    you only get 50% not 100% interest you
  • 00:06:51
    still have the same thing that interest
  • 00:06:53
    you know earn some of it yourselves own
  • 00:06:54
    50% and you get 50% of that let's see we
  • 00:06:58
    had brown coming up ended 2% of that
  • 00:07:01
    grey coming up so you still get a
  • 00:07:04
    pattern but it's a little bit reduced
  • 00:07:05
    and the idea is oh in this case it would
  • 00:07:07
    actually be e to the point 5 because
  • 00:07:10
    it's 50% interest and then times the
  • 00:07:13
    amount of time that you want to you take
  • 00:07:15
    it for so you could have 50% interest
  • 00:07:17
    for 2 years and it actually would be the
  • 00:07:19
    same thing as 100 percent interest for 1
  • 00:07:21
    year because 50% times 2 equals 100
  • 00:07:27
    percent times 1 so the cool thing about
  • 00:07:30
    these is really flexible you can adjust
  • 00:07:32
    the rate in time and that works because
  • 00:07:34
    it's based on very universal principles
  • 00:07:36
    100 percent growth and it's continuous
  • 00:07:39
    so there's no kind of jagged edges so
  • 00:07:42
    that's sort of the key insight about e
  • 00:07:43
    the article has some kind of
  • 00:07:45
    computations but here's one little cool
  • 00:07:47
    fact you can actually model the
  • 00:07:50
    staircase like growth so if we have this
  • 00:07:52
    kind of jumpy growth right that's sort
  • 00:07:54
    of appearing out of nowhere e can
  • 00:07:57
    actually match that to some smooth curve
  • 00:08:00
    so that will be e to some interest rate
  • 00:08:03
    I don't know let's call it Q or
  • 00:08:05
    something right so e to the Q that can
  • 00:08:07
    actually match up with any kind of
  • 00:08:09
    jagged interest rate that you want into
  • 00:08:11
    the hood it'll hit all the points so the
  • 00:08:14
    neat thing is that any kind of growth
  • 00:08:16
    can be considered on its own is this
  • 00:08:18
    jagged growth or it can be considered as
  • 00:08:21
    e raised to some interest rate and the
  • 00:08:23
    idea here is that you can actually use
  • 00:08:25
    the natural logarithm to figure out what
  • 00:08:27
    interest rate that is but we'll get to
  • 00:08:28
    that separately so again the ideas
  • 00:08:30
    behind e 100% growth is growing by
  • 00:08:33
    itself you start with something it grows
  • 00:08:35
    and it earns interest it is itself which
  • 00:08:38
    earn some interest on its own and so on
  • 00:08:41
    and so on and also it's continuous it's
  • 00:08:44
    smooth it's not jagged like this it's a
  • 00:08:46
    smooth curve and because of that you can
  • 00:08:48
    vary this growth from 100 percent to 50
  • 00:08:51
    percent growth which actually changes
  • 00:08:53
    the curve to be
  • 00:08:54
    lower and depending on how much you
  • 00:08:55
    angle it you can actually model any type
  • 00:08:58
    of growth that you want so that's my aha
  • 00:09:00
    moment about it's not this magic number
  • 00:09:01
    it's not a magic spell it's just taking
  • 00:09:04
    the idea of growth and simplifying it to
  • 00:09:06
    have this few assumptions as you can be
  • 00:09:08
    growing by yourself a hundred percent
  • 00:09:10
    and don't be jagged and with that you
  • 00:09:12
    get happy mess
Tags
  • e
  • eksponentiel vækst
  • kontinuerlig vækst
  • naturlige processer
  • finansiel vækst
  • 100% vækst
  • renter
  • matematik
  • formler