Pi Function (Euler Factorial Function)

Рет қаралды 11,556

Күн бұрын

In this video, I showed the relationship between the Pi Function and Gamma Function. I explained that the Pi function is just a shifted version of the Gamma function. I also used the Pi function to show that 0 factorial = 1
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Пікірлер: 40

@flowingafterglow629 5 ай бұрын

This is a lot of fun, with the back and forth and your quick responses to the comments. You posted a video yesterday in response to comments from the day before, and this video is now including a response to comments from literally yesterday. Very enjoyable!

@PrimeNewtons 5 ай бұрын

Glad you enjoyed it!

@souravgarain2835 5 ай бұрын

You are an amazing teacher

@adibjauhari Ай бұрын

I like how you explain math concepts. With such a fun expression, it makes me watch this video till the end. Thanks 😄

@kartikbhardwaj2604 5 ай бұрын

Bring more real analysis vids❤

@ProactiveYellow 5 ай бұрын

0 as a member of the natural numbers depends on the field. In axiomatic set theory, specifically using Von Neumann's construction of the natural numbers, the natural numbers are constructed as follows: Let 0 be defined as the empty set ∅, and let the function s(n) on the natural numbers be defined as s(n)=n U {n} where U is the union of sets. The set of natural numbers is therefore the intersection of all sets that satisfy the axiom of infinity. That is, the natural numbers are precisely the set of 0 and all its successors. This has the benefit of any natural number n containing precisely n elements, and for all n and m, n

@allozovsky 5 ай бұрын

As a compromise solution, we may use *ℕ* for {1, 2, 3, ...}and *ℕ₀* for {0, 1, 2, 3, ...}, though ISO 80000 defines the set of natural numbers as the set of positive integers and zero, and denotes it by simply *ℕ* (without a subscript).

@patrickfrei9322 5 ай бұрын

Came here for this, thank you! It really puzzles me when people just claim that 0 isn't part of the naturals 😅

@EvilSandwich 5 ай бұрын

Man, when you set out to correct a faux pas, you don't mess around. lol

@prakrit1280 5 ай бұрын

Sir's teaching methodology is awesome 😎👍Plus Sir is genious - math mentor, youtuber, singer 😎😇👍 Always looking forward to your upcoming videos😁

@anglaismoyen 5 ай бұрын

The factorial plot thickens... by the way, what watch do you wear?

@jlmassir 5 ай бұрын

You're a great singer too! May I suggest two more reasons for 0! = 1! = 1? First, it allows a uniform formula for the binomal coefficients. Second, n! is the number of permutations of n elements, and there is only one permutation of 1 or 0 elements (a permutation of 0 elements is somewhat spooky, but it is a real thing in combinatorics). Please do a video about this!

@allozovsky 5 ай бұрын

Also 0! is an _empty product,_ which (by convention) is equal to the neutral element of multiplication (multiplicative identity), that is 1. Just like 0⁰ is also an empty product and is equal to 1 for the same reason (when treated as exponentiation with a _natural_ exponent).

@jlmassir 5 ай бұрын

@@allozovsky So this argument resolves the controversy about 0^0?

@allozovsky 5 ай бұрын

@@jlmassir Yeah, that is how exponentiolation with a _natural_ exponent is normally defined: *a⁰ = 1* (for any base *a,* as an empty product), and *aⁿ⁺¹ = aⁿ·a,* which gives us *0⁰ = 1, 0¹ = 0⁰·0 = 1·0 = 0,* and so on. No division by zero is needed to define 0⁰ with this approach, so it is well-defined under exponentiation with a natural exponent.

@Gremriel 5 ай бұрын

"There's no mystery," he says. Me, watching this: "uh huh".

@lucdutreiz5135 3 ай бұрын

Loved when you sang

@surendrakverma555 5 ай бұрын

Very good lecture Sir. Thanks 🙏

@slavinojunepri7648 Ай бұрын

Excellent 👌

@bowlineobama Күн бұрын

Mr. Newtons, since you and I are big fans of Euler, I would like to send you a gift. Where can I send it? No kidding.

@prabhatrexkira398 5 ай бұрын

Amazing....Never Stop Learning, Never Stop Roasting 😅😅😅😅

@mrvortex8885 5 ай бұрын

Anyone know how to solve 12.8(x+2^x)=x^x

@allozovsky 5 ай бұрын

12.8 = 256/20 = 4⁴/20 4⁴·(x+2ˣ) = xˣ·20 x = 4 But we need to prove there are no other solutions (real or complex).

@DragomirGąsieniec 5 ай бұрын

Never stop teaching, cause those who stop teaching stop developing the world 😁

@sphakamisozondi 5 ай бұрын

Euler, Gauss and the Bernoulli brothers are in the Mount Rushmore of mathematics. Modern mathematicians, I gotta give it to David Hilbert and Henri Poincaré

@patrickfrei9322 5 ай бұрын

There were more than just 2 berboullis, whole family of geniuses 😄

@haotj 5 ай бұрын

At 5:59 shouldn't it be t^(-1) since PI(x) = Gamma(x+1)?

@Ahmad-yi6d 5 ай бұрын

❤

@valemontgomery9401 5 ай бұрын

Do we really need the symbols for pi and gamma in the first place? We already have the integral and the ‘!’ for the factorial function, what’s the point of using pi just to say the same thing?

@mikefochtman7164 5 ай бұрын

Just a shorthand way of writing an often used function. Like the upper-case 'L' used for LaPlace function, or many others. Saves a little space for a well documented and often used idea.

@wkmartins 5 ай бұрын

Integral notation might be a littlle too long. But i agree just using '!' would be better because we don't invent a new symbol when we extend multiplication to the reals

@derzahlenmensch6089 5 ай бұрын

It's more of a formal thing. Usually, factorials using the "!"- notation are only defined for natural numbers (n! = n(n-1)!) while the Γ - and the Π - function extend that concept to any complex number z. So it's basically just used to make clear that you're trying to calculate the factorial of a complex number (not specifically a natural number). But yes, you're right you could use the "!"-notation for any factorial (as long as it's clear that the integral definition can be used which every person knowing that definition should think of at first though).