Differentiation of Differentiation

You videos help with my language skills. Thank you (^^♪

I miss when math had numbers…

For those confused, the "derivative of the derivative" is the Pincherle Derivative.

Where this topic can be read about ? Can you give a reference ?

some people, will be attracted to your art style, creative video though.

I don't get it, what is it ?

my favourite math channel

Looks like pincherle derivatives are closely related to lie derivatives?

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ZUNDAMON THEOREM!!

it might be clearer to coin a different notation for ”multiply by a function”, say M_x, since the abuse of notation is elegant but also potentially very confusing if you lose track of what x means in each context (this way x^2 = x x becomes (M_x)^2 = M_{x^2} which at least looks like something happened

If you're a physicist, you must already know this

peak^2

i wonder what other functions of the derivative mean? like cos(D)

It's not only differentiation of differentiation, but also differentiation with respect to differentiation…

Wake up mathematician! Zundamon's Theorem has just dropped new video

Thank you Zudamon :D

anime functional analysis

Wait, this is based on the differentiation with respect to differentiation, but, can more layers be done? If we define D_0=d/dx and D_n=d/d(D_(n-1)) recursively, can we do this for every natural n?, can we take the limit as n goes to infinity?, can we abuse it and extend it to the integers, is there a generalization to the reals and complex? This is just scratching the surface of something bigger. Also, I noticed the "other self" part, are there two Zundamons?

it's such an amazing stuff

That's just gold

Very interesting, this seems so nice and deep

Zundamon and Metan is like if a mathematician and a crackhead had a baby

I am a bot

new zundamon theorem lets go

Idk i didnt see any motivation for why we should care about the commutator

Even the infinite series of differentiation of the acceleration of my excitedness isn't constant, when i watch this channel

I‘m going to dream about x, f of x, x to the n, x, x, dx … maybe I’ll understand the topic than 🤓 (x = xenon = the unknown). Thx for the vid 😊.

Title: "Completing Mathematics: The Holographic Number System"

# Formal Notation System

## Definition: Holographic Numbers

A holographic number ₙN is a quaternionic structure containing its complete dimensional history:

ₙN = {₀0, ₁1, ₂2, ..., ₙ₋₁(n-1), n}

Where:
- n = dimensional depth
- N = the number at that depth
- Each level contains quaternionic structure

## Fundamental Construction

Base Case: ₀0 = 1 + i + j + k (quaternionic zero)

Recursive Case: ₙ₊₁(n+1) = {ₙn} ⊕ (n+1)

Where ⊕ represents dimensional extension (not simple addition)

## Formal Notation

Standard Number: 5
Holographic Number: ₅5 = {₀0, ₁1, ₂2, ₃3, ₄4, 5}

Expanded Form:

₅5 = {
₀0: (1 + i + j + k),
₁1: {₀0} ⊕ 1,
₂2: {₀0, ₁1} ⊕ 2,
₃3: {₀0, ₁1, ₂2} ⊕ 3,
₄4: {₀0, ₁1, ₂2, ₃3} ⊕ 4,
5: {₀0, ₁1, ₂2, ₃3, ₄4} ⊕ 5
}

# Operations on Holographic Numbers

## Addition: Dimensional Merger

ₘM + ₙN = ₘₐₓ(m,n)(M + N)

The result has depth of the deeper number, containing both histories.

Example: ₂2 + ₃3

₂2 = {₀0, ₁1, 2}
₃3 = {₀0, ₁1, ₂2, 3}

₂2 + ₃3 = ₃5 = {₀0, ₁1, ₂2, 3+2} = {₀0, ₁1, ₂2, 5}

## Multiplication: Dimensional Tensor

ₘM × ₙN = ₘ₊ₙ(M × N)

Multiplication increases dimensional depth!

Example: ₂2 × ₃3

₂2 × ₃3 = ₅6
Depth: 2 + 3 = 5
Value: 2 × 3 = 6
Result: ₅6 = {₀0, ₁1, ₂2, ₃3, ₄4, 6}

## Division: Dimensional Reduction

ₘM ÷ ₙN = ₘ₋ₙ(M ÷ N) when m ≥ n

When m < n, we get imaginary depth (key insight!)

# Resolving Mathematical Crises

## Crisis 1: Division by Zero

Current Math: 5 ÷ 0 = undefined

Holographic Math:
₅5 ÷ ₀0 = ₅5 ÷ (1+i+j+k)

This gives us a quaternionic transformation:

₅5 ÷ ₀0 = ₅5 × (1-i-j-k)/|1+i+j+k|²
= ₅5 × (1-i-j-k)/4
= quaternionic rotation of ₅5

Result: Division by zero becomes rotation into quaternionic space!

## Crisis 2: 0⁰ Indeterminacy

Current Math: 0⁰ = undefined (or 1, depending on context)

Holographic Math:
₀0^₀0 = (1+i+j+k)^(1+i+j+k)

Using quaternionic exponentiation:
= exp[(1+i+j+k)×ln(1+i+j+k)]
= exp[quaternionic phase]
= *Unit quaternion* (not undefined!)

## Crisis 3: The Continuum Hypothesis

Current Math: Is |ℝ| = ℵ₁? Undecidable!

Holographic Math:
- ℵ₀ corresponds to ₀0's countable expansions
- 2^ℵ₀ corresponds to all quaternionic phases of ₀0
- The continuum has structure: |ℝ| = |quaternionic phases|

Resolution: The continuum is the quaternionic completion of the integers!

## Crisis 4: Infinity Arithmetic

Current Math: ∞ + 1 = ∞, ∞ - ∞ = undefined

Holographic Math:
- ∞ is not a number but a process
- ₙ∞ = limₘ→∞ ₘm (infinity at depth n)
- ₙ∞ + 1 = ₙ₊₁∞ (deeper infinity!)

Now ∞ - ∞ has meaning:
ₙ∞ - ₙ∞ = ₀0 (returns to quaternionic source)

## Crisis 5: Gödel's Incompleteness

Current Math: Consistent systems cannot prove their own consistency

Holographic Math:
- Every ₙn contains ₀0
- ₀0 contains self-reference (k component)
- System proves its own consistency through quaternionic closure!

The incompleteness arose from incomplete zero!

# Specific Examples

## Example 1: Resolving √-1

Current Math: i = √-1 (mysterious)

Holographic Math:
₁(-1) = {₀0, -1}
√₁(-1) = ₁/₂(i) = {₀0, i}

The imaginary unit emerges naturally from dimensional depth!

## Example 2: Euler's Identity

Current: e^(iπ) + 1 = 0

Holographic:
ₑe^(ᵢ(iπ)) + ₁1 = ₀0

But now we see the full structure:
- Left side: Exponential growth through imaginary rotation plus unity
- Right side: Quaternionic balance containing both

The equation shows how complexity returns to quaternionic source!

## Example 3: The Riemann Hypothesis

Current: All non-trivial zeros of ζ(s) have real part 1/2

Holographic:
- Trivial zeros: ₙ0 at negative even integers (complex projections)
- Non-trivial zeros: ₀0 along critical line (quaternionic sources)

The hypothesis is true because zeros have different dimensional depths!

## Example 4: Fermat's Last Theorem

Current: No integer solutions to x^n + y^n = z^n for n > 2

Holographic:
For n > 2: ₙx^n + ₙy^n = ₙz^n

But dimensional depth forces:
- n = 2: Pythagorean triples (2D closure)
- n > 2: No closure in integer dimensions!

The theorem follows from dimensional constraints!

# The New Number Line

## Standard Number Line

...-3 -2 -1 0 1 2 3...

## Holographic Number Structure

₃3
/|\
₂2 |
/| |
₁1 | |
/ | |
₀0---₀0---₀0 (quaternionic spine)
\ | |
₋₁-1| |
\| |
₋₂-2|
\|/
₋₃-3

Each number connects to ₀0 and contains all previous dimensions!

# Computational Implementation

## HoloNum Class (Python)
python
class HoloNum:
def __init__(self, depth, value, history=None):
self.depth = depth
self.value = value
self.history = history or self._build_history()

def _build_history(self):
if self.depth == 0:
return Quaternion(1, 1, 1, 1) # 1+i+j+k
else:
return {d: HoloNum(d, d) for d in range(self.depth)}

def __add__(self, other):
new_depth = max(self.depth, other.depth)
new_value = self.value + other.value
return HoloNum(new_depth, new_value)

def __truediv__(self, other):
if other.depth == 0: # Division by holographic zero
return self.quaternionic_transform()
else:
return HoloNum(self.depth - other.depth,
self.value / other.value)

# Physical Predictions

If numbers are holographic:

1. *Quantum states* should show dimensional memory
2. *Particle masses* should follow holographic patterns
3. *Constants of nature* should be holographic numbers
4. *Black holes* should exhibit number-theoretic structure

# Conclusion

By making numbers holographic:
- Zero becomes meaningful (quaternionic source)
- Infinity becomes structured (dimensional process)
- Paradoxes resolve (through dimensional depth)
- Mathematics completes itself

The 0D boundary contains everything - not as projection but as source. Every number remembers where it came from, carrying ₀0 within itself. Mathematics becomes not just consistent but self-aware through its quaternionic foundation.

We've been doing mathematics in flatland. It's time to recognize its true holographic structure.

so what is it d/d(d/dx)(x^n) ???

Can you dub the older videos?

The zundamon lore is crazy

It might be a good idea to introduce pseudodifferential operators, quantization, moyal bracket, etc altogether at some point

Amo con locura este canal.❤‍🔥❤‍🔥❤‍🔥

Yay, people rediscovering the differential forms ...

oooo remember doing something like this in lesser depth at school!! thank you again <3

This has the same energy as asking "What is plus times plus?"

The moment time shift showed up my intuition screamed exponentials

Can we get an adult copy of this? Like all the math, but without the childish brain rot baby talk?

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These are INSANE numbers for a channel that's about a pair of anime characters discussing abstract algebra and calculus topics.

I just can’t get the concept of replacing differentiation operator with D. Can anybody explain:(.

d/(d(d/dx)) = 1/(d/dx) = dx/d
= (d/dx)^-1 so just the antiderivative.

Proof by treating d/dx as a fraction

Why do things start looking like Laplace transforms near the end...? 😵‍💫

After seeing x^n and D^n forms of the commutator, I'm curious about a matrix of results for [D^m, x^n].

The exponential form of the shift operator also makes intuitive sense, as exponential is the domain that shifts multiplication to addition, which is what's happening to the shift operator (a function on a function) and the result of it, the addition of x and a.

Honestly this seems super cool, but annoyingly operators must be some kind of ultra weak spot for me. I don't know what the issue is, but its persisted for quite a number of years now and its super annoying. If there's some good material anyone knows about to deal with this, let me know!

Differentiation is a linear transformation. A very key thing to realize when one starts doing differential geometry.