𝟎: The Hollow Core of Mathematics

Zero is nothing. And yet without it, there is no number, no symmetry, no algebra, no calculus.

Zero is not just a placeholder. It is the place itself.

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➕ 𝑎 + 0 = 𝑎

The most elemental truth: adding nothing changes nothing.
This is not merely a trick of definition: it is a declaration of identity.

Zero is the silent partner in every sum.
It asks nothing, gives nothing, changes nothing, and yet, it is always there, invisible.

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🧮 𝑎⁰ = 1

Let's look at the descent of exponents:

𝑎² = 𝑎⋅𝑎 = 𝟷·𝑎·𝑎
𝑎¹ = 𝑎   = 𝟷·𝑎
𝑎⁰ = ?   = 𝟷

Why?
From the recursive rule:

$$ 𝑎ⁿ = (𝑎ⁿ⁻¹)⋅𝑎 $$

A number raised to the power of nothing still holds a trace of itself: unity.

So, $𝑎⁰ = 1$ for any $𝑎 ≠ 0$.

❗ 0! = 1 — The Empty Product

The factorial grows by multiplying down:

𝑎! = (𝑎-1)!·𝑎 

3! = 2!·3 
2! = 1!·2 
1! = 0!·1

To keep this chain unbroken, we must define:

0! = 1

This is no trick. In combinatorics, 0! counts the number of ways to arrange zero items: exactly one way: by doing nothing.

Emptiness, too, has structure. Even nothing can be organized.

⁉ 0⁰ — Undefined?

Mathematicians hesitate here. Why?

Take the function 𝑥^𝑥. As 𝑥 → 0⁺, we get:

x	x^x
1	1
0,9	0,9095325760829
0,8	0,8365116420730
0,7	0,7790559126704
0,6	0,7360219228178
0,5	0,7071067811865
0,4	0,6931448431551
0,3	0,6968453019359
0,2	0,7247796636776
0,1	0,7943282347242
0,01	0,9549925860214
0,001	0,9931160484209
0,0001	0,99907938998
0,00001	0,999884877372
0,000001	0,999986184584

㏐ 𝑥ˣ
𝑥⟶0
1
┿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁
│⠀⠑⠀⠀⠀⠀⠀⠊
│⠀⠀⠡⠀⠀⠀⠌
│⠀⠀⠀⠑⠤⠊
│
┼──────────╂
           1
x= {1 ..-0.1.. 0}
x= {1 ..÷10..  0}
x^x = {1 … 1}

• 0^n = 0
• n^0 = 1
• So what should 0^0 be?

Some contexts define it as 1 (like combinatorics), others leave it undefined, honoring the ambiguity.

Is 0⁰ defined? It is where rules meet their edge.

It seems to approach 1, but:

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🚫 Division by Zero — The Forbidden Operation

Multiplication is repeated addition:
 3 × 5 = 3 + 3 + 3 + 3 + 3

Division is its inverse, repeated subtraction:
 15 ÷ 5 = 3 because
 15 - 5 - 5 - 5 = 0

But what about 10 ÷ 0?

How many zeros must you subtract to reach 10?
You could subtract forever, and never move.

15÷5 = 3  ⟺  15 -5-5-5 = 0
                 ╰─╮ ╭─╯
                    3

10 ÷ 0 : 1 -0-0-0-0-0-0-0... 
           ╰────╮ ╭──╌╌┄┄┈┈
                 ∞

10 ÷ 0 = ? → Infinite subtraction of 0 still gives 10 → So no number satisfies it

x	1/x
1	1
0,5	2
0,1	10
0,01	100
0,001	1000
0,0001	10000
0,00001	100000
0,000001	1000000

But

x	1/x
-1	-1
-0,5	-2
-0,1	-10
-0,01	-100
-0,001	-1000
-0,0001	-10000

㏐ 1÷𝑥  = ∞
𝑥⟶0⁺

㏐ 1÷𝑥  = -∞
𝑥⟶0⁻

∄ ㏐ 1÷𝑥
  𝑥⟶0

As x → 0, 1÷x → ∞ or -∞ depending on the direction. But at zero itself?

There is no number that multiplied by zero gives one.
Division by zero is not wrong—it is meaningless.

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🌀 Zero: The Sign

• It is absence, but essential for place value in numbers.
• It is identity for addition, neutral in subtraction, annihilation in multiplication, but forbidden in division.
• It is empty, but not void. It holds the entire structure of arithmetic together.

Zero is not just a number. It's a puzzle!