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Appendix: A proof that becomes simple

Nested radicals with visible cancellation

For students, teachers, and the simply curious. No prior knowledge of logarithms needed — if you can do 3 + 5, you can follow this to the end.

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Appendix: A proof that becomes simple

Note — This section is for maths teachers and maths enthusiasts. Share it with them — this notation often surprises even experienced maths teachers, and the proof tends to spark a good discussion.

Wikipedia has a page about nested radicals — expressions where a square root contains another square root. That page is full of advanced maths — but we're only looking at one small corner of it, and we'll keep it simple.

√(3 + 2√2) = 1 + √2
Wikipedia says: "It is not immediately obvious" that these two are equal.

In school notation, proving this requires you to know special rules about roots. But with our notation, it's just a puzzle. Let's solve it — you only need one thing you probably already know:

(a + b) ↑ 2 = a ↑ 2 + 2 × a × b + b ↑ 2
The "square of a sum" rule, which works in any notation.

Translating to our notation

We write the square root as ↓2. Let's translate both sides, piece by piece.

The right side is easy. It says: 1 + √2. The only thing to translate is √2, the square root of 2:

√2 → 2 ↓ 2
Read: "2 down 2". The first number (2) is what you're taking the root of. The second number (2) says it's a square root.

So the right side becomes: 1 + 2 ↓ 2.

Now the left side. It says: √(3 + 2√2). That's a root of something, and that "something" itself contains another root. We work from the inside out:

Inner √2 → 2 ↓ 22 × √2 → 2 × (2 ↓ 2)3 + 2 × √2 → 3 + 2 × (2 ↓ 2)√(3 + 2 × √2) → (3 + 2 × (2 ↓ 2)) ↓ 2
The outer √ becomes ↓2 at the very end — it takes the root of everything inside.
Left side: (3 + 2 × (2 ↓ 2)) ↓ 2Right side: 1 + 2 ↓ 2
This is what we want to show: left side = right side.

The plan

Look at the left side: it says (something) ↓ 2. That means: "the square root of what's inside." The "something" inside is 3 + 2 × (2 ↓ 2).

If we square the right side (↑ 2), and end up with exactly 3 + 2 × (2 ↓ 2), then we know the left side equals the right side. Why? Because ↑ 2 and ↓ 2 are inverses — they cancel each other.

Step by step

Let's square the right side:

(1 + 2 ↓ 2) ↑ 2
We use the "square of a sum" rule with a = 1 and b = 2 ↓ 2.
(1 + 2 ↓ 2) ↑ 2((1 plus 2 down 2) up 2)

Expanding with the rule gives us three pieces:

1 ↑ 2 + 2 × (2 ↓ 2) + (2 ↓ 2) ↑ 2
a ↑ 2 + 2 × a × b + b ↑ 2, where a = 1 and b = 2 ↓ 2.
1 ↑ 2 + 2 × (2 ↓ 2) + (2 ↓ 2) ↑ 2(1 up 2 plus 2 times (2 down 2) plus (2 down 2) up 2)

Now comes the magic. Look at the last piece: (2 ↓ 2) ↑ 2. That's ↓ followed by ↑, with the same number (2). They're inverses — they cancel! So (2 ↓ 2) ↑ 2 = 2. And 1 ↑ 2 is just 1. So we get:

1 + 2 × (2 ↓ 2) + 2
The inverses cancelled: (2 ↓ 2) ↑ 2 became just 2.
1 + 2 × (2 ↓ 2) + 2(1 plus 2 times (2 down 2) plus 2)

Rearranging 1 + 2 = 3:

3 + 2 × (2 ↓ 2)
That's exactly what's inside the ↓ 2 on the left side!
3 + 2 × (2 ↓ 2)(3 plus 2 times (2 down 2))

We squared the right side and got exactly the left side's contents. That means:

(3 + 2 × (2 ↓ 2)) ↓ 2 = 1 + 2 ↓ 2 ✓
Both sides are equal. Done!
(3 + 2 × (2 ↓ 2)) ↓ 2((3 plus 2 times (2 down 2)) down 2)

Why this matters

The entire proof hinged on one moment: seeing that (2 ↓ 2) ↑ 2 = 2, because ↓ and ↑ cancel. In school notation, that same step would be written as (√2)² = 2. Can you see the cancellation there? Not really — √ and ² look nothing alike. But ↓ and ↑? They're the same arrow, pointing opposite ways. The cancellation is staring you in the face.

That's the point of the whole article. Better notation doesn't just look nicer — it makes hard things easy to see.

You can both sit back down now.

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2 ↑ 3 = 82 = 8 ↓ 3left unknown? use down3 = 8 ⇓ 2right unknown? use double-down