TL;DR: Instead of labouriously computing , we can mentally calculate using the alpha-max plus beta-min algorithm, by estimating

and this will be very close to the actual . This is useful for adding up sources of variance, or figuring out radiuses, or other such things.

Background

The mathematical relationship is surprisingly common. It happens among other things in

• geometry (Pythagorean theorem);
• statistics (sources of variance add up);
• physics (the energy–momentum relation).

When it shows up, it's often because one of the variables is unknown, i.e. we have either

• ​ or
• ​.

The annoying part is that these are hard to mentally calculate, even when one is good at estimating squares and square roots (e.g. because of previous logarithm practice) because numbers grow large when squared.

Insight

I just had a flash of insight. Maybe the problem is thinking of this as three separate operations (square, add, take the root). What if instead we think of it as one fundamental, composite operation? We could call it ⊞ (Unicode name apt: squared plus), and define it as

and then we could use spaced repetition to train ourselves in evaluating it mentally, much like we would do with multiplication tables [...]

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Outline:

(00:28) Background

(01:08) Insight

(02:10) Prior art

(03:37) Inverting

(03:54) Example

The original text contained 2 footnotes which were omitted from this narration.

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First published:

Source:

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Narrated by TYPE III AUDIO.