Improving our units and notation

Many of the systems and ideas we use every day, like the way we count and measure things, were developed over centuries. They have managed to “stand the test of time”, which is certainly an advantage; but they also suffer a form of “historical inertia”, which can prevent subsequent insights from gaining traction.

For example, we still talk about 90°, 120° and 180° instead of ¼, ⅓ and ½ turns; because the Babylonians didn’t have fractions 5000 years ago. We also have redundant units like the tonne (Megagram) and litre (milli-cubic-metre); we waste time learning tricky, niche concepts (like subtraction and pseudovectors) rather than more elegant, general alternatives (like negatives and rotors, in the cases mentioned).

These pages are my ongoing attempt to find such situations, where we can replace complicated approaches with simpler ones; combine a bunch of ad-hoc choices into some smaller set of consistent rules; and avoid some notions entirely (turning them into niche historcal curiosities, rather than foundational curricula taught to every child).

Note that the truly important parts, the underlying ideas, are generally not my own; although their unorthodox presentation will include some originality; e.g. to avoid historical baggage and focus on coherence. Many of these are whole subjects which will not be described in full detail, or with the level of rigour some might desire; this is largely due to my own ignorance, but also to focus on equipping individuals (mostly myself!) with a ‘core’ of good ideas, which can be taken further if desired. I’ll include links to Wikipedia, etc. for those who are curious for more!

Each idea should be mostly stand-alone, although they will be presented in a cohesive way (for example, long-subtraction uses the notation from negatives); I’ll try to cross-link such cases.


Quick wins lists some ideas which are relatively easy to adopt right now. More specific, in-deph pages are collected below, into rough categories.

Units of Measure

The metric system is far better than the various ‘imperial’ systems, although not for the reasons usually given.

We can make life even simpler by limiting ourselves to the SI subset of metric.


‘Minus’ is ambiguous and should be avoided. Instead, I find over-bar notation composes better than minus signs

Over-bar notation can also be extended to place-value notation, which makes long-subtraction obsolete.

Bars can also be used to keep track of matching parentheses.


Subtraction is ambiguous, conflating multiple distinct ideas:

It can be useful to distinguish between absolute values and the relative differences between them. This is pretty straightforward in one dimension, and lets us find the difference between Natural numbers (resulting in an Integer).

This is closely related to vectors, and extends to signed areas (bivectors), signed volumes (trivectors), etc. in higher dimensions. Those concepts are core to Geometric Algebra, but also appear in fields like Algebraic Calculus. I’m not sure if they’re directly compatible, but it would be interesting to find out!



A work in progress