In Mag World, the mechanics of doing math is called logarithmetic. When dealing with large and small numbers, logarithmetic is much simpler and easier than linear arithmetic. You just need to know a few basics and then you can practice.
How would you convert 93 million miles into a Mag Number?
There are two ways. The precise scientific way is to convert the number into SI units and then take the log base 10, rounding to the nearest whole number (or half, depending).
With logarithmetic, that level of precision is not at all necessary.
A number with N digits is ^N. If the number starts with 1 or 2, use N-1. If it starts with 3-7, use N-½. So “high 6-figures” is ↑6 while “low 6-figures” is ↑5.
You should memorize these metric prefixes if you don’t know them already:
Here you need to have a sense of the units. Each component is added to the total, and if it’s a rate (“per hour”), subtract the time unit.
So add ↑3 to to ↑8 (as above), and you can guess that 93 million miles is ↑11 distance.
↑2 ↑6 ↑3 = ↑11Here’s the real answer with all of its glorious precision:
>>> meters_per_mile = 1609.344
>>> log10(93_000_000*meters_per_mile)
11.17513183375531Of course, 93 million miles is 1 AU (Astronomical Unit), commonly known as the distance from the Earth to the Sun. So if you’ve already internalized mag distance, you might just know that 1 AU is ↑11 distance, the same way you might know that the Sun is ↑9 size.
The world used about 180,000 TWh in 2023.
Large-scale energy usage is measured in “terawatt-hours per year”, but it should seem weird that this unit has two different time units embedded in it. Energy usage is not quite the same as power, which is one reason why TWh/year is used instead of watts; but can we get this number in watts equivalent?
It’s actually pretty easy:
Add them all up to get ↑13 (10 terawatts). Compare this to a Kardashev Type I civilization which uses ↑16 power (10 petawatts), or 1000x as much power.
The speed of light in a vacuum is ↑8.5 speed. A year is ↑7.5 time. Smoosh them together to get a light-year of ↑16 distance.
The “real” answer:
>>> log10(9_460_730_472_580_800)
15.975924670013155To get the square of a mag number is really easy: multiply by 2. Or the square root, divide by 2. Cube and cube root, multiply or divide by 3. You probably won’t need any exponents other than 2 or 3, but if you do, multiply or divide it just the same.
So if you have a town of 100 km², then that’s ↑2 (100) ↑3 (km) x 2 (²; note that the kilo in kilometer is squared, but the coefficient is not) = ↑8 area. And to cross that town lengthwise would be the square root of that area, which is half its mag number, or ↑4 distance (10 km).
This also works for circles; you can take the square root of the area to estimate the diameter, or square the diameter to estimate the area (and either is only off by about 10%). So a circle with diameter ↑3 would have ↑6 area.
A cube of ↑6 size has ↑18 volume–this is just 3x the mag size. A sphere of ↑6 size has ↑17.7 volume–this is 3x-.3, because a sphere has about half the volume of the cube it fits in. (You can probably get away with just 3x though.)
Usually we only need to focus on the integer order-of-magnitude and we can ignore any precision beyond that. But sometimes you do need a bit more precision.
Fortunately there are basically only 3 fractional mag numbers you need to know:
You can think of ^.5 as either 3 or pi (because the square root of 10 ≈ π), since for mag purposes they’re the same.
So to halve a mag number, for instance to convert from diameter to radius, subtract .3. Conversely, to double a mag number, add .3.
This also means that ↑9.3 is 2 billion (↑9 is a billion, .3 is 2x). and ↑3.7 is 5000 (↑3 is a thousand, .7 is 5x).
If you’re serious about developing facility with mag numbers/logarithms, internalize these. You should have a deep-seated sense that halfway between 100 and 1000 is 300 (actually 316).
Math Olympiads have an entire Fermi Question section, which logarithmetic is perfect for–they even want an integer for the answer! You can do quite well completely ignoring any precision beyond half an exponent.
In daily life, noting the mag level of every quantity you come across is very useful to establish common ground. And especially if you have data over time, noting when the magnitudes shift can be really powerful.
With time and practice, converting from linear numbers and doing math with mag numbers and then converting back, will become like second nature.