Mag Math is easy

In Mag World, the mechanics of doing math is called logarithmetic, and it was discovered around 1600 by John Napier, who also invented the decimal point notation we use today for fractions.

The secret of logarithmetic is that multiplication becomes addition, and division becomes subtraction. This one property makes them so convenient for computation, that a mere 2 years after Napier published his logarithm tables, William Oughtred invented the slide rule, which remained the most commonly used calculation tool in science and engineering for over 300 years (until the handheld scientific calculator was introduced in 1972). The “slide” of one ruler against another does a kind of analog addition, which becomes multiplication since both rulers have logarithmic scale markings.

But we are getting ahead of ourselves. First, we need to learn how to enter and leave Mag World.

Converting Into Mag Numbers

How would you convert “93 million miles” into its mag Number?

There are two ways. The precise scientific way is to convert the number into standard metric units and then use a calculator to get the base 10 logarithm, including one digit past the decimal. You can use this method if you’re ever unsure.

But once you’ve gotten the hang of mag math, that level of rigor is usually not necessary. Here’s how to calculate the magnitude of a number in your head:

1. Convert numbers

If a number is a power of 10, like 1000, count the number of zeroes. For 1000, this is ↑3.

If it’s close to a power of 10, like 900 or 1500, you can just round it to the nearest magnitude, in this case 1000 (which would also be ↑3, since in Mag World they’re both “about a thousand”).

Otherwise you can add a single decimal, based solely on the first digit:

• 2000: ↑3.3
• 3000: ↑3.5
• 5000: ↑3.7
• 8000: ↑3.9

You can interpolate if you want, but generally just “rounding” to the nearest half is good enough for government work. For instance something in the range of 500 to 5000 might be rounded to ↑3 or ↑3.5 or ↑4.

2. Convert multipliers

You should memorize these 8 standard metric prefixes if you don’t know them already:

• ↑-12: pico- (trillionth)
• ↑-9: nano- (billionth)
• ↑-6: micro- (millionth)
• ↑-3: milli- (thousandth)
• ↑3: kilo- (thousand)
• ↑6: mega- (million)
• ↑9: giga- (billion)
• ↑12: tera- (trillion)

Also, be aware that there’s a British million which is ↑9 and a British billion which is ↑12, although they are much less frequently used today.

3. Convert Units

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.

By the way, if we use a scientific calculator to multiply 93,000,000 miles by 1609.344 (the number of meters in a mile) and then take the log base 10, we get the answer with all of its glorious precision: ↑11.17513183375531 meters. We could include more precision in our answer, by rounding to ↑11.2 or ↑11.18 or ↑11.175, but in truth no one will be any wiser.

By the way, 93 million miles happens to be the distance from the Earth to the Sun, commonly known as 1 Astronomical Unit (AU). Making note of anchor points like these (↑11 meters = 1AU) helps internalize the mag scale.

Powers and Roots

To 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) × ↑3² (km²) = ↑8 area. Note that “kilometer squared” doubles the kilo (↑3² = ↑6) but not the coefficient. 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.)

Estimate like Fermi

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.

Example: Energy Usage

The world used about 180,000 TWh in 2023.

Large-scale energy usage is measured in “terawatt-hours per year”, but it’s a bit weird that this has two different time units in it. Of course, energy usage is not quite the same as instantaneous 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:

• ↑5 (“180,000”)
• ↑12 (“tera”)
• ↑0 (“watt”)
• ↑3.5 (“hours”; 1 hour = ↑3.5)
• ↑-7.5 (“per year”; 1 year = ↑7.5, subtract because of “per”)

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.

Example: light-year

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: ↑15.975924670013155. ↑16 is close enough for almost all purposes.