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The rest of the world wonders why the United States still uses their old system of measurement which has a more complicated scale than using the metric system, employing a more standard and uniform scale of measurement. The British implemented the imperial system in America when they invaded but after America gained independence, why didn't they change after the metric system was created. Well, part of it is because of the history between the Americans and the French, who created the metric system. Despite that though, the US seriously tried converting to the metric system but the reason why they still aren't using it as the standard measurement is pretty simple.

Switching to metric is, in a sense, like switching to another language. If you’re not American, picture this: how would you feel if your government enacted a new rule that forced you to switch to the imperial system? Yes, the metric is simpler and uses fewer units, but rational reasons aside, you’d be furious simply because you’d have to change the frame of reference you’ve been using to all your life. The UK switched to metric in 1965, and this happened only because the industry forced it. UK companies were simply having too much a hard time trading with European countries. Even 50 years later, many Britons still refuse to move entirely to metric. Distances are still measured in miles, yards and inches, weight in pounds and stones; liquids in pints and gallons.

(Image credit: Wikimedia Commons)

Your last paragraph is parallel to why the field I work in uses eV for temperature.

A select few special people get rather indignant if you mention that scientists use any non-SI units. It seems amusing some would pick that hill to die on.
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I rarely have had to know more than pi~3 for the vast majority of my work. 3.14 or 22/7 or 3+1/8 (what your binary works out to) is about the limit of what most people would need if doing multiplication with paper and pencil, or on a slide rule. Base 7 would be far better since the classic approximation 22/7 -> 3.1, but that seems penny wise pound foolish.

I find the obsession some have with memorizing pi a bit odd. Nothing wrong with it, but ultimately impractical. If someone were worried about the desert island scenario where they don't have access to a book, calculator or computer, I think they would be far more versatile with a bit of calculus, as you could then figure out how to calculate pi, do it once, and write it down.

As far as actual constants with units, they are arbitrary and you can change more than one of them to be convenient whenever you want. General relativity is commonly done with what we called theorist units, where c and G are set equal to one simply to avoid writing those lets in equations a lot (and suddenly time, distance, mass and energy are all now the same unit value...). Otherwise the updated SI just picks these values to be consistent with what we already use (including Avogadro's constant which will have an exact value).
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In my limited, biased personal experience, a lot of Americans have a rough idea of metric units (except maybe air temperature), but I've heard complaints from people from outside of North America about how it is simply impossible to convert units without a computer.

My issue with converting is not signage and labels, but screw threads. I use metric threads for most projects, but some still need imperial threads and I try not to mix. For example, 1/4-28 is used on vacuum parts, and I don't want an M6 anywhere near that. It is one thing to change a product over when it is stand alone and after market parts are willing to change for new models (e.g. for a car), but harder when you have to interface with existing equipment. Even then, there is a lot of science equipment that is spec'ed metric only, but comes only in 12.7 and 25.4 mm sizes.
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Switching base isn't a unit change but rather a change of notation. Switch to base-2 and you'll still have that the circumference of a circle is 2*π*r (or τ*r for those so inclined). If you decide to switch to base-π for your numbering system then how many sides does a triangle have? "3.14" base 10 at only 3 significant digits is more accurate than "11.00100" base 2 (3.125 base 10). "3.24" base 16 is even more accurate. That's because the significance of a significant figure depends on the base. Trigonometry in the SI system is based on radians = units of π, not base-60/360 degrees. The decimal compass system is called the "gradian", with 100 gradians in a right angle. My calculator in school supported it.
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I think the big difference in the USA with regard to Metric units is that, although they are official, there is usually no legal requirement that enforces their use by parties who wish to transact based upon the old units. If the farmer at the market wants to advertise that her produce costs a certain amount "per pound", they may do so, and if "per kilo" that is fine also according to the law. Where laws exist that insist upon display of measures, such as 21 CFR 101, they require both the currently common and the S.I. units.

So no "Metric Martyrs" in the USA, they have the liberty to describe distances in "smoots" and speeds in "furlongs per fortnight" if they choose to do so.

Also the need to standardize for commerce (and calculating tarriffs!) that existed between pre-metric European states doesn't exist so urgently because of the vast size of the US market with its pre-existing standard of measurement already settled.

I find any efforts to describe a country's traditional units as "arbitrary" and SI units as "scientific" as pretty amusing. In many cases it is a Chesterton's fence issue - for example the Fahrenheit scale has a unit size that is equivalent to the "just noticeable difference" in temperature of the average human and the range of 0°F to 100°F is adequate to describe outdoor temperature in all but the most extreme weather in populous places. There is often a lot of cultural knowledge and utility buried in tradition.
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