why imperial fraction is better for binary than metric system

I found a metric system-promoting website, where they had a question-and-answer page trying to cut down all the arguments against converting to the metric system

Why Not Metric For Us?

Here is one of them that pertains to my thinking. It is the idea that the binary can be easily converted to octal or hexadecimal numbers, so we should use base 8 or base 16 for calculations.

Base 12 or 16 is better than 10, as 12 divides by 2,3,4 and 6, 16 by 2,4,8 and 16, while 10 divides only by 2 and 5.

1. Ok, sure, the bigger numbers you choose, by the more other numbers you can divide them. Unfortunately for a larger base you would need a larger set of symbols, too. So are you seriously suggesting a base 12 or base 16 system? While you are defending American culture and tradition with the English system, you are asking people to give up such a tradition as the decimal system and use an arbitrary base-12 or base-16 notation for measuring? Please: Keep the decimal system, but abolish the English system!
2. How about dividing 16 by 3 or 12 by 5? The fractional beauty lasts not too far. You can always find numbers that your base does not divide by. With decimals, you can represent any number to any accuracy.
3. And after all: Base 12 or 16 would not help the English system but only make it even more convoluted! Base 12 is useful for inch-foot conversion only. For merely all other conversions it’s useless! Same for base 16, which is useful maybe for oz-pound conversion, but an even greater nightmare for everything else!

I finally decided to look for where Professor Roger Doering said made metric system bad. Here is the lecture he gave:

Professor: If you want to deal with a .8, this becomes an interesting process. .8 is nasty to represent a binary. It won’t be clean. It’s an infinite series. Like representing 1/3 as a decimal number. It’s not clear. We can change it and make it clean but maybe we should work on cleaning until you see what’s going on. We’ll convert the .8, add the binary point. We’ll start with .8 and we’ll double .8. So I’m going to double that. It becomes 1.6. At that point, I take the digit in front of the binary point and I write it up here. Okay. So we’re starting with this digit. [on board]. Then we get to subtract that so it’s now .6. Double that and becomes 1.2. And write the 1. Okay. Then I take off 1, take the .2 and double it to .4. Write the 0. I double . 4. It becomes .8, write a 0 and go oops I’m back to where I started. It’s an infinite repeating binary fraction. Not very pretty, is it. So yeah?

Male Student: So from a different value it’d terminate right?

Professor: If we put 3/4, it’d terminate very nicely. If we choose .75, double that, we get 1.5. Which will be a 1 there. Take away the 1. Take .5, double it, get another 1. And we’re done. We have a 1/2 plus a quarter. That’s how the digits work. If you look at the value of digits in decimal, this is 2, 1, 4, 8… [on board]. Going the other direction, this is 1/2, 1/4, 1/8. Those are the values of the digits. So numbers like 1/2 a 1/4 and 3/16 are wonderful in the binary system. People with the metric system really hurt us. I’m not kidding. Fractions don’t work well on the computer. The old fractions we used did. So not a wonderful change.

So, what he seemed to have said was that .8 was not easy to represent in binary without getting trapped into loops and approximation. We can say he doesn’t like .8 or .9, he prefers .75, .5, .25, and anything of those nature divided by 2.

And that was how he defended how we divided the inch, always by 1/2, 1/4, 1/8, 1/16, 1/32. In fact, those numbers are exactly 2 to the power of a negative number. 2^-1, 2^-2, 2^-3, etc. and is just reverse of 2 to the power of a positive number.

I was wrong, however, in thinking that the inch was also divided by 1/3. It’s not. It’s only divided by 1/2 repeatedly. So, there is something graceful about 1/256, 2/256, 3/256, that make binary less nasty.

It looks like from the lecture, .75 is presented as 0.11

.25, double it, and get .5, so that’s a 0, double it again, and get a 1, so I assume it’s represented as 0.01

.5, double it, and get 1, so it would be 0.10

Very neat idea.

How about 15/16, which is .9375. double it, and get 1.875

.875 * 2 = 1.75

.75 * 2 = 1.5

.5 * 2 = 1

Therefore, you get 0.1111

Whereas, 3/16, which is 0.1875

.1875 * 2 = .375

.375 * 2 = .75

.75 * 2 = 1.5

.5 * 2 = 1

Therefore, you get 0.0011, which is quite right and elegant because 00 = 0, 01 = 1, 10 = 2, and 11 = 3.

Beyond the inch, however, the imperial system is indefensible. 12 inches in a foot, 3 feet in a yard, etc.

Alan

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