Base 10 and Binary

Patterner

I can't help but wonder if any of this is of interest outside of my own head. :rofl: But anyway...

I was just watching this video about Perfect Numbers. I didn't get very far, because I went off on a tangent when he showed what the first four PNs are in binary. I understand the concept of different bases easily enough. I remember a night class in college right out of high school (43 years ago) where we converted numbers into Base 16 (with A-E representing 11-15) and into binary.

I was somewhat lost back then, just floundering, and simply stopped going to most of my classes after the first few. I don't know if there was any attempt to teach us how to do things in the other bases. So here I am tonight, teaching myself how to multiply in binary, and I have some thoughts.

First, multiplying in binary is really easy. The funny thing is, although I know the answer is correct, I don't know what it is. Just making random strings of 1s and 0s, I came up with the two factors of 110100101 and 11011. Multiplying verticality, I got 10110001100111.

I don't use binary, so I don't recognize what even the smaller factor is. It only takes a few seconds to figure out that it is 27. I didn't want to risk adding wrong in my head when converting the larger factor, so I put 256 + 128 + 32 + 4 + 1 into the calculator. It's 421. I didn't even bother trying that with the product. I just went to a site that translated binary to Base 10. It's 11,367.

And yes, 421 x 27 = 11,367. I multiplied it correctly in binary. But I had no idea what the numbers were.

My second thought is a question. Do people who work with binary often enough think in binary the way the rest of us think in Base 10? It's definitely more cumbersome, because you can't easily see what column all of the numbers are in a string like 10110001100111. It's probably easy to see it as 101100001100111, or 101110001100111, or even 110110001100111. I assume they use commas, so maybe it's 10,110,001,100,111. Someone who works in binary a lot must surely be able to see which columns all the 1s are in this example. But its gotta be difficult for a number like 111,101,011,000,100,100,011,001,111,001,000. And that's only 8,238,805,960.

But still, do such people actually think in binary? Or do they just more easily convert it to Base 10 to make it understandable? Because I would think that thinking in binary means the numbers are called different things. 21, 31, 41, 51, and 61 are called what they are because of the base they are in. You wouldn't call 101001 "forty one" if you thought in binary. Do we have non-Base 10 words for numbers in binary?

My third thought is another question. Why do we use Base 10? Doesn't it make more sense to go to the next value after you have used up all your fingers? I hold up fingers for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, then my friend holds up one finger for 11. Although I guess I should rewrite that. My tenth finger could be *. Then we would write:
1, 2, 3, 4, 5, 6, 7, 8, 9, *, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1*, 20, 21...
And, again, we'd need new words, because 11 in Base 11 is not this many Xs: XXXXXXXXXXX
It is this many: XXXXXXXXXXXX

My fourth thought is wondering if people using different mathematical bases would have different thinking in general. Maybe not. Maybe it wouldn't go much beyond the math.

Leontiskos

I would suggest doing some historical reading on the ways that different cultures used different bases. For example, "History of Bases used in Ancient Civilizations."

noAxioms

I multiplied it correctly in binary. But I had no idea what the numbers were. — Patterner

You did know what the numbers were since you had the answer. You just didn't know the base-10 representation of that answer any more than you knew the base-13 representation of it, but none of those representations is 'what the number is'.

And that's only 8,238,805,960.

Most people have little idea what that number is either. Sure, it's a string of digits, but enough to be little more than an abstraction corresponding to no relatable quantity. It's probably big in most contexts, but how big? Words fail to convey. You can talk about human population, but having little real concept how many people there are, what does that tell me?

My second thought is a question. Do people who work with binary often enough think in binary the way the rest of us think in Base 10?

Not often, but I did work in hex enough to memorize the base-16 multiplication table. I hate octal, but hex is your friend. Somehow I find it far more friendly.

But yea, computers mostly do math in binary. My old TI-59 calculator did it internally in decimal. It was wicked slow compared to the HP which stored its numbers more efficiently in binary. But the TI needed a far simpler circuit to put the answer up on the display.
The HP taught me the power of RPN, and was a precursor to how programming should be done.

You wouldn't call 101001 "forty one" if you thought in binary. Do we have non-Base 10 words for numbers in binary?

Sure. Put commas in there to group them 3-4 at a time, just like we do with decimal. The word 'hundred' and such are applicable, but it's probably more terse to just say "OneOhOne, One". I do that with decimal all the time, skipping the optional words.

My third thought is another question. Why do we use Base 10?

Duh... Count the fingers. In binary, a dexterous person can count to a thousand with those fingers. I regularly hold counts up to 100 with just that (One digit per hand, thumbs count as a 5), but I cannot do the binary thing with any efficiency.

My fourth thought is wondering if people using different mathematical bases would have different thinking in general. Maybe not. Maybe it wouldn't go much beyond the math.

Never mind that. You want a debate? How about the whack job (some Arabic nut case) that invented big-indian numbers, a travesty that Motorola/unix decided to continue, and Microsoft/Intel descarded in favor of little-indian (actually little Endian for those unfamiliar). The former is the source of so many computer bugs, but of course the Arabic guys (and the Romans as well) didn't think of that at the time. With the Romans, you at least know when the number was going to end. Not so with the Arabic system, where you knew nothing until you were told that the number was done. A big-vs-little endian debate is a worthy one, and one which I absolutely have an opinion.

Patterner

Duh... Count the fingers. — noAxioms

You don't understand my point. which is darn likely my fault, since I never tried to express it before.

I multiplied it correctly in binary. But I had no idea what the numbers were.
— Patterner
You did know what the numbers were since you had the answer. You just didn't know the base-10 representation of that answer any more than you knew the base-13 representation of it, but none of those representations is 'what the number is'. — noAxioms

The number had no meaning to me in binary. I could do the math, but there was no meaning. I find that an odd situation. Whenever I do math in Base 10, I know what the numbers mean. When I do math in binary, I'm just like a calculator, or like the Chinese Room, or the Game of Life, simply manipulating things according to rules.

Never mind that. You want a debate? — noAxioms

I do not. I wasn't aware I said anything that would invite debate. I'm just wondering how much of our thinking that isn't about math is, nevertheless, affected by math, and how different it would be if we thought in binary.

noAxioms

You don't understand my point. which is darn likely my fault. — Patterner

OK, re-reading, I think I see your point. Never mind the friend, since with enough friends, counting becomes base ∞, which reduces to simply tally marks.
So your point is that 11 values (0-10) can be represented with 10 fingers, so '10' should instead be the next number beyond that, rather than the redundant <all fingers> is the same value as (1 in the next column). Interesting point, but the next column is a count of handfuls, not a count of 'more than handfuls'. Best I can think of.

Nobody seems to do binary arithmetic. It's a cute exercise, but awkward for people. The c language lets you directly specify constants in base 10, 8, or 16, but not binary. There's just no call for it.

The number had no meaning to me in binary. I could do the math, but there was no meaning.

Well that's mostly because you don't work with binary regularly, so you haven't learned the feel of what the numbers are. The example of the big decimal number was my example of us not really having much of a feeling for big decimal numbers either. After a while, it's just a string of digits.

To illustrate, Tegmark computed the distance to the nearest exact copy of you. It was a huge number.10**(10**28). What units you might ask? With a number like that, units don't matter. Angstroms? Parsecs? It's the same number either way. If you don't believe that, try to convert that number of angstroms to the value in parsecs.

Never mind that. You want a debate? — noAxioms

I do not. I wasn't aware I said anything that would invite debate.

It was a rhetorical question on my part. It's one of those topics that divide people who care, kind of like the Monty Hall problem.

I'm just wondering how much of our thinking that isn't about math is, nevertheless, affected by math

Most thinking has been at least partially about math, long before numbers were discovered. Consider the calculus needed to throw a spear accurately.

and how different it would be if we thought in binary.

Most math isn't done discreetly at all, so the base is irrelevant. But the math we learn in school would admittedly be pretty awful if it was done in binary. We'd need more than just commas to help parse the strings of digits.

Patterner

OK, re-reading, I think I see your point. — noAxioms

Which may not be a legitimate point. I go back and forth on it. but I kind of think that, when you run out of fingers, you have to move up to the next value.

The number had no meaning to me in binary. I could do the math, but there was no meaning.
-Patterner

Well that's mostly because you don't work with binary regularly, so you haven't learned the feel of what the numbers are. — noAxioms

I don't know if it's possible, though, between the length strings can be making it impossible for people to know what the number is, and the fact that we don't have a naming convention for anything but base 10. we always convert back to base 10, which means we're not thinking in base two. That would be the case with any other base. If we worked in base 11 a lot, we would quickly come to convert 23 to our 25. so 23×4 = 91. But that's doing the multiplication in base 10 in my head, then converting the answer to base 11.

It was a rhetorical question on my part. — noAxioms

Ok. Wasn't sure if I had said something that seemed like a challenge. :lol:

Patterner

I would suggest doing some historical reading on the ways that different cultures used different bases. For example, "History of Bases used in Ancient Civilizations." — Leontiskos

Interesting. Thanks!

Base 10 and Binary

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