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  • As part of doing bits and bytes and fairly low-level stuff, we have mentioned

  • the possibility of a thing called 'binary coded decimal' and that, effectively, IBM

  • in their mainframe days, led the charge on this. Now I've done a video on this and how

  • it led, eventually, to extended binary coded decimal (EBCDIC). I would like to talk about

  • what is BCD, why is it necessary and in particular that's to answer a question

  • that I did, vaguely, think about in my early days of doing heavyweight, macho

  • calculations [on a computer] overnight. It's effectively: "How does this stuff ever get printed out?"

  • I thought, well, what's the biggest ratio between actual computation and fairly

  • quick printout that I can think of. Not my quantum calculations, they're

  • minor-league. Those of you who are familiar with Douglas Adams and "Hitchhiker's

  • Guide to the Galaxy", which was made into a BBC series here and also became a

  • movie, I think, will know that Douglas, who was a very far-seeing guy, had this

  • idea of the ultimate computer called Deep Thought. It was [an idea] years ahead of its

  • time, because I'm pretty sure that Deep Blue - IBM's chess machine; Deep Mind - the AI

  • machine that's doing GO, is it, and so on. And you know Deep Learning,Deep

  • everything. I'm not at all sure it isn't all down to Douglas Adams! >> Sean: I just have to

  • point out here that the ultimate computer was the Earth! >> DFB: Oh in the very end ... ?

  • >> Sean: And there will be some fans pointing out...I >> DFB: Oh! I see. Nevertheless, Deep Thought

  • was asked: "Deep Thought: what is the answer to Life the Universe and

  • Everything?: And several million years later, presumably even with quantum assistance.

  • [Music] >> Deep Thought: "You're really not going to like it!" >> High Priests: "Tell us!"

  • The answer to Life the Universe and Everything is: [voices on film] 42 42 42 >> DFB: 42 ! And my first thought

  • when I was laughing out loud at that was, I thought: "What's 42 in binary ?!" Is there

  • something special about it? And, yeah, how would you convert the binary string for 42

  • into being literally 4 and 2 on a piece of line-printer output? 42 in

  • binary is 101010. How do we know that that's 42?

  • Well, starting at the right its powers of two. So, 2^0 is units ,then 2s

  • then 4s then 8s then 16s and 32s. Armed with that knowledge we say OK

  • it's a 2 plus an 8 which is 10 plus a 32 is 42. Pretty good look! A six-bit

  • representation for a 2-digit decimal number. But if you feed that into a

  • printer you're not going to get 42 printed out, because by and large they do

  • not interpret binary. So, where does the printable form of '42' come from? And the

  • answer is it goes way back to the fact that IBM and their computing machines -

  • way back at the start of the 20th century they were the Hollerith company. You

  • know, doing census details; controlling elections from punch cards and some IBM

  • early electronic computers I think - the IBM 650 was it? - actually did work

  • internally on decimal. Just like Tommy Flowers [and Colossus]. I think they both used bi-quin.

  • But the pressure was on, from binary-based hardware to say: "This [binary] is more

  • efficient by far. You'd be mad not to use it". OK?,says IBM, but although it's got

  • to be a binary representation we want to directly link it to the decimal that

  • will ultimately be printed out. Now think about it, that's easy to do but

  • what would this binary-coded decimal be for 42? Every decimal position can range

  • from 0 to 9. So, how many bits do we need to represent 9? Well, we know that 3

  • bits isn't enough. 3 bits goes up to 111, which is 7, but you need a 8 and 9. So

  • you've got to go to that 4th bit. And you end up, of course, with 8 being 1000

  • and 9 being 1001. Ah! - I hear several of you saying - but you don't stop there, you

  • can go on to hexadecimal?! And this is precisely the thing with BCD. You must

  • *not* let it go over into the hexadecimal range from 10 - 16 [correction: 10 - 15] because the average

  • person wants their answers out in decimal - not in hexadecimal. So, here we go then. Look 0100

  • taken as a grouping of bits, on its own, that's nothing in the units column;

  • 0 in the 2's column; 1 in the 4s column; 0 in the 8s column. That

  • represents decimal 4. Right next door to it, is a separate 4-bit entity. If I write

  • down 0010 that represents a 2. In its simplest form that is what binary

  • coded decimal is. And you just use them in four-bit nibbles. Now we all know a

  • nibble is half a bite. A byte equals 8 bits - well it does in the modern world.

  • So half a bite? Well, the name 'nibble' caught on for

  • obvious reasons. A nibble being a small bite and I'll use an 'i', but some people

  • like to extend the joke as much as possible and actually spell nybble with

  • a 'y'. I don't mind! And then a nibble can hold a hex digit. You might say: Oh! well -

  • that's it then - if I fed 0100 down a serial line, into

  • a printer, it would cough into life and print 4 ?" No. Not quite. But we're getting

  • close. Because what we've got to ask ourselves is ...

  • this whole print-out thing is treating decimal digits as characters. They're not

  • being thought of in their numeric sense at all. It's just any other character. It's

  • like an 'A', a 'B', a'Z', a '!' or whatever. The ASCII committee in the

  • 1960s didn't just work in a vacuum. They knew what IBM - who they loved

  • and hated - had been doing for years. And it basically said: "It is so much easier

  • if you base what's printed ultimately on a BCD representation, but put a special

  • marker at the front of the BCD to pad it out to an 8-bit byte. But do it in such

  • a way that, in a sense, the codes that are going to do 0 - 9 are in a 'sensible' place.

  • Now, what does that mean. Well, let's do ASCII first, even though historically

  • it was second. In ASCII the digits occupy from 30 (hexadecimal) to 39 (hexadecimal)

  • What it means is, if you have

  • got, shall we say this 4 (0100). That is your hex nibble for 4. All you have

  • to do, to make it printable, as an ASCII 4, is to glue on the left hand end

  • hexadecimal three (which is the same as decimal three) which is 0011. So 0011

  • prepended to 0100 gives you an 8-bit entity which, if you fire that down at a

  • line printer it'll cough into action and print a 4.

  • Notice that when I say "glue on at the front", in order for it to be efficient

  • you don't want to be adding on something that will cause carries, if you see

  • what I mean. You need to park it [the ASCII boundary where digits start] on a multiple of 16,

  • and then what will happen is that whatever way you convert your 4 into

  • being 34 (hex) it will not cause ripple carries, which are inefficient. OK, say the

  • IBM mainframe types. Why not (historically) tell the youth of today what IBM did originally?

  • Same idea! ASCII learned a lot from what

  • IBM did. In IBM EBCDIC you don't prepend a 3 (0011 hex), you prepend an F (1111 hex)

  • But the principle is the same. Yeah? it's cleanly on that boundary.

  • So, that's an absolute crucial fact in making BCD-to-printed-out-results be fast and

  • efficient. You need to be able to put something in there that's low cost

  • because you're going to be doing millions of these BCD to ASCII/EBCDIC

  • conversions. So that is a very sort of crucial fact to get hold of to try and

  • reduce the conversion burden before print out. So, in many ways then that's

  • much of the story. If you once got it into BCD then both

  • ASCII or (if you're still on IBM mainframes) EBCDIC, do make it easy for

  • you to get into a printable form very quickly. And of course the I/O routines

  • hidden underneath Fortran and C will be well aware of this. Converting from pure

  • binary 101010 to 42 BCD (0100 0010), they look very different! When you get

  • onto telling us how to do it, it's not going to be a cheap operation is it?

  • Because that's the hard bit. It's all very well saying [that] getting from BCD to ASCIIk

  • is a piece of cake, but what about getting from binary to BCD? And the answer is

  • there's bound to be overheads there and it's bound to be something that you

  • probably want to eliminate? And to that effect, some people in the

  • commercial computing set-up said: "Look, binary to BCD will turn out to be

  • expensive. Tell you what - If what we do is totally trivial [arithmetically] and really isn't rocket

  • science, wouldn't we be better off to try and devise software, or

  • even hardware assistance circuits, to do all the arithmetic on a BCD notation?

  • Never convert it into pure binary! Because if all you're doing is adding up how

  • many voters in, y'know, Norwalk County or something, have voted for such-and-such,

  • you don't need all these huge great binary things that numerical analysts use

  • You're just counting numbers. And even if you're looking at somebody's balance,

  • in dollars and cents, surely that's simple enough arithmetic. It's better to

  • perhaps pay a little penalty for doing BCD arithmetic because you can overcome

  • that penalty, a bit, by having specialized hardware. Why not do it all in BCD and

  • then you don't have the binary to BCD overhead? All you have to do after you've

  • done your BCD arithmetic is slip a FFFF [correction: 1111] on the front end if it's EBCDIC

  • a 3 (hexadecimal) if its ASCII, and that's it. [It] prints out. There is one

  • classic one where you really can pick up that it's very advantageous to do it in

  • BCD. If you think about the number 0.10, which might represent ten

  • cents shall we say. You know, your bank account's been drained down. It's

  • down to its last 10 cents. So here you've got 0.10.

  • That's .1 (in decimal). What is it in binary? Oh dear!

  • You look at what it is in binary: [looks at notes] 0.000110011001100 ....

  • And it goes on for ever! There is no exact representation of .1 (decimal)

  • as a binary expansion. It just doesn't stop. And accountants and actuaries get

  • paranoid about that: "Oh! I know it'll never happen but I hate the idea that

  • the rounding might go wrong and imy client's balance might drop to

  • 9 cents instead of 10 cents (!) How about some other examples of things that use

  • BCD, just to finish off with? I don't know if I've got it here ...

  • to dig it out of there, Sean. eventually [reaches for desk drawer] It's a little [electronic] hand calculator. What better

  • place to use BCD? It's utterly display- dominated. Apart from things like square

  • root - that's probably about the most complex thing that you can ask a simple

  • four-function calculator to do. But mostly it's additions, subtractions

  • simple divides and so on. They use BCD! Another example of simple

  • devices [that] you see in a shopping mall, or whatever, that could use BCD. Digital

  • clocks. It makes eminent sense to use BCD

  • because it's the display, and the change in the display, is happening all the

  • time - every second = but the actual incrementing is trivial. You don't need

  • to convert into binary to add one second to a digital display,

  • you really don't. Live with the BCD. Focus on the display, because that's what

  • matters above all else. i

As part of doing bits and bytes and fairly low-level stuff, we have mentioned

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二進制編碼十進制(BCD)與道格拉斯-亞當斯的42 - Computerphile (Binary Coded Decimal (BCD) & Douglas Adams' 42 - Computerphile)

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    林宜悉 發佈於 2021 年 01 月 14 日
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