Code

Author Charles Petzold Pages 400 Disk N/A Level All Levels Published 10/11/2000 ISBN 9780735611313 Price $17.99 To see this book's discounted price, select a reseller below.	More Information About the Book Table of Contents Sample Chapter Index Related Series Related Books About the Author Support: Book & CD Rate this book

Chapter 9. Bit by Bit by Bit (Continued)

The flip side of this is that any information that can be reduced to a choice among two or more possibilities can be expressed using bits. Needless to say, there are plenty of forms of human communication that do not represent choices among discrete possibilities and that are also vital to our existence. This is why people don't form romantic relationships with computers. (Let's hope they don't, anyway.) If you can't express something in words, pictures, or sounds, you're not going to be able to encode the information in bits. Nor would you want to.

A thumb up or a thumb down is one bit of information. And two thumbs up or down—such as the thumbs of film critics Roger Ebert and the late Gene Siskel when they rendered their final verdicts on the latest movies—convey two bits of information. (We'll ignore what they actually had to say about the movies; all we care about here are their thumbs.) Here we have four possibilities that can be represented with a pair of bits:

00 = They both hated it.

01 = Siskel hated it; Ebert loved it.

10 = Siskel loved it; Ebert hated it.

11 = They both loved it.

The first bit is the Siskel bit, which is 0 if Siskel hated the movie and 1 if he liked it. Similarly, the second bit is the Ebert bit.

So if your friend asked you, "What was the verdict from Siskel and Ebert about that movie Impolite Encounter?" instead of answering, "Siskel gave it a thumbs up and Ebert gave it a thumbs down" or even "Siskel liked it; Ebert didn't," you could have simply said, "One zero." As long as your friend knew which was the Siskel bit and which was the Ebert bit, and that a 1 bit meant thumbs up and a 0 bit meant thumbs down, your answer would be perfectly understandable. But you and your friend have to know the code.

We could have declared initially that a 1 bit meant a thumbs down and a 0 bit meant a thumbs up. That might seem counterintuitive. Naturally, we like to think of a 1 bit as representing something affirmative and a 0 bit as the opposite, but it's really just an arbitrary decision. The only requirement is that everyone who uses the code must know what the 0 and 1 bits mean.

The meaning of a particular bit or collection of bits is always understood contextually. The meaning of a yellow ribbon around a particular oak tree is probably known only to the person who put it there and the person who's supposed to see it. Change the color, the tree, or the date, and it's just a meaningless scrap of cloth. Similarly, to get some useful information out of Siskel and Ebert's hand gestures, at the very least we need to know what movie is under discussion.

If you maintained a list of the movies that Siskel and Ebert reviewed and how they voted with their thumbs, you could add another bit to the mix to include your own opinion. Adding this third bit increases the number of different possibilities to eight:

000 = Siskel hated it; Ebert hated it; I hated it.

001 = Siskel hated it; Ebert hated it; I loved it.

010 = Siskel hated it; Ebert loved it; I hated it.

011 = Siskel hated it; Ebert loved it; I loved it.

100 = Siskel loved it; Ebert hated it; I hated it.

101 = Siskel loved it; Ebert hated it; I loved it.

110 = Siskel loved it; Ebert loved it; I hated it.

111 = Siskel loved it; Ebert loved it; I loved it.

One bonus of using bits to represent this information is that we know that we've accounted for all the possibilities. We know there can be eight and only eight possibilities and no more or fewer. With three bits, we can count only from zero to seven. There are no more 3-digit binary numbers.

Now, during this description of the Siskel and Ebert bits, you might have been considering a very serious and disturbing question, and that question is this: What do we do about Leonard Maltin's Movie & Video Guide? After all, Leonard Maltin doesn't do the thumbs up and thumbs down thing. Leonard Maltin rates the movies using the more traditional star system.

To determine how many Maltin bits we need, we must first know a few things about his system. Maltin gives a movie anything from 1 star to 4 stars, with half stars in between. (Just to make this interesting, he doesn't actually award a single star; instead, the movie is rated as a BOMB.) There are 7 possibilities, which means that we can represent a particular rating using just 3 bits:

"What about 111?" you may ask. Well, that code doesn't mean anything. It's not defined. If the binary code 111 were used to represent a Maltin rating, you'd know that a mistake was made. (Probably a computer made the mistake because people never do.)

You'll recall that when we had two bits to represent the Siskel and Ebert ratings, the leftmost bit was the Siskel bit and the rightmost bit was the Ebert bit. Do the individual bits mean anything here? Well, sort of. If you take the numeric value of the bit code, add 2, and then divide by 2, that will give you the number of stars. But that's only because we defined the codes in a reasonable and consistent manner. We could just as well have defined the codes this way:

This code is just as legitimate as the preceding code so long as everybody knows what it means.

If Maltin ever encountered a movie undeserving of even a single full star, he could award a half star. He would certainly have enough codes for the half-star option. The codes could be redefined like so:

But if he then encountered a movie not even worthy of a half star and decided to award no stars (ATOMIC BOMB?), he'd need another bit. No more 3-bit codes are available.

The magazine Entertainment Weekly gives grades, not only for movies but for television shows, CDs, books, CD-ROMs, Web sites, and much else. The grades range from A+ straight down to F (although it seems that only Pauly Shore movies are worthy of that honor). If you count them, you see 13 possible grades. We would need 4 bits to represent these grades:

We have three unused codes: 1101, 1110, and 1111, for a grand total of 16.

Whenever we talk about bits, we often talk about a certain number of bits. The more bits we have, the greater the number of different possibilities we can convey.

It's the same situation with decimal numbers, of course. For example, how many telephone area codes are there? The area code is three decimal digits long, and if all of them are used (which they aren't, but we'll ignore that), there are 103, or 1000, codes, ranging from 000 through 999. How many 7-digit phone numbers are possible within the 212 area code? That's 107, or 10,000,000. How many phone numbers can you have with a 212 area code and a 260 prefix? That's 104, or 10,000.

Similarly, in binary the number of possible codes is always equal to 2 to the power of the number of bits:

Number of Bits	Number of Codes
1	21 = 2
2	22 = 4
3	23 = 8
4	24 = 16
5	25 = 32
6	26 = 64
7	27 = 128
8	28 = 256
9	29 = 512
10	210 = 1024

Every additional bit doubles the number of codes.

If you know how many codes you need, how can you calculate how many bits you need? In other words, how do you go backward in the preceding table?

The method you use is something called the base two logarithm. The logarithm is the opposite of the power. We know that 2 to the 7th power equals 128. The base two logarithm of 128 equals 7. To use more mathematical notation, this statement

2⁷ = 128

is equivalent to this statement:

log₂128 = 7

So if the base 2 logarithm of 128 is 7, and the base 2 logarithm of 256 is 8, then what's the base 2 logarithm of 200? It's actually about 7.64, but we really don't have to know that. If we needed to represent 200 different things with bits, we'd need 8 bits.

Bits are often hidden from casual observation deep within our electronic appliances. We can't see the bits encoded in our compact discs or in our digital watches or inside our computers. But sometimes the bits are in clear view.

Here's one example. If you own a camera that uses 35-millimeter film, take a look at a roll of film. Hold it this way:

Click to view graphic

You'll see a checkerboard-like grid of silver and black squares that I've numbered 1 through 12 in the diagram. This is called DX-encoding. These 12 squares are actually 12 bits. A silver square means a 1 bit and a black square means a 0 bit. Square 1 and square 7 are always silver (1).

What do the bits mean? You might be aware that some films are more sensitive to light than others. This sensitivity to light is often called the film speed. A film that's very sensitive to light is said to be fast because it can be exposed very quickly. The speed of the film is indicated by the film's ASA (American Standards Association) rating, the most popular being 100, 200, and 400. This ASA rating isn't only printed on the box and the film's cassette but is also encoded in bits.

There are 24 standard ASA ratings for photographic film. Here they are:

25	32	40
50	64	80
100	125	160
200	250	320
400	500	640
800	1000	1250
1600	2000	2500
3200	4000	5000

How many bits are required to encode the ASA rating? The answer is 5. We know that 24 equals 16, so that's too few. But 25 equals 32, which is more than sufficient.

The bits that correspond to the film speed are shown in the following table:

Square 2	Square 3	Square 4	Square 5	Square 6	Film Speed
0	0	0	1	0	25
0	0	0	0	1	32
0	0	0	1	1	40
1	0	0	1	0	50
1	0	0	0	1	64
1	0	0	1	1	80
0	1	0	1	0	100
0	1	0	0	1	125
0	1	0	1	1	160
1	1	0	1	0	200
1	1	0	0	1	250
1	1	0	1	1	320
0	0	1	1	0	400
0	0	1	0	1	500
0	0	1	1	1	640
1	0	1	1	0	800
1	0	1	0	1	1000
1	0	1	1	1	1250
0	1	1	1	0	1600
0	1	1	0	1	2000
0	1	1	1	1	2500
1	1	1	1	0	3200
1	1	1	0	1	4000
1	1	1	1	1	5000

Most modern 35-millimeter cameras use these codes. If you must set the exposure manually on your camera, or if your camera has a built-in light meter but you must set the film speed manually, your camera doesn't use these codes. If your camera uses these codes and you take a look inside where you put the film, you should see six metal contacts that correspond to squares 1 through 6 on the film canister. The silver squares are actually the metal of the film cassette, which is a conductor. The black squares are paint, which is an insulator.

The electronic circuitry of the camera runs a current into square 1, which is always silver. This current will be picked up (or not picked up) by the five contacts on squares 2 through 6, depending on whether the squares are bare silver or are painted over. Thus, if the camera senses a current on contacts 4 and 5 but not on contacts 2, 3, and 6, the film speed is 400 ASA. The camera can then adjust film exposure accordingly.

Inexpensive cameras need read only squares 2 and 3 and assume that the film speed is 50, 100, 200, or 400 ASA.

Most cameras don't read or use squares 8 through 12. Squares 8, 9, and 10 encode the number of exposures on the roll of film, and squares 11 and 12 refer to the exposure latitude, which depends on whether the film is for black-and-white prints, for color prints, or for color slides.