Data Representation CS280 – 09/13/05

Binary (from a Hacker’s dictionary) A base-2 numbering system with only two digits, 0 and 1, which is perfectly suited for electronic operations since it can be expressed by power states (on/off), voltage levels (high/low) or charge (positive/negative), but is less than ideal for humans, who find it awkward to say things like “It’s a Catch – situation,” “He’s the pound gorilla,” and “That’s the dollar question.” “There are only 10 kinds of people in the world…those that understand binary and those that do not.” (from the ACM CS t-shirt).

Looking at more data Character representations

Data and Data Representation So how is this data that we operate on stored in the computer?

Let’s start with numbers Binary codes are Base 2 We “think” and operate in Base 10. What does this mean?

Counting Base 10 has 10 digits to represent different numbers of things Base 2 has only 2 digits available. Counting

Base then we run out of unique digits. So we move to a positional system. 10 – means we have ten things – a 1 in the 10’s position and no more things in the 1’s position.

Binary counting 0 1  then we run out of digits 10 This represents the number 2. A 1 in the 2’s position and a 0 in the 1’s position.

Positional notation 10’s positions represented 1 * 10 0 = 1 1 * 10 1 = 10 1 * 10 2 = * 10 3 = 1,000 1 * 10 4 = 10,000 1 * 10 5 = 100,000 1 * 10 6 = 1,000,000 2’s positions represented 1 * 2 0 = 1 1 * 2 1 = 2 1 * 2 2 = 4 1 * 2 3 = 8 1 * 2 4 = 16 1 * 2 5 = 32 1 * 2 6 = 64

How does the computer then store numbers? Let’s say we want to represent the number 53 in binary = Why? See chart next page.

Converting from binary to decimal Use chart 1 * 2 0 = 1 1 * 2 1 = 2 1 * 2 2 = 4 1 * 2 3 = 8 1 * 2 4 = 16 1 * 2 5 = 32 1 * 2 6 = 64 1 * 2 7 = * 28 = decimal Must use 7 bits xxxxxxx 75 – 64 = 9 1xxxxxx 32 and 16 are not used 100xxxx 9 – 8 = xxx 4 and 2 are not used last digit is What is the general subtraction algorithm to convert from binary to decimal number?

Converting binary to decimal 274 – decimal 4 * 10 0 = 4 7 * 10 1 = 70 2 * 10 2 = 200 total – binary 1 * 2 0 = 1 1 * 2 1 = 2 0 * 2 2 = 0 1 * 2 3 = 8 total11 What is the general algorithm for taking a number in base X and converting it to its base 2 equivalent?

Number representation Numbers are represented by their corresponding binary representation We are disregarding sign We are disregarding floating point What about other kinds of data?

Think about the binary values as a kind of code.

The binary values represent codes How many different values can be stored in 1 bit? How many in 2 bits? How many in 4 bits? How many in a byte?

General form encoding If you have x possible unique symbols, and y positions for any one of those symbols, then the general number of unique codes is x y Example, you have 2 dice each of which has 6 different face values, so there are 36 or 6 2 possible unique codes.

ASCII codes represent characters of data Use 1 byte or 8 bits Unicode extends the Ascii codes by another byte. ASCII can form most of the characters used by “Western” languages along with punctuation symbols. Unicode allows for special symbols and symbols in other languages like Japanese, Chinese, Arabic

Figure 8.7. ASCII, The American Standard Code for Information Interchange (page 220)

Reading the chart Left column is the left side of the byte (group of 8 bits) (another term is the high order) Right column is the right side of the byte. Value is the corresponding binary code.

Binary to hex Hexidecimal (base 16) codes can be used to represent groups of 4 binary digits. Hexidecimal counting: A B C D E F A = 10Binary 1010 B = C = D = E = F =

So the letter Z can be abbreviated in binary 5 A in hex Commonly binary numbers are represented in groups of 4 numbers with the leading 0’s used as placeholders. Hex numbers are shown as 2 digit with a space in between each group of two.

Encoding – character string Text or character strings are typically contiguously stored in memory. Assume that each character takes up one byte of space, how many bytes would be required for a phone number (we are using a slightly different example than the book. Note the hyphens and spaces:

568 – 8771 – requires 10 bytes E

In class assignment Using the chart on page 220, what is your first (or nick) name in ASCII binary codes? Work with your partner. Write the first name (spread out). Write the binary code for each letter of your name based on the ASCII chart. Convert at least one of those binary codes to the decimal (base 10) equivalent.

What about other kinds of data? Chapter 11 material

Pixels A pixel is like a dot. Your computer screen is composed of thousands of pixels. How many? Settings – Control Panel – Display – Settings Screen area is the dimensions expressed in terms of pixels. Higher the number the better the resolution.

Each pixel Has a color associated with it. Colors are a combination of red, green, and blue light – RGB The intensity of the particular color defines how much of that color contributes to the overall color displayed. Each color is associated with a 1 byte code. In one byte we can have values from 0 (no color) to 1 (full intensity).

Color See example in Word document Black is coded red green blue White is coded We will also use this feature when we code HTML colors.

Sound Analog – real world – infinitely continuous Digital – representation - discrete Sound is a continuous series of sound waves. To digitize we cannot capture every infinite value that hits our ears. But we can sample the values.

Figure Sound wave. The horizontal axis is time; the vertical axis is sound pressure.

Figure Two sampling rates; the rate on the right is twice as fast as that on the left.

Figure (a) Three-bit precision for samples requires that the indicated reading be approximated as +10. (b) Adding another bit makes the sample twice as accurate.

Figure Schematic for analog-to- digital and digital-to-analog conversion.

Sampling While we lose some information in this process, it is usually negligible in terms of our ability to perceive the sounds.

But to produce sounds Requires a large amount of data. For example, at a 16 bit representation of each sound, it would take 10 megabytes to reproduce 1 minute of a song. Compression – Remove the parts of the sound that we cannot hear. – MP3 format.

Images Images have the same problem. If each image is made up of thousands of pixels, and each pixel requires 3 bytes of data, then each image is huge. JPEG format compresses the digital representation to remove the differences in hues of a picture that we cannot perceive. Then we can compress by using run-length compression to code the remaining bits.

Run-length compression If my bit pattern is: We can code a value to indicate that we have: 24 0’s followed by 14 1’s followed by 16 0’s, etc. When we have many changing values in the pattern, it will not save us much space, but by making patterns of identical pixels, you can save a good deal of data space.

Lossy vs lossless conversion Lossless – no loss of data in the conversion Lossy – there is loss of data Run-length coding is lossless. You can convert the original to a compressed form and recover it exactly. Compression that removes some of the detail (things that we cannot perceive) is lossy. You cannot reproduce exactly the same sound/picture.