1. Chapter 2: How are data represented?

2. Why we care? The accuracy of our results The accuracy of our results The speed of processing The speed of processing The range of alphabets available to us The range of alphabets available to us The size of the files we must store The size of the files we must store The quality of graphics on screen and on paper The quality of graphics on screen and on paper The time it takes for Internet download The time it takes for Internet download

3. Why computers work in binary? Cheapest and simplest in design and engineering Cheapest and simplest in design and engineering Switch: on  1; off  0 Switch: on  1; off  0 Circuit: voltages Circuit: voltages –1.7 volts – higher  1 –0.0 volts - 1.3 volts  0 –Voltages (1.3 to 1.7) are avoided in design Mathematics: binary numbers Mathematics: binary numbers –Using digits 0 and 1 only.

4. Decimal vs. Binary Decimal # system Decimal # system –10 symbols: 1, 2, 3,…9, 0 –Base = 10 (We have 10 fingers) –Decimal number 2324 reads “2 thousands 3 hundreds twenty four”. Binary # system Binary # system –2 symbols: 0 and 1 –Base = 2 –Binary number 1101 = ?

5. Decimal vs. Binary 4232. 2*1000 3*1002*104*1Each digit represents: 1000100101Position values: 10 3 Position values (base):10 2 10 1 10 0 Decimal # System: 1110. 1*8 1*40*21*1Each digit represents: 8421Position values: 2323 Position values (base):2 2121 2020 Binary # System: Value in Decimal:2*1000+3*100+2*10+4*1 = 2324 D Value in Decimal:1*8+1*4+0*2+1*1 = 13 D

6. Why do computer work in binary? Binary digits – bits Binary digits – bits 8 bits = 1 byte 8 bits = 1 byte 2 10 bytes = 1024 bytes =1 kilobytes = 1KB 2 10 bytes = 1024 bytes =1 kilobytes = 1KB 2 20 bytes = 2 10 KB = 1 megabytes = 1MB 2 20 bytes = 2 10 KB = 1 megabytes = 1MB 2 30 bytes = 2 10 MB = 1 gigabytes = 1GB 2 30 bytes = 2 10 MB = 1 gigabytes = 1GB 2 40 bytes = 2 10 GB = 1 terabytes = 1TB 2 40 bytes = 2 10 GB = 1 terabytes = 1TB

7. Types of data Instructions Instructions –Computer instructions are coded in sequences of 0’s and 1’s Numbers Numbers –2324, -34.35, 34567890123.12345 Characters and symbols Characters and symbols –A, B, C, … Z, a, b, c,… z, –0, 1, 2, 3 … 9, +, -, ), (, *, &, etc Images Images –Photos, charts, drawings Audio Audio –Sound, music, etc Video Video –Video clips and movies

8. Representation of Numbers Fixed-size-storage approach: Fixed-size-storage approach: –Computers allocate a specified amount of space for a number Integers Integers 1 bit: 0 to 1 1 bit: 0 to 1 2 bits: 00, 01, 10, 11  0 to 3 2 bits: 00, 01, 10, 11  0 to 3 4 bits: 0000, 0001, 0010, … 1111  0 to 15 4 bits: 0000, 0001, 0010, … 1111  0 to 15 1 byte: 0 to 255 1 byte: 0 to 255 2 bytes: -32768 to +32767 2 bytes: -32768 to +32767 4 bytes: -2,147,648 to +2,147,483,647 4 bytes: -2,147,648 to +2,147,483,647 Note: with 4 bytes for integers, any number smaller than -2,147,648 or larger than 2,147,483,647 would be incorrectly represented.,

9. Representation of Numbers 1110. 1*2 0*11*0.51*0.25Each digit represents: 211/21/4Position values: 2121 Position values (base):2020 2 -1 2 -2 Binary # System: Value in Decimal: 2 + ½ + ¼ + 1/8 = 2.875 D 1 1*0.125 1/8 2 -3 Binary representation of real numbers

10. Representation of Numbers Floating-point numbers for real numbers Floating-point numbers for real numbers –Three parts of representation: 1. Sign (always 1 bits: 0 for + and 1 for -) 2. Significant digits (e.g., six bits) 3. the power of 2 for the leftmost digit (e.g., 3 bits) –Example for binary -1111.01 Sign: 1 (negative) Sign: 1 (negative) Significant digits: 111101 B Significant digits: 111101 B Power of 2: 011 B Power of 2: 011 B –Example for binary +100.1101 B Sign: 0 (positive) Sign: 0 (positive) Significant digits: 110110 B Significant digits: 110110 B –Note: the last digit is lost, which is 1/16 in decimal Power of 2: 010 B Power of 2: 010 B

11. Representation of Numbers Single-precision floating-point numbers Single-precision floating-point numbers 1.Sign (always 1 bits: 0 for + and 1 for -) 2.Significant digits: 23 bits 3.exponent: 8 Double-precision floating-point numbers Double-precision floating-point numbers 1.Sign (always 1 bits: 0 for + and 1 for -) 2.Significant digits: 52 bits 3.exponent: 11 What you should know? What you should know? –Computers can represent numbers only in limited accuracy. E.g., when you enter a 20 digit decimal # into a program that uses single-precision, only about 7 digits are actually stored, the rest are lost. E.g., when you enter a 20 digit decimal # into a program that uses single-precision, only about 7 digits are actually stored, the rest are lost. –Real examples: Designing aircraft on p.35 Designing aircraft on p.35 The Vancouver Stock Exchange Index on pp. 38-39 The Vancouver Stock Exchange Index on pp. 38-39

12. Representation of Numbers // file: public_html/2005f-html/cil102/accuracy.c #include #include int main() { int x, y, result;// x, y, and result all use 32 bits to represent integers (-2,147,648 to +2,147,483,647) int x, y, result;// x, y, and result all use 32 bits to represent integers (-2,147,648 to +2,147,483,647) char op; char op; int i; int i; for (i = 0; i < 100; i++) { for (i = 0; i < 100; i++) { printf("please enter an expression:\n"); printf("please enter an expression:\n"); scanf("%d %c %d", &x, &op, &y); scanf("%d %c %d", &x, &op, &y); if (op == '+') if (op == '+') result = x + y; result = x + y; else if (op == '-') else if (op == '-') result = x - y; result = x - y; else { else { printf("Invalid operator!!"); printf("Invalid operator!!"); break; break; } printf("%d %c %d = %d\n", x, op, y, result); printf("%d %c %d = %d\n", x, op, y, result); }} // When you enter 2000000000 + 500000000, the result is -1794967296

13. Representation of Numbers Variable-size-storage approach: Variable-size-storage approach: –Allow a wide-range of numbers to be stored accurately –Needs significant more time to process –Fixed-size approach is used more common than variable-size approach.

14. Representation of characters There are no visual letters A, B, C, etc stored in computers like we have in mind. There are no visual letters A, B, C, etc stored in computers like we have in mind. Letters and symbols are encoded in 8 bits – one byte - of 0’s and 1’s. Letters and symbols are encoded in 8 bits – one byte - of 0’s and 1’s. –Keyboard converts keys A, B, C etc to their corresponding codes and –monitor converts the code into visual letters A, B, C etc on screen. Two commonly used coding schemes: Two commonly used coding schemes: –ASCII: American Standard Code Information Interchange –EBCDIC: Extended Binary Coded Decimal Interchange Code

15. Representation of characters CharacterEBCDICASCIIA1100000101000001 B1100001001000010 a1000000101100001 b1000001001100010 01111000000110000 11111000100110001 21111001000110010, (comma) 0110101100101100 - (dash) 0110000000100101

16. Representation of characters Foreign characters – two approaches Foreign characters – two approaches –Use one byte per char Ex., Ex., –ISO-8859-1 for Western (Roman) –ISO-8859-7 for Greek –ISO-2022-CN for simplified Chinese Webpage: using “META charset=…” to specify which encoding is used. Webpage: using “META charset=…” to specify which encoding is used. –Use two bytes per char/symbols 16 bits have 65,536 combinations (characters) 16 bits have 65,536 combinations (characters) Unicode coding system Unicode coding system

17. Representation of Images A picture is treated as a matrix of dots, called pixels.

18. Representation of Images The pixels are so small and close together we cannot really see them as separate dots. The pixels are so small and close together we cannot really see them as separate dots. Resolution: dots per inch (dpi) Resolution: dots per inch (dpi) –72 dpi for Web images –600 or 1200 dpi for professional printers or home photo printers

19. Representation of Images The color of each pixel is represented using bits. The color of each pixel is represented using bits. Black/White: one bit per pixel Black/White: one bit per pixel –1-white and 0-black Gray scale: one byte per pixel Gray scale: one byte per pixel –256 different degrees of gray (00000000 to 11111111) –00000000 black, 01111111 intermediate gray, 11111111 white Color: three bytes per pixel Color: three bytes per pixel –Red, green, blue color –One byte for the intensity of each of the three color –256 possible red, 256 green, 256 blue Pure red: 11111111 for red byte, 00000000 for green and blue Pure red: 11111111 for red byte, 00000000 for green and blue White: 11111111 for all three bytes White: 11111111 for all three bytes Black: 00000000 for all three bytes Black: 00000000 for all three bytes

20. Representation of Images Image storage -- size Image storage -- size Gray scale: one byte per pixel Gray scale: one byte per pixel E.g., A 3 X 5 picture with 300 dpi resolution 3 * 300 = 900 pixels per column 3 * 300 = 900 pixels per column 5 * 300 = 1500 pixels per row 5 * 300 = 1500 pixels per row 900 * 1500 = 1,350,000 pixels/picture 900 * 1500 = 1,350,000 pixels/picture Needed storage = 1,350,000 bytes/picture = 1MB/picture Needed storage = 1,350,000 bytes/picture = 1MB/picture Color: three bytes per pixel Color: three bytes per pixel E.g., A 3 X 5 picture with 300 dpi resolution 3 * 300 = 900 pixels per column 3 * 300 = 900 pixels per column 5 * 300 = 1500 pixels per row 5 * 300 = 1500 pixels per row 900 * 1500 = 1,350,000 pixels/picture 900 * 1500 = 1,350,000 pixels/picture Needed storage = 3 (bytes per pixel) * 1,350,000 Needed storage = 3 (bytes per pixel) * 1,350,000 = 4,050,000 bytes/picture = 4,050,000 bytes/picture = 4MB/picture --- TOO BIG = 4MB/picture --- TOO BIG

21. Representation of Images Image compression Image compression Color table Color table –Most pictures contain a small # of different colors –Use a table to define colors that are actually used in the picture –Each pixel has an index to the color table. –Each image contains a color table and table indices –Example For a picture with 100 different colors, the color table would contain 100 entries, three bytes each entry for each color. One byte can be used as index to the table for each pixel.

22. Representation of Images Drawing commands Drawing commands –Draw picture using basic commands –Just as artists draws using a pencil or a brush and other basic movements –Example, A house is drawn by sketching various elements (doors, windows, walls), adding color to them, and moving to the desired position. A house is drawn by sketching various elements (doors, windows, walls), adding color to them, and moving to the desired position.

23. Representation of Images Data averaging or sampling Data averaging or sampling –Condense the size by selecting a smaller collection of information to store. –Many different ways of sampling and data averaging –An example: choose to store only every other pixel in an image (sampling)– reducing the size to half. To display the full picture, the computer need to fill in the missing data with, for example, the average of neighboring pixels (data averaging) –The resulting picture cannot be as sharp as the original –Lossy data compression

24. What are “.gif”, “.ps”, “.jpg”, “.bmp” formats? Commonly used image file formats -1 Commonly used image file formats -1 –Bitmap (.bmp) Pixel-by-pixel storage of all color information for each pixel. Pixel-by-pixel storage of all color information for each pixel. Lossless representation Lossless representation Files are huge. Files are huge. –Graphics Interchange Format (.gif) Use one or more color tables – the color table technique Use one or more color tables – the color table technique Each table contains 256 colors. Each table contains 256 colors. Suitable for pictures with a small # (<256) of different colors (e.g., organization charts) Suitable for pictures with a small # (<256) of different colors (e.g., organization charts) Not suitable for pictures with shading (e.g., photos) Not suitable for pictures with shading (e.g., photos)

25. What are “.gif”, “.ps”, “.jpg”, “.bmp” formats? Commonly used image file formats - 2 Commonly used image file formats - 2 –PostScript (.ps) Employ the drawing commands technique Employ the drawing commands technique “moveto” draws a line from current position to a new one and “arc” draws an arc given its center, radius, etc “moveto” draws a line from current position to a new one and “arc” draws an arc given its center, radius, etc General shapes can be used in multiple places General shapes can be used in multiple places Fonts can be reused. Fonts can be reused. Useful when the picture can be rendered as a drawing or its contains many of the same elements (e.g., text of the same fonts) Useful when the picture can be rendered as a drawing or its contains many of the same elements (e.g., text of the same fonts) –Joint Photographic Experts Group (JPEG) (.jpg) use the data averaging and sampling on 8*8 pixel blocks use the data averaging and sampling on 8*8 pixel blocks User determines the level of details and clarity User determines the level of details and clarity High-quality image – 8*8 blocks maintain their contents High-quality image – 8*8 blocks maintain their contents Low-quality image – info in 8*8 blocks is discarded  smaller files Low-quality image – info in 8*8 blocks is discarded  smaller files

26. Comparison b/w jpg, gif, and ps Pictures in the textbook Pictures in the textbook http://www.cs.grinnell.edu/~walker/fluency- book/figures/chapter2/fig-2-overview.html http://www.cs.grinnell.edu/~walker/fluency- book/figures/chapter2/fig-2-overview.html Comparison of.jpg and.gif Comparison of.jpg and.gif http://www.siriusweb.com/tutorials/gifvsjpg/ More on.jpg and.gif More on.jpg and.gif http://www.wfu.edu/~matthews/misc/jpg_vs_gif/Jpg VsGif.html http://www.wfu.edu/~matthews/misc/jpg_vs_gif/Jpg VsGif.html

27. Summary – chapter 2 Computers work in binary Computers work in binary Integers may be constrained in size Integers may be constrained in size Real numbers may have limited accuracy Real numbers may have limited accuracy Computations may produce roundoff errors, affecting accuracy Computations may produce roundoff errors, affecting accuracy Characters and languages are encoded in binary Characters and languages are encoded in binary Pictures are displayed pixel by pixel Pictures are displayed pixel by pixel Color table, draw commands, and data averaging and sampling compression techniques Color table, draw commands, and data averaging and sampling compression techniques.bmp, jpg,.gif,.ps formats.bmp, jpg,.gif,.ps formats

28. Terminology Binary vs. decimal Binary vs. decimal Position value Position value The base of a # system The base of a # system Bit/byte/KB/MB/GB/TB Bit/byte/KB/MB/GB/TB Integer binary #s Integer binary #s Real # in binary Real # in binary Floating point numbers Floating point numbers Representational error Representational error Roundoff errors Roundoff errors ASCII/EBCDIC/Unicode ASCII/EBCDIC/Unicode Pixels Pixels Dots per inch (dpi) Dots per inch (dpi) Bitmap Bitmap Color table Color table Data averaging Data averaging Data sampling Data sampling Data compression Data compression.jpg,.bmp,.gif,.ps.jpg,.bmp,.gif,.ps