This document discusses how digital technologies like bytes, kilobytes, megabytes, and gigabytes can be used to motivate mathematics concepts. It provides examples of the storage required for different types of digital files and exercises involving calculating file sizes and download speeds. The document also discusses how digital images can be represented using matrices and manipulated using matrix algebra operations.

1. Mathematics in the Digital Age Reva Narasimhan Kean University, NJ

2. Introduction Today’s student has access to a wide variety of digital technologies Leverage this knowledge to develop interest Motivate the mathematics content with modern topics

3. Powers of 10 Byte [ 8 bits] 1 byte: a single character; Kilobyte [ 1,000 bytes OR 10 3 bytes] 2 Kilobytes: A typewritten page; 10 Kilobytes: static web page; 100 Kilobytes: A low-resolution photograph;

4. Powers of 10 Megabyte [ 1,000,000 bytes OR 10 6 bytes] 2 Megabytes: A high resolution photograph; 5 Megabytes: The complete works of Shakespeare OR 30 seconds of TV-quality video; 10 Megabytes: A minute of high-fidelity sound

5. Powers of 10 Gigabyte [ 1,000,000,000 bytes OR 10 9 bytes] 1 Gigabyte: a symphony in high-fidelity sound OR a movie at TV quality; 20 Gigabytes: A good collection of the works of Beethoven

6. Exercises If the size of a digital photograph is 400 KB, how many photos of that size can fit in a 1GB flash drive, assuming the entire capacity of the drive can be used? If the download speed of a DSL modem is 1.0Mbps (Megabits per second), how long will it take to download a four-minute song of size 4MB? (Source: apple.com)

7. Solution 10 9 /(4*10 5 )=2500 1.0Mbps is equal to 125 KB /sec (8 bits = 1 byte) So 4*10 6 /(1.25 *10 5 B/sec) = 32 seconds

8. 3. Exponential Growth In 1965, Gordon Moore, then director of Intel research, conjectured that the number of transistors which fit on a computer chip doubles every few years. This has come to be known as Moore's Law . Analysis of data from Intel Corporation yields the following model of the number of transistors per chip over time: s(t) = 2297.1e 0.3316t where s(t) is the number of transistors per chip and t is the number of years since 1971. (Source: Intel Corporation) (a) What is the number of transistors per chip in 1971 according to this model? (b) How long does it take to double the number of transistors?

9. Digital pictures Elementary approach – each pixel represents a color coded in RGB – red, blue, green components Each color component varies from 0 to 255 (=2 8 possibilities). Occupies one byte of storage. Make your own digital “picture” with M&M’s

10. Digital “pictures” from M&M’s Each M&M represents 1 pixel Your picture can be stored as 3 separate matrices, one each for red, blue, and green Find the RGB value for each M&M; fill in the Red, Blue and Green matrices How much storage is required for your picture? What is the resolution of your picture?

11. Digital Pictures Matrix algebra plays an important role in manipulating digital images. Underlying algorithms in software such as Adobe Photoshop are constructed from matrix mathematics . Also, images are usually compressed to save space. Algorithms for image compression use advanced mathematical techniques.

12. Images in MATLAB A=imread('spring_bulbs.jpg'); Name Size Bytes A 480x320x3 460800 Three dimensional array to store RGB value Grand total is 460800 elements using 460800 bytes

13. Read Image from Matrix The following command displays the image stored in the matrix A: » imagesc(A) Further refinements require image processing toolbox in MATLAB

14. Megapixel numbers and digital cameras How much bigger can I print a 10-megapixel photo than a 5-megapixel photo? (Source: David Pogue, The New York Times) (Hint: The megapixel numbers refer to total area covered by the pixels)

15. Where are the extra MP’s? 5 MP: area of 1944 x 2592 pixels. Printed at 180 dots per inch, that’s about 11 by 14 inches. 10MP: area 2736 x 3648 pixels. An 180-dpi print that’s about 15 by 20 inches—under three inches more on each margin (Source: David Pogue, The New York Times )

16. Digital Animation Digital animation, at its core, consists of using transformations on a set of points Points are created in 3-D space and manipulated by matrix transformations. Pixar Animation created a software program called Renderman to do this. A simple example of tranformation of points can be shown through this Excel file

17. Video Game Design Uses methods of computational geometry Example of use of dot product A portion of a computer video game consists of a ball colliding with a wall. The origin is taken to be the left bottommost corner of the computer screen. The ball's location is given by the vector v = <6, 10> and the wall makes an angle of 45 o with the horizontal. What is the perpendicular distance from the ball to the wall?

18. Solution The solution to this problem uses a vector projection along the direction of the wall and calculating the perpendicular component. These types of computations are used extensively in creating video games. (Source: www.gamasutra.com)

19. References D. Pogue, Deconstructing the Megapixel Myth , www.nytimes.com, February 2007 R. Narasimhan, College Algebra and Trigonometry , Houghton Mifflin C. Watson, An Image Processing Tutorial , http://www.cs.washington.edu/research/metip/tutor/tutor.html Download files related to this presentation at www.collegemath.info