Mathematics in the Digital Age
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Transcript of Mathematics in the Digital Age
Mathematics in the Digital Age
Reva Narasimhan
Kean University, NJ
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
Powers of 10
Byte [ 8 bits] 1 byte: a single character;
Kilobyte [ 1,000 bytes OR 103 bytes] 2 Kilobytes: A typewritten page; 10 Kilobytes: static web page; 100 Kilobytes: A low-resolution photograph;
Powers of 10
Megabyte [ 1,000,000 bytes OR 106 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
Powers of 10
Gigabyte [ 1,000,000,000 bytes OR 109 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
Exercises
1. 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?
2. 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)
Solution
1. 109/(4*105)=2500
2. 1.0Mbps is equal to 125 KB /sec
(8 bits = 1 byte)
So 4*106/(1.25 *105 B/sec) = 32 seconds
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) = 2297.1e0.3316t
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?
Digital pictures
Elementary approach – each pixel represents a color coded in RGB – red, blue, green components
Each color component varies from 0 to 255 (=28 possibilities). Occupies one byte of storage.
Make your own digital “picture” with M&M’s
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?
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.
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
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
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)
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)
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
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 45o with the horizontal. What is the perpendicular distance from the ball to the wall?
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)
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