mathematics in the digital age

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Mathematics in the Digital Age Reva Narasimhan Kean University, NJ

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Presents ideas on how math is used in digital applications such as digital cameras.

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Page 1: Mathematics in the Digital Age

Mathematics in the Digital Age

Reva Narasimhan

Kean University, NJ

Page 2: Mathematics in the Digital Age

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

Page 3: Mathematics in the Digital Age

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;

Page 4: Mathematics in the Digital Age

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

Page 5: Mathematics in the Digital Age

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

Page 6: Mathematics in the Digital Age

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)

Page 7: Mathematics in the Digital Age

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

Page 8: Mathematics in the Digital Age

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?

Page 9: Mathematics in the Digital Age

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

Page 10: Mathematics in the Digital Age

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?

Page 11: Mathematics in the Digital Age

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.

Page 12: Mathematics in the Digital Age

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

Page 13: Mathematics in the Digital Age

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

Page 14: Mathematics in the Digital Age

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)

Page 15: Mathematics in the Digital Age

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)

Page 16: Mathematics in the Digital Age

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

Page 17: Mathematics in the Digital Age

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?

Page 18: Mathematics in the Digital Age

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)

Page 19: Mathematics in the Digital Age

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