## Interactive Astronomy using 3D Computer Graphics

This is a school project on using 3D Computer Graphics. It explores a phenomenon whereby a sundial can actually go backwards in the tropics!

Abstract:

Understanding spherical astronomy requires good spatial visualization. Unfortunately, it is very hard to make good three dimensional (3D) illustrations and many illustrations from standard textbooks are in fact incorrect. There are many programs that can be used to create illustrations, but in this report we have focused on TEX-friendly, free programs. We have compared MetaPost, PSTricks, Asymptote and Sketch by creating a series of illustrations related to the problem of why sundials sometimes go backwards in the tropics.

In it, the Hezekiah Phenomenon is being discussed.

Quote: First, we would like to explain where the name Hezekiah Phenomenon comes from. In the Bible there is a story about God making the shadow of the sundial move backward as a sign for King Hezekiah.

The Bible gives two versions of the story of King Hezekiah and the sundial. First in 2 Kings, Chapter 20.

8 And Hezekiah said unto Isaiah, What [shall be] the sign that the LORD will heal me, and that I shall go up into the house of the LORD the third day? 9 And Isaiah said, This sign shalt thou have of the LORD, that the LORD will do the thing that he hath spoken: shall the shadow go forward ten degrees, or go back ten degrees? 10 And Hezekiah answered, It is a light thing for the shadow to go down ten degrees: nay, but let the shadow return backward ten degrees. 11 And Isaiah the prophet cried unto the LORD: and he brought the shadow ten degrees backward, by which it had gone down in the dial of Ahaz. (2 Kings 20: 8–11, King James Version)

## Tung Soo Hua 董素华 Honours Bachelor of Science degree in Mathematics

Tung Soo Hua (Chinese: 董素华; pinyin: Dǒng Sùhúa, Dong Suhua) is an award-winning television news anchor and current affairs presenter with MediaCorp TV Channel 8 and Channel U. (Wikipedia)

Tung won the Best Chinese-language News Presenter award for Star Awards in 2004, 2005, 2006, 2007, 2009 and 2011 Star Awards.

Tung studied in Nanyang Girls’ High School, and graduated with a Masters degree in Social Sciences (International Studies) from the National University of Singapore, after obtaining her first Honours Bachelor of Science degree in Mathematics.

Outstanding Science Alumni Award 2011 TUNG Soo Hua
BSc (Hons) 1997, M.Soc.Sci. 2007
Presenter/Senior Producer, Chinese News, MediaCorp Pte Ltd

Ms Tung Soo Hua is an award-winning television news and current affairs presenter with MediaCorp, Singapore’s leading media company. Currently, she co-hosts “Evening News at 10pm” on the most watched Mandarin channel in Singapore, Channel 8, and fronts “Money Week”, a weekly financial programme on Channel U. She started her journalism career in MediaCorp as a Chinese-language news producer in 1997.

Ms Tung was named the “Best News/Current Affairs Presenter” for six times between 2004 and 2011 in “Stars Awards”, which is MediaCorp’s gala event recognising its talents for their excellence. She graduated with a Masters degree in Social Sciences (International Studies) from the National University of Singapore, after obtaining her first Honours Bachelor of Science degree in Mathematics.

Recommended Math book:

Currently in its eighteenth printing in Japan, this best-selling novel is available in English at last. Combining mathematical rigor with light romance, Math Girls is a unique introduction to advanced mathematics, delivered through the eyes of three students as they learn to deal with problems seldom found in textbooks. Math Girls has something for everyone, from advanced high school students to math majors and educators.

## What happens if light slows down

This is an essay I wrote many years ago on an introduction to Special Relativity.

I repackaged it into a book and it is now available on Lulu.com:

There is also an Ebook version:

Excerpt:

## What happens if light slows down – A Beginner’s Guide to Relativity and Light

In the beginning God created the heavens and the earth. And God said, “Let there be light,” and there was light. Light is one of the most ubiquitous things that we see, and it is also one of the oldest – it existed since the beginning of mankind. However, light is also mysterious in that no one really understands what it is and how it is rectilinearly propagated. Nevertheless, the speed of light plays an important part in physics, and it is one of the more often quoted constant. What will happen then, if the speed of light suddenly changes from 300000000m/s to a fraction of its original self –3000 m/s? (It is theoretically possible to slow down light to such a speed, by shining a beam of light through a medium with a refractive index of 100,000.)

## Sum of roots and Product of roots of Quadratic Equation

Given a quadratic equation $ax^2+bx+c=0$ with roots $\alpha$ and $\beta$, we have:

$\displaystyle\boxed{\alpha+\beta=\frac{-b}{a}}$

$\displaystyle\boxed{\alpha\beta=\frac{c}{a}}$

How do we prove this? It is actually due to the quadratic formula!

Recall that the quadratic formula gives the roots of the quadratic equation as: $\displaystyle\boxed{x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$

Now, we can let

$\displaystyle \alpha=\frac{-b+\sqrt{b^2-4ac}}{2a}$

$\displaystyle \beta=\frac{-b-\sqrt{b^2-4ac}}{2a}$

Hence,

$\displaystyle \alpha+\beta=\frac{-2b}{2a}=\frac{-b}{a}$

$\begin{array}{rcl} \displaystyle \alpha\beta&=&\frac{-b+\sqrt{b^2-4ac}}{2a}\times\frac{-b-\sqrt{b^2-4ac}}{2a}\\ &=&\frac{b^2-(b^2-4ac)}{4a^2}\\ &=&\frac{4ac}{4a^2}\\ &=&\frac{c}{a}\end{array}$

In the above proof, we made use of the identity $(A+B)(A-B)=A^2-B^2$

The above formulas are also known as Vieta’s formulas (for quadratic). There we have it, this is how we prove the formula for the sum and product of roots!

# Manga guide to Math series

Check out the following interesting comic books explaining Math (Calculus, Linear Algebra, and Statistics) in a fun and enjoyable way.

In The Manga Guide to Calculus, you’ll follow along with Noriko as she learns that calculus is more than just a class designed to weed out would-be science majors. You’ll see that calculus is a useful way to understand the patterns in physics, economics, and the world around us, with help from real-world examples like probability, supply and demand curves, the economics of pollution, and the density of Shochu (a Japanese liquor).

Mr. Seki teaches Noriko how to:

• Use differentiation to understand a function’s rate of change
• Apply the fundamental theorem of calculus, and grasp the relationship between a function’s derivative and its integral
• Integrate and differentiate trigonometric and other complicated functions
• Use multivariate calculus and partial differentiation to deal with tricky functions
• Use Taylor Expansions to accurately imitate difficult functions with polynomials

Whether you’re struggling through a calculus course for the first time or you just need a painless refresher, you’ll find what you’re looking for in The Manga Guide to Calculus.

Follow along in The Manga Guide to Linear Algebra as Reiji takes Misa from the absolute basics of this tricky subject through mind-bending operations like performing linear transformations, calculating determinants, and finding eigenvectors and eigenvalues. With memorable examples like miniature golf games and karate tournaments, Reiji transforms abstract concepts into something concrete, understandable, and even fun.

As you follow Misa through her linear algebra crash course, you’ll learn about:

• Basic vector and matrix operations such as addition, subtraction, and multiplication
• Linear dependence, independence, and bases
• Using Gaussian elimination to calculate inverse matrices
• Subspaces, dimension, and linear span
• Practical applications of linear algebra in fields like computer graphics, cryptography, and engineering

But Misa’s brother may get more than he bargained for as sparks start to fly between student and tutor. Will Reiji end up with the girl—or just a pummeling from her oversized brother? Real math, real romance, and real action come together like never before in The Manga Guide to Linear Algebra.

The Manga Guide to Statistics

This manga textbook is written for those interested in understanding principles of statistics. Each of the seven chapters is organized into four sections: a cartoon, a text explanation to supplement the cartoon, an exercise that includes the answer, and a summary. Readers can learn much about the subject by just reading the cartoon, but they will gain a more thorough understanding by working through the other three sections in each chapter. Yamamoto provides Rui with easy-to-understand examples and graphic illustrations, making the subject less intimidating.

# Multiply by 9999 trick

Here is a nice trick to multiply a 4 digit number by 9999.

For instance, lets try multiplying 1729 by 9999.

First, we reduce 1729 by 1.

1729-1=1728

Then, we subtract each of the above digits from 9 to get 8271.

(9-1=8, 9-7=2, 9-2=7, 9-8=1)

In conclusion, 9999 x 1729=1728,8271.

Impressive isn’t it?

This trick works for many 9s too, for example multiplying by 99999999.

To multiply 9999 with a number with less digits, for instance, 12, simply pad zeroes in front of the number, to become 0012.

Then, using the above method, 9999 x 0012 = 0011,9988=119988.

Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks

# The Alexa Toolbar for Internet Explorer

## Features:

• Alexa Traffic Rank: See how popular a website is.
• Related Links: Find sites that are similar to the site you are visiting.
• Wayback: See how a site looked in the past.
• Hot Pages & Searches: See what’s popular on the web right now.

Alexa Internet, Inc. is a California-based subsidiary company of Amazon.com which provides commercial web traffic data. Founded as an independent company in 1996, Alexa was acquired by Amazon in 1999. Its toolbar collects data on browsing behavior and transmits it to the Alexa website, where it is stored and analyzed, forming the basis for the company’s web traffic reporting. As of 2013, Alexa provides traffic data, global rankings and other information on 30 million websites,[3] and its website is visited by over 8.5 million people monthly. (Wikipedia)