# New Kindle Release: Math Shorts – Exponential and Trigonometric Functions

I’m on a roll here … another math book, comin’ right up … I’m happy to announce that today I’m expanding my “Math Shorts” series with the latest release “Math Shorts – Exponential and Trigonometric Functions”. This time it’s pre-calculus and thus serves as a bridge between my permanently free e-books “Algebra – The Very Basics” and “Math Shorts – Derivatives”. Without further ado, here’s the blurb and table of contents (click cover to get to the product page on Amazon):

Blurb:

Before delving into the exciting fields of calculus and mathematical physics, it is necessary to gain an in-depth understanding of functions. In this book you will get to know two of the most fundamental function classes intimately: the exponential and trigonometric functions. You will learn how to visualize the graph from the equation, how to set up the function from conditions for real-world applications, how to find the roots, and much more. While prior knowledge in linear and quadratic functions is helpful, it is not necessary for understanding the contents of the book as all the core concepts are developed during the discussion and demonstrated using plenty of examples. The book also contains problems along with detailed solutions to each section. So except for the very basics of algebra, no prior knowledge is required.

Once done, you can continue your journey into mathematics, from the basics all the way to differential equations, by following the “Math Shorts” series, with the recommended reading being “Math Shorts – Derivatives” upon completion of this book. From the author of “Great Formulas Explained” and “Statistical Snacks”, here’s another down-to-earth guide to the joys of mathematics.

1. Exponential Functions
1.1. Definition
1.2. Exercises
1.3. Basics Continued
1.4. Exercises
1.5. A More General Form
1.6. Exercises

2. Trigonometric Functions
2.1. The Sine Function
2.2. Exercises
2.3. The Cosine Function
2.4. Exercises
2.5. Roots
2.6. Exercises
2.7. Sine Squared And More
2.8. The Tangent Function
2.9. Exercises

3. Solutions to the Problems

# Amazon Plans to Use Drones to Deliver Packages

Usually I don’t post news in my blog, but this sounds like a fantastic idea. Amazon is testing drones that could deliver up to 5 pounds per flight (which covers 86 % of all Amazon sales). The service, called Prime Air, could be available within five years if the ongoing series of tests is successful and the necessary FAA permissions are obtained. As an ebook author, I wonder though what impact this will have on the ebook market. Will people go back to print?

Customer: “Your damned drone put my package on the roof again!”

Here’s a picture of Amazon’s “Octocopter”:

(Taken from regmedia.co.uk)

# The Emerging World Of Ebooks

As we all know, the Internet changed everything. The Net and the Web have brought the world closer together. Many older means of communication have either been replaced or changed so as to co-exist with, and complement, electronic communication. Email, for example, has replaced almost all business and a lot of personal letter writing, though our mailboxes remain filled with lots of mail, most unwanted. A lot of printed newspapers and magazines still exist, but their content is now also available on websites, and the websites are timelier and often offer more detailed information. Printed media is not dead by any means, as millions of people still prefer to curl up with a good book or grab a paper on their way to work. There have been many efforts to popularize ebooks, downloadable books in digital form, but their acceptance remains in its infancy.

But that won’t stay that way. Ebooks make sense. Since books are almost all text, an ebook download is very fast and hundreds of ebooks can fit onto a small storage card. Ebooks do not contribute to cutting down forests, they do not need to be trucked across the country, they do not produce waste, and they are usually a lot less expensive than printed books. Ebooks also have many other advantages. Depending on your ebook reader software, an ebook can be annotated, bookmarked and searched. The latter is especially useful; I often want to go back to a certain quote or paragraph in a book, and electronic search is so much easier than leafing through a printed book.

One of the problems ebooks face is that people do not know how to use them. They are confused by the many different ebook formats or think they need a particular piece of hardware to read them. In fact, the formats are not really a problem. Most computers can read popular ebook formats and ebook reader software is freely available. Hardware is a bit more of an issue. Hardcovers and paperbacks are awfully convenient and they don’t need batteries, so a lot of people shy away from reading on a computer screen or spending the money for a dedicated ebook reader.

This is really too bad as ebooks are clearly the way of the future. They just make too much sense. Those who dismiss ebooks are missing out on a great and increasingly attractive alternative to the printed page. Those who are willing to give ebooks a chance are rewarded with lower costs and the ability to carry an entire library on a device of their choice, be that a notebook computer, a Tablet PC, a dedicated ebook reader, a PDA or even a smartphone. And they have access to a potentially much larger variety of books. That’s because ebooks make self-publishing easy and lots of authors who don’t have a chance of getting picked up by traditional print publishing houses can distribute their books electronically. Best of all, there is no waste and there will never be unsold books that end up on a bargain table or in a landfill.

My advice is to give ebooks a chance. Download a free ebook. Look for sites dedicated to ebooks, especially those with a website design that is appealing. See what format you prefer, and what device you like to read on. But be warned: you may get hooked. Once you get into them, downloading and reading ebooks can become a passion.

# Quantitative Analysis of Top 60 Kindle Romance Novels

I did a quantitative analysis of the current Top 60 Kindle Romance ebooks. Here are the results. First I’ll take a look at all price related data and conclusions.

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• Price over rank:

There seems to be no relation between price and rank. A linear fit confirmed this. The average price was 3.70 \$ with a standard deviation of 2.70 \$.

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• Price frequency count:

(Note that prices have been rounded up) About one third of all romance novels in the top 60 are offered for 1 \$. Roughly another third for 3 \$ or 4 \$.

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• Price per 100 pages over rank:

Again, no relation here. The average price per 100 pages was 1.24 \$ with a standard deviation of 0.86 \$.

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• Price per 100 pages frequency count:

About half of all novels in the top 60 have a price per 100 pages lower than 1.20 \$. Another third lies between 1.20 \$ and 1.60 \$.

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• Price per 100 pages over number of pages:

As I expected, the bigger the novel, the less you pay per page. Romance novels of about 200 pages cost 1.50 \$ per 100 pages, while at 400 pages the price drops to about 1 \$ per 100 pages. The decline is statistically significant, however there’s a lot of variation.

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• Review count:

A little less than one half of the top novels have less than 50 reviews. About 40 % have between 50 and 150 reviews. Note that some of the remaining 10 % more than 600 reviews (not included in the graph).

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• Rating over rank:

There’s practically no dependence of rank on rating among the top 60 novels. However, all have a rating of 3.5 stars or higher, most of them (95 %) 4 stars or higher.

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• Pages over ranking:

There’s no relation between number of pages and rank. A linear fit confirmed this. The average number of pages was 316 with a standard deviation of 107.

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• Pages count:

About 70 % of the analyzed novels have between 200 and 400 pages. 12 % are below and 18 % above this range.

# Typical Per-Page-Prices for Ebooks

I did a little analysis of ebook prices per 100 pages for different categories in the Amazon Kindle store. In each category I looked at the top 12 paid books. This data can help readers to judge prices and authors to set them. Here are the results in increasing order:

Erotica: 1.7 \$ per 100 pages (ranging from 1.0 – 3.1 \$ per 100 pages)
Sci-Fi and Fantasy: 1.8 \$ per 100 pages (ranging from 0.8 – 4.4 \$ per 100 pages)
Short Stories: 2.0 \$ per 100 pages (ranging from 0.5 – 4.2 \$ per 100 pages)
Self-Help: 3.6 \$ per 100 pages (ranging from 1.3 – 6.7 \$ per 100 pages)
Applied Math: 4.0 \$ per 100 pages (ranging from 0.9 – 7.9 \$ per 100 pages)
Economy / Business: 7.2 \$ per 100 pages (ranging from 3.3 – 17.2 \$ per 100 pages)

Typical (and in my opinion fair) prices seem to be 2 \$ per 100 pages for fiction and 4 \$ per 100 pages for non-fiction. In the special case of business books, prices of 7 \$ per 100 pages seem common.

# Missile Accuracy (CEP) – Excerpt from “Statistical Snacks”

An important quantity when comparing missiles is the CEP (Circular Error Probable). It is defined as the radius of the circle in which 50 % of the fired missiles land. The smaller it is, the better the accuracy of the missile. The German V2 rockets for example had a CEP of about 17 km. So there was a 50/50 chance of a V2 landing within 17 km of its target. Targeting smaller cities or even complexes was next to impossible with this accuracy, one could only aim for a general area in which it would land rather randomly.

Today’s missiles are significantly more accurate. The latest version of China’s DF-21 has a CEP about 40 m, allowing the accurate targeting of small complexes or large buildings, while CEP of the American made Hellfire is as low as 4 m, enabling precision strikes on small buildings or even tanks.

Assuming the impacts are normally distributed, one can derive a formula for the probability of striking a circular target of Radius R using a missile with a given CEP:

p = 1 – exp( -0.41 · R² / CEP² )

This quantity is also called the “single shot kill probability” (SSKP). Let’s include some numerical values. Assume a small complex with the dimensions 100 m by 100 m is targeted with a missile having a CEP of 150 m. Converting the rectangular area into a circle of equal area gives us a radius of about 56 m. Thus the SSKP is:

p = 1 – exp( -0.41 · 56² / 150² ) = 0.056 = 5.6 %

So the chances of hitting the target are relatively low. But the lack in accuracy can be compensated by firing several missiles in succession. What is the chance of at least one missile hitting the target if ten missiles are fired? First we look at the odds of all missiles missing the target and answer the question from that. One missile misses with 0.944 probability, the chance of having this event occur ten times in a row is:

p(all miss) = 0.94410 = 0.562

Thus the chance of at least one hit is:

p(at least one hit) = 1 – 0.562 = 0.438 = 43.8 %

Still not great considering that a single missile easily costs 10000 \$ upwards. How many missiles of this kind must be fired at the complex to have a 90 % chance at a hit? A 90 % chance at a hit means that the chance of all missiles missing is 10 %. So we can turn the above formula for p(all miss) into an equation by inserting p(all miss) = 0.1 and leaving the number of missiles n undetermined:

0.1 = 0.944n

All that’s left is doing the algebra. Applying the natural logarithm to both sides and solving for n results in:

n = ln(0.1) / ln(0.944) = 40

So forty missiles with a CEP of 150 m are required to have a 90 % chance at hitting the complex. As you can verify by doing the appropriate calculations, three DF-21 missiles would have achieved the same result.

Liked the excerpt? Get the book “Statistical Snacks” by Metin Bektas here: http://www.amazon.com/Statistical-Snacks-ebook/dp/B00DWJZ9Z2. For more excerpts see The Probability of Becoming a Homicide Victim and How To Use the Expected Value.