Learning a Second or Third Language

Learning languages is a great way to learn about another culture and give yourself access to a whole new world through the wealth of literature. I believe some people have an innate ability to learn new languages faster than others but anyone can learn a new language if they really want to and they put in the required effort to acquire the new language.

In this day and age, learning some languages has become very easy. Some of the more popular languages have tons of material available to help new learners acquire the language. Language learning has become more easy and affordable due to the Internet and mobile applications. One can download a free app and learn any language without leaving their home. Duolingo and Memrise are two such apps that offer a ton of languages and all for free or almost free.

In order to learn a language, motivation of some kind is needed. It could be anything from a desire to speak the language with natives while traveling or to get a job or study abroad. It could be as simple as wanting to enjoy the literature or other media in the local language. Whatever your motivation is, it is important because motivation makes the task less daunting. Depending on the motivation, you’ll have a different approach to learning. If you’re really interested in the literature, you can study the language from that aspect and not have to worry about the conversational part of it which could be quite different. Furthermore, the speaking & listening skills are very different from reading & writing abilities when it comes to languages.

I have been learning Arabic as a second language for a few years now. My motivation has been to learn Arabic to be able to understand the Quran and Hadith in its original text, without having to rely on translations. My focus has been literature rather than speaking.

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Learning the Quran & Teaching Others

Learning the Quran should be a priority for every Muslim. Not only is the Quran a source of knowledge, it is also a source of countless blessings. It is the word of Allah. The Quran speaks to us to inform us of what Allah requires from us. Imagine that. The Creator of everything telling us why He has created us and what He wants us to do and what He wants us not to do! All of this because He wants us to succeed. It is therefore upon us to do our utmost to find out what Allah says in the Quran.

Since the Quran is in the Arabic language, it is important that we acquire as much Arabic language as necessary to understand the Quran in its original Arabic. It is not an impossible task for anyone. There are tons of resources available online these days to learn it ourselves or enroll in a class and learn it locally in our own city/country. If we put our minds to it, we can all do it relatively easily inshaaAllah!

خيركم من تعلم القرأن وعلمه

From the meaning of this well known Hadith, we learn that the best of people are those who learn the Quran and teach it to others. It should be the goal of all Muslims to try to achieve the status of the best of people by striving hard to learn the Quran properly, to read it with Tajweed, to learn the Arabic necessary to understand the meanings in the original Arabic.

Furthermore, it should be our goal to put what we learn into practice in our daily lives. The first step after understanding is to internalize the teachings of the Quran by implementing them into our lives to the best of our ability.

Finally, it should be our aim to teach what we learn to others. With the help of Allah we can become the best of people by learning the Quran and then teaching it to others.

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Reading As a Habit

Reading is a habit that everyone should try their best to acquire. It is one of the easiest ways to gain knowledge that one can do on their own long after having left school. If you’re not reading regularly, you’re doing yourself a disservice. There is so much good reading material out there on every subject for every type of person and interest. Reading is a great way to learn from a large number of people who have taken the time and exerted a lot of effort to write quality books that we can all benefit from.

It is not a difficult habit to acquire and cherish our entire lives. It does require though that we find the right kinds of books to get the most benefit out of them. It is easy to fall into the trap of reading junk. Reading does transform our thinking to varying degrees, depending on what and how much one is reading. It does become quite important therefore to find the useful books and stay away from useless and specially damaging books. Do some research and find good books (reader ratings are a good way to check), get recommendations from your friends & family. Even libraries have recommendations. You can also create a profile on one of the online reading communities (such as goodreads.com or librarything.com or other such resources).

Make reading a lifelong thing, a lifestyle if you will. It doesn’t matter if you go through phases of your life where you’re not regularly reading a lot but always try to be on the lookout for a good book. Make a list of books you’d like to read and try to get those books and allocate time in your life to read them. Your future self will thank you.

Make yourself a goal: read at least one new book every month (or more or less) depending on what you’d like to get out of this habit.

It doesn’t matter which language we read in as long as we are reading quality books. Books provide good perspectives from various people, so try to read books from various authors not just one.

Aside from traditional books, there are audio and eBook options available. I enjoy all types of medium as long as the content is good and well written and in an area that I find interesting. So get out there and read something today!

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Amazing Math

1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321

1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111

9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888

Brilliant, isn’t it?

And look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111 = 12345678987654321

Now, take a look at this…

From a strictly mathematical viewpoint:

What Equals 100%?
What does it mean to give MORE than 100%?

We have all been in situations where someone wants you to

How about ACHIEVING 101%?

What equals 100% in life?

Here’s a little mathematical formula that might help
answer these questions:

Is represented as:

Therefore, one can conclude with mathematical certainty that:

get you there, It’s the Love of God that will put you over the top!

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Top 10 Reasons to get an Online Education

Online education is increasingly becoming available for various fields where it makes sense. Business, Mathematics, Humanities and some theoretical courses from physical sciences (Physics, Chemistry, Biology etc.) can easily be taught online. Of course, online education is simply not adequate for some disciplines, but it’s only a matter of time when virtually everything could be taught online.

1-Availability: Online education has matured over the years and is now available from some of the top universities in the world. For instance, check out the following:

2-Affordability: It’s a much more affordable option when compared to the traditional brick and mortar institutions. Some even offer FREE courses online Here is a short list of some of the providers.

3-Flexibility: You can study anywhere, anytime. This is a definite edge for those who wish to acquire knowledge despite their busy lives. If you’re traveling, you can take the class from your hotel room. Or watch class videos offline or write your paper while you’re flying over the Atlantic.

4-Global Collaboration: Your virtual classroom can have students from around the world from various backgrounds. Students can come together to exchange knowledge and benefit from a variety of backgrounds and experiences.

5-Fastest way to learn new technologies: As fast as technologies are evolving, it’s hard for most people to keep up. Offering online education and updating it as soon as changes occur is much more feasible than offering traditional offline instructor lead courses.

6-Way more choices for online courses: There is a dizzying array of courses available online from institutions from all over the world. If you find a subject hard to understand, you can even take the course from, say a university in the US, then follow it up with the same or similar course from the UK or Singapore or Canada or Denmarkor any university from anywhere in the world offering online courses.

7-Easier to acquire marketable skills: Some of the most marketable skills don’t seem to be available in every traditional school. If you need to learn the latest programming language or acquire Mandarin conversational skills.

8-Introverts, stay at home mothers and, in some cases, even prisoners can study: Online education is potentially available to anyone with a device and an Internet connection. You don’t even have to get out of the house much less travel to another city or country to acquire education via online outlets.

9-Great use of new technology: Online education technology has come a long way and matured enough to be very effective both for the providers enabling them to reach a much wider audience and for the students to have much bigger choice of courses available. Furthermore, courses can be provided in mixed media – audio, video, text, live sessions, discussion forums for students and teachers and more. This makes the experience for everyone involved much richer than the standard medium of face-to-face interaction, in some ways.

10-Continue acquiring knowledge long after your college days are over: Traditional education may be stopped after graduating from the university. However, online education can continue and anyone, young and old, can acquire new knowledge and skills.

What is missing from online education is the element of direct human interaction, in class rooms and on campus. This should be compensated for by having offline meet ups between local students and instructors where possible or simply taking some classes in traditional school settings. A hybrid online/offline model is probably the best option to get the best outcome.

In conclusion, do not dismiss online education. It’s real and it’s here to stay. Embrace it now as a viable alternative to traditional ways of learning or risk being left behind.

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3 Websites You Can Use to Make the World of Difference

by - Friday, November 28, 2014

It is almost without doubt that the vast majority of the public within every country desire greater global equality. According to a recent report by Oxfam though, the 85 richest people in the world are now as wealthy as the poorest half of the world combined.
This is a sobering fact and shows that the world is continuing to become less equal. There are also many other issues happening across the world such as climate change, job losses, bankers´ bonuses, and even the recent ruling in Ferguson in the US that can lead to a sense of disillusionment and helplessness. In the face of the government, authorities and powerful corporations, it can often seem impossible to make a difference. In fact through simple online actions you can bring about positive change!

3 Websites You Can Use to Make the World of Difference:

"Change.org is the world's largest petition platform, empowering people everywhere to create the change they want to see."

"Avaaz empowers millions of people from all walks of life to take action on pressing global, regional and national issues, from corruption and poverty to conflict and climate change. Our model of internet organising allows thousands of individual efforts, however small, to be rapidly combined into a powerful collective force." Avaaz currently has over 40 million members, and is growing rapidly.

"ONE is an international campaigning and advocacy organization of more than 6 million people taking action to end extreme poverty and preventable disease, particularly in Africa… because the facts show extreme poverty has already been cut by 60% and can be virtually eliminated by 2030, but only if we act with urgency now."
"Cofounded by Bono, we raise public awareness and work with political leaders to combat AIDS and preventable diseases, increase investments in agriculture and nutrition, and demand greater transparency so governments are accountable to their citizens. ONE does not raise money itself to build schools, hospitals and the like, but does its work by advocacy and campaigning so that government funds continue to flow to programs that make a difference in people’s lives. ONE works closely with African activists and policymakers as they fight corruption, promote poverty-fighting priorities, monitor the use of aid, and help build civil society and economic development. ONE’s work is strictly politically non-partisan."
These sites all have a number of things in common. They all have millions of members with rapidly growing userbases, they all harness social media such as Twitter and Facebook to share their campaigns, and they have all had huge successes which couldn´t have been achieved without the internet. The massive victories achieved by each site can be read about in depth on each of their websites. These include:

"Meriam Yehya Ibrahim saved from execution in Sudan" on change.org. "One million people from around the globe signed Emily Clarke's petition, joining a chorus of international voices demanding freedom and justice for Meriam, a Sudanese woman who was sentenced to death because of her faith."

"Playing a crucial role in the G8’s historic commitment in 2005 to double funding to fight poverty and disease in Africa – and holding their feet to the fire ever since." One.org
The victories and facts on each of these sites speak for themselves. You can make a difference no matter how difficult the odds facing you may seem. The power to change things is shifting and all it can take is enough people taking the time to sign a petition online.
If you liked this blog post and feel others might be interested in the websites I´ve mentioned please share it below. I would love to know any other websites that are great for online activism. Let me know in the comments. Thanks for reading!

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So today is my first full day back in the states and I just don’t feel the same here as I used to.  Yes, at first it was really nice and comforting to go through the US Citizens line and be back in Phoenix (even with the 110 degree weather that we arrived to) but now the odd feeling has set in.  I look at things that I used to pass by daily and they look a ton different.  The poor people on Mill seem like they have it better than they used to, the grocery store that I go to and can stock up on whatever I want seems to be FILLED with way too much food and even my instant Internet access now seems a little compulsive.

This trip really opened my eyes to how the “other” part of the world lives and how much I used to take things for granted.  The people in SA taught me also another life lesson, happiness is what you make of it.  Living in a camp with almost no food, using a metal port-a-john as your toilet, living in tents and showering in garbage bags; still somehow has made these people happy.  THey do not like their situations but they have found a way to make the best of it.

There are a million and a half lessons that I will take with me from this trip but the largest is to be grateful for everything that I have because like these people, it could be gone in just one night.


Using type="image" will create an upload field so the administrator can upload an image from their own computer. By using the optional variables shown below you can specify resizing and cropping for these images giving you control over what is displayed on the page.

Image upload attributes

Attribute Values and description
width Integer. The maximum width allowed, in pixels. The image will be proportionally scaled down if the width exceeds the given value.
height Integer. The maximum height allowed, in pixels. The image will be proportionally scaled down if the height exceeds the given value.
crop True or false. Whether to crop the image to a fixed size. Requires both width and height to be set.
quality Compression quality for uploaded images. Values are between 0 (very compressed) to 100 (top banana). The default is 85.
output The information to output with the tag. Optional, and defaults to path. See below.
density Integer. The pixel density of the image. Default is 1. A value of 2 would scale the image to 2x the dimensions given, but display the image at the size specified (e.g. for retina displays).
sharpen 0-10. The amount of sharpening to apply to images. Default is 4.
bucket The name of a resource bucket to store the image files in.
<perch:content id="logo" type="image" label="Logo">

Displaying images

It is important to note that if you simply use an image tag like the one above, this will print out the path to the image on your page, not a complete HTML image tag.

To create an image tag you would use this path in the src attribute of an HTML image tag, as shown below. I have also used an additional field with a type of text so that the administrator can add some alt text for the image.

<img src="<perch:content id="logo" type="image" label="Logo">" alt="<perch:content id="logoalt" type="text" label="Alt text">">

Changing image sizes in the template after upload

Perch resizes the images on upload when you add them to the site, therefore if you change the size of the images in your template you will need to re-upload the images to generate them at the correct size.

Sharpening transparent PNGs

The underlying PHP GD image library does a bad job of applying sharpening to transparent PNGs, resulting in unpleasant edge artifacts. There’s nothing Perch can do to fix this, it’s an underlying weakness in that part of PHP. If you need to add transparent PNGs, you have two options for avoiding this issue:

  1. Disabling sharpening with sharpen="0" on the template tag
  2. Switch to using ImageMagick, which does a much better job

Output options

As well as outputting the image path, you can use an image tag to output other types of information about the image. This is done with the optional output attribute.

Values for the output attribute

Value Description
path Default. The path to the image file.
w The width of the image in pixels.
h The height of the image in pixels.
size The file size in bytes. Can be used with the format attribute (e.g. MB).
tag Output a basic HTML image tag.
mime Output the mime type of the image, if known.

For example, to output the width of an image:

<perch:content id="logo" type="image" label="Logo" output="w">


Electronic Thesis and Dissertation Repository

Doctor of Philosophy

Dr. Daniel Ansari

Humans possess the remarkable ability to process numerical information using numerical symbols such as Arabic digits. A growing body of neuroimaging work has provided new insights into the neural correlates associated with symbolic numerical magnitude processing. However, little is known about the cortical specialization underlying the representation of symbolic numerical magnitude in adults and children. To constrain our current knowledge, I conducted a series of functional Magnetic Resonance Imaging (fMRI) studies that aimed to better understand the functional specialization of symbolic numerical magnitudes representation in the human brain.

Using a number line estimation task, the first study contrasted the brain activation associated with processing symbolic numerical magnitude against the brain activation associated with non-numerical magnitude (brightness) processing. Results demonstrated a right lateralized parietal network that was commonly engaged when magnitude dimensions were processed. However, the left intraparietal sulcus (IPS) was additionally activated when symbolic numerical magnitudes were estimated, suggesting that number is a special category amongst magnitude dimensions and that the left hemisphere plays a critical role in representing number.

The second study tested a child friendly version of an fMRI-adaptation paradigm in adults. For this participant’s brain response was habituated to a numerical value (i.e., 6) and signal recovery in response to the presentation of numerical deviants was investigated. Across two different brain normalization procedures results showed a replication of previous findings demonstrating that the brain response of the IPS is modulated by the semantic meaning of numbers in the absence of overt response selection.

The last study aimed to unravel developmental changes in the cortical representation of symbolic numerical magnitudes in children. Using the paradigm tested in chapter 2, results demonstrated an increase in the signal recovery with age in the left IPS as well as an age-independent signal recovery in the right IPS. This finding indicates that the left IPS becomes increasingly specialized for the representation of symbolic numerical magnitudes over developmental time, while the right IPS may play a different and earlier role in symbolic numerical magnitude representation.

Findings of these studies are discussed in relation to our current knowledge about symbolic numerical magnitude representation.


Geometric Proofs of Trigonometric Identities

Sparked by a conversation this past weekend about the usefulness of the half-angle identities, I constructed geometric proofs for \sin(x/2) and \cos(x/2). Since I’ve never seen these anywhere before, I thought I’d share.

And while I was at it, I thought I’d share all my other geometric proofs, so here they are, posted mostly without comment.

Some of these are so well-known as to be not worth mentioning. Many of them have been stolen from Proofs Without Words I or Proofs Without Words II. I came up with a few of them myself. Frustratingly, almost none of them are to be found in Precalculus textbooks, where they might be learned and appreciated.

Pythag 1

Pythag 2

Pythag 3

Pythag 4

Sincos of a sum 1

_______________________________________________________________Sincos of a sum 2

sincos of a diff 1

sincos of a diff 2

Though this one is my favorite:

sine and cosine of a sum best 1

sine and cosine of a sum best 2

Partially because of the way it naturally generalizes into the proof of the derivative of sine. If you just let \beta approach 0, \cos(\beta) approaches 1 and that point in the interior of the circle ends up on the circle, where \sin(\beta) merges with \beta itself.

Proof of derivative of sinx

double angle 1

double angle 2

double angle 3

half angle 1

half angle 2

half angle 3

half angle 4____________________________________________________________________________________

And finally, one that shows that the sum of a sine and cosine function of the same argument is also a sinusoid. Since I lost the original picture and don’t feel like remaking it, you’ll have to complete the proof on your own!

sum of sine and cosine

Update: After some feedback on twitter, I’ve decided to add a few more diagrams. Tim Brzezinski sent me a link to his website of geometric proofs of trig identities and he had some that I’ve never seen before.

Check it out!

I’ve taken two of his diagrams and added them below.

tan of a sum 1

tan of a sum 2

tan of a diff 1

tan of a diff 2

Extraneous Solutions – Part 2 of 3

Solving an Equation as a Sequence of Equation Replacement Operations

Part 1 was so long because I wanted to be extremely thorough and to present things to an audience that perhaps hadn’t thought much about the logic of equation solving at all. Since we’re now all experts, perhaps it’s worth it to summarize everything very succinctly.

Given an equation in one free variable, we want to find the solution set. To do this, we replace that equation with an equivalent equation whose solution set is more obvious.

(1) 8x - 5 = 5x + 1

(2) 8x = 5x + 6

(3) 3x = 6

(4) x = 2

If in the transition from (1)-(2), from (2)-(3), and from (3)-(4) we are careful to replace each equation with an equivalent equation, then by the transitivity of equivalence, the original equation and terminal equation are guaranteed to be equivalent. Since the solution set of the terminal equation is obvious, we know the solution set of the original equation, as well. Thus solving an equation requires establishing that certain equation replacement operations are indeed equivalence preserving and having the creativity and experience to know which ones to apply and in what order.

What are the Equivalence-Preserving Operations on Equations?

If a = b, then f(a) = f(b) for any well-defined function f. If a and b are expressions containing a free-variable, then any value of that variable which satisfies a = b will also satisfyf(a) = f(b). In other words, if you find it useful, feel free to replace any equation with a new equation which is the result of applying any function to both sides of the original equation. Any solution to the original equation will also be a solution to the new equation.

If the function f is also one-to-one, then by definition, f(a) = f(b) \Rightarrow a = b so any solution of f(a) = f(b) will also be a solution to a = b. Thus applying f to both sides of an equation is equivalence-preserving. If f is not one-to-one, then in general, the operation is not equivalence-preserving.

In solving equation (1), we applied f(n) = n + 5, g(n) = n - 5x and h(n) = n/2 in that order. Since all three of the functions are one-to-one, we are assured that (1) and (4) are equivalent. If we had cause to apply a non-one-to-one function, then we should be vigilant for extraneous solution.

A More Interesting Example

(5) \sqrt{6x-2} - \sqrt{x+1} = 2

As I mentioned in the other post, these square roots are begging to be squared, but since there are two of them, one squaring will not be enough. Even though it’s not necessary to do so, it’s helpful to move one radical expression to the other side.

(6) \sqrt{6x-2} = 2 + \sqrt{x+1}

(7) 6x - 2 = 4 + 4\sqrt{x + 1} + x + 1 We squared!

(8) 5x - 7 = 4\sqrt{x+1}

(9) 25x^2 - 70x + 49 = 16x + 16 We squared again!

(10) 25x^2 - 86x + 33 = 0

(11) (25x - 11)(x - 3) = 0

So x \in \{\frac{11}{25}, 3\}

Since in the transition from (6)-(7) and again in the transition from (8)-(9) we had reason to apply the non-one-to-one function f(n) = n^2, we should be vigilant for extraneous solutions. [Note: since both sides of (6) are necessarily positive, applying f(n) = n^2 is equivalence-preserving, so no extraneous roots will be created there.] By checking back in the original equation, we see that 3 is a solution, but \frac{11}{25} is not. I am more or less content to leave it at that. But some may ask for more clarity as to exactly what happened and when, so let’s indulge them.

I will now list each equation in reverse order along with its solution set:

(11) (25x - 11)(x - 3) = 0                            \{\frac{11}{25}, 3\}

(10) 25x^2 - 86x + 33 = 0                             \{\frac{11}{25}, 3\}

(9) 25x^2 - 70x + 49 = 16x + 16                 \{\frac{11}{25}, 3\}

(8) 5x - 7 = 4\sqrt{x+1}                                     \{3\}

Since 5\cdot\frac{11}{25} - 7 = \frac{11}{5} - \frac{35}{5} = -\frac{24}{5} \neq 4\sqrt{\frac{11}{25} + 1} = 4\sqrt{\frac{11}{25} + \frac{25}{25}} = 4\sqrt{\frac{36}{25}} = 4\cdot\frac{6}{5} = \frac{24}{5}

So we have isolated the precise moment when the extraneous solution x = \frac{11}{25} is created and it appears exactly where we would expect it, in the transition from (8) to (9) as we replaced (8) with the result of applying the non-one-to-one function f(n) = n^2 to both sides.

More specifically, if x = \frac{11}{25}, (8) reads - \frac{24}{5} = \frac{24}{5}, which is false, but (9) reads (- \frac{24}{5})^2 = ( \frac{24}{5})^2, which is true. For this particular value of x, we squared both sides and replaced a false statement with a true statement. In retrospect, we can say that x =\frac{11}{25} is not a solution to (8) or to any previous equation in the solving sequence, but is a solution to (9) and thus to all subsequent equations in the solving sequence.

(7) 6x - 2 = 4 + 4\sqrt{x + 1} + x + 1                          \{3\}

Since both sides of (7) are positive when x = 3, it does not surprise us that,

(6) \sqrt{6x-2} = 2 + \sqrt{x+1}                               \{3\}

(5) \sqrt{6x-2} - \sqrt{x+1} = 2                                \{3\}

By fully analyzing the logic behind each step of our equation replacement sequence, we not only:

  • confirm that x = 3 is a solution and that x = \frac{11}{25} is not and
  • understand that squaring both sides may produce an extraneous solution

but also

  • isolate the precise step in the solving sequence in which this extraneous solution was created answering the why, how, and when for this problem
  • confirm that the non-solution status of x = \frac{11}{25} is not merely due to an error of algebra or arithmetic, but is a direct result of that fact that this value produces an equation (8) of the form a = -a

That last point is crucial in distinguishing the phenomenon of extraneous roots from the phenomenon of user error in algebra or arithmetic. If our equation solving sequence consists solely of equivalence-preserving operations, we do not even need to check to see if solutions to our terminal equation are also solutions to our original equation. If we do decide to check, perhaps out of an abundance of caution, and find a discrepancy, then user error must be to blame.

On the other hand, if a solver does employ solution-set-enlarging operations in the solving sequence and finds that a solution to the terminal equation is not a solution to the original equation, is this because the solution is extraneous or due to user error? One could perform an analysis like I did above and confirm that the non-solution is not due to user error, but instead to the logic of the process.

Extraneous Solutions – Part 1 of 3?

Solving an Equation with a Radical Expression

Consider the following equation:

(4) x+8 = 9

Squaring Both Sides

A Silly Equation Leads to Extraneous Solutions

Consider the equation,

(6) x = 4

Another Radical Equation

(11) x+8 = 9

(12) x = 1

Those Who are Still Not Satisfied

A Trigonometric Equation which Yields an Extraneous Solution

For parallelism, consider the (somewhat artificial) equation:

-Will Rose

Proving identities – what’s your philosophy?

What happens in your classroom when you give students the following task?

Prove 1+\frac{1}{\cos{\theta}}=\frac{\tan^2{\theta}}{\sec{\theta}-1}.

Sometimes the command is Verify or Show instead of Prove, but the intent is the same.

Two non-examples

Here are two ways that a student might work the problem.

Method 1

Method 2

How do you feel about these methods? In my opinion, both methods represent a fundamental misunderstanding of the prompt. Method 1 is especially grotesque, but Method 2 also leaves a lot to be desired. Let me explain. And if you think the above methods are perfectly fine, please be patient and hear me out.

This is the crux of the issue:

The prompt was to prove the statement. But if the first line of our work is the very thing we’re out to prove, then we are already assuming the thing we want to prove. We’re Begging the Question.

It’s as if someone demands,

“Prove Statement X, please!”

and we reply,

“Well, let’s first start by assuming that Statement X is true.”

This is nonsense.

What went wrong?

So what is the proper way to engage this proof? Let’s roll back a bit.

The error in these approaches seems to stem from a desire to perform algebraic operations on both sides of an equation in the same way that you might if you were solving an equation.

When we “do algebra” and write Equation B below another Equation A without any words, we always mean that Equation A implies Equation B. That is, when we write

Equation A

Equation B

Equation C

we mean that Equation C follows from Equation B, which follows from Equation A.

Some might claim that each line should be equivalent to the last. But, again, when we “do algebra” by performing algebraic manipulations to both sides of an equation to transform it from equation A into equation B, we always mean A\Rightarrow B, we don’t mean A\iff B. Take, for example, the following algebra which results in an extraneous solution:

x=2 \text{ or } x=-1

In this example, each line follows from the previous, however reversing the logic doesn’t work. But we accept that this is the usual way we do algebra (A\Rightarrow B\Rightarrow C\Rightarrow \cdots). Here the last line doesn’t hold because only one solution satisfies the original equation (x=2). Remember that our logic is still flawless, though. Our logic just says that IF \sqrt{x+2}=x for a given xTHEN (\sqrt{x+2})^2=x^2.

As we move through the algebra line by line, we either preserve the solution set or increase its size. In the case above, the solution set for the original equation is {2}, and as we go to line 2 and beyond, the solution set is {2,-1}.

For more, James Tanton has a nice article about extraneous solutions and why they arise, which I highly recommend.

So if this is the universal way we interpret algebraic work, which is what I argue, then it is wrong to construct an argument of the form A\Rightarrow B\Rightarrow C in order to prove statement A is true from premise C. The argument begs the question.

Both Method 1 and Method 2 make this mistake.

How does a proof go again?

I want to actually make a more general statement. The argument I gave above regarding how we “do algebra” is actually how we present any sort of deductive argument. We always present such an argument in order, where later statements are supported by earlier statements.

ANY time we see a sequence of statements (not just equations) A, B, C that is being put forward as a proof, if logical connectives are missing, the mathematical community agrees that “\Rightarrow” is the missing logical connection.

That is, if we see the proof A,B,C as a proof of statement C from premise A, we assume that the argument really means A\Rightarrow B\Rightarrow C.

This is usually the interpretation in the typical two-column proof, as well. We just provide the next step with a supporting theorem/definition/axiom, but we don’t also go out of our way to say “oh, and line #7 follows from the previous lines.”

Example: Given a non-empty set E with lower bound a and upper bound b, show that a\leq b.

1. E is non-empty and a and b are lower and upper bounds for E. (given)
2. Set E contains at least one element x. (definition of non-empty)
3. a\leq x and x\leq b. (definitions of lower and upper bound)
4. a\leq b. (transitive property of inequality)

Notice I never say that one line follows from the next. And also notice that it would be a mistake to interpret the logical connectives as biconditional.

The path of righteousness

I encourage my students to work with only ONE side of the expression and manipulate it independently, in its own little dark box, and when it comes out into the light, if it looks the same as the other side, you’ve proved the equivalence of the expressions.

For example, to show that \log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)=-1+\log\left(\frac{t}{t-2}\right) for t>2, I would expect this kind of work for “full credit”:

\text{LHS }=\log\left(\frac{1}{t-2}\right)-\log\left(\frac{10}{t}\right)

=\text{ RHS}

Interestingly, I WOULD also accept an argument of the form A\iff B\iff C as justification for conclusion A from premise C, but I would want a student to say “A is true if and only if B is true, which is true if and only if C is true.” Even though it provides a valid proof, I discourage students from using this somewhat cumbersome construction.

So let’s return to the original problem and show a few ways a student could do it correctly.

Three examples

Method A – A direct proof by manipulating only one side

Method B – A proof starting with a known equality

Method C – Carefully specifying biconditional implications

\text{if and only if}

\text{if and only if}

\text{if and only if}

\text{if and only if}

While all of these are now technically correct, I think we all prefer Method A. The other methods are cool too. But please, please, promise me you won’t use Methods 1 or 2 which I presented in my introduction.

In conclusion

Some might argue that the heavy criticism I’ve leveled against Methods 1 and 2 is nitpicking. But I disagree. This kind of careful reasoning is exactly the business of mathematicians. It’s not good enough to just produce “answers,” our job is to produce good reasoning. Mathematics, remember, is a sense-making discipline.

Thanks for staying with me to the end of this long-winded post. Can you tell I’ve had this conversation with a lot of students over the last ten years?

Further reading

  1. Dave Richeson has a similar rant with a similar thesis here.
  2. This article was originally inspired by this recent post on Patrick Honner’s blog. A bunch of us fought about this topic in the comments, and in the end, Patrick encouraged me to write my own post on the subject. So here I am. Thanks for pushing me in the right direction, Mr. Honner!

Happy π day!

Today at our school we had to have the obligatory π day celebrations. Here are the ways we observed π at RM:

  • To raise money for the math honors society, students purchased π-grams for one another and a piece of pi with a note were delivered to the recipients during first period.
  • There was a pie eating contest on “main street”, also sponsored by the math honors society, and guess who won? Check out my crown!!

Get the joke?? On the back it says "We are the 3.14%."

Action shot

Mr. Chase wins!

  • In each of my classes kids bring in food and we have a π day party. I pass out a sign up sheet a few days before and kids agree to bring all sorts of round food. Here’s the sign up sheet. Feel free to use it, steal it, modify it. Also available in pdf format. Here’s one of my favorite food items from this year:

  • I showed these youtube videos.
  • I did a mini lecture on the history of π and gave some interesting facts about the number.
  • Students got to find out at what digit of π their birthday appears. You can too!

One more thing you can still do, if you haven’t yet observed π day:

Happy π day!!!

Also, on an unrelated note, today’s Google logo is great. If you’re interested in the mathematics of origami, you probably know who Robert Lang is. Today’s Google logo is an origami piece created by Lang in honor of the late Akira Yoshizawa, world famous origami artist.

Pi R Squared

[Another guest blog entry by Dr. Gene Chase.]

You’ve heard the old joke.

Teacher: Pi R Squared.
Student: No, teacher, pie are round. Cornbread are square.

The purpose of this Pi Day note two days early is to explain why \pi is indeed a square.

The customary definition of \pi is the ratio of a circle’s circumference to its diameter. But mathematicians are accustomed to defining things in two different ways, and then showing that the two ways are in fact equivalent. Here’s a first example appropriate for my story.

How do we define the function \exp(z) = e^z for complex numbers z? First we define a^b for integers a > 0 and b. Then we extend it to rationals, and finally, by requiring that the resulting function be continuous, to reals. As it happens, the resulting function is infinitely differentiable. In fact, if we choose a to be e, the \lim_{n\to\infty} (1 + \frac{1}{n})^n \, not only is e^x infinitely differentiable, but it is its own derivative. Can we extend the definition of \exp(z) \, to complex numbers z? Yes, in an infinite number of ways, but if we want the reasonable assumption that it too is infinitely differentiable, then there is only one way to extend \exp(z).

That’s amazing!

The resulting function \exp(z) obeys all the expected laws of exponents. And we can prove that the function when restricted to reals has an inverse for the entire real number line. So define a new function \ln(x) which is the inverse of \exp(x). Then we can prove that \ln(x) obeys all of the laws of logarithms.

Or we could proceed in the reverse order instead. Define \ln(x) = \int_1^x \frac{1}{t} dt . It has an inverse, which we can call \exp(x) , and then we can define a^b as \exp ( b \ln (a)). We can prove that \exp(1) is the above-mentioned limit, and when this new definition of a^b\, is restricted to the appropriate rationals or reals or integers, we have the same function of two variables a and b as above. \ln(x) can also be extended to the complex domain, except the result is no longer a function, or rather it is a function from complex numbers to sets of complex numbers. All the numbers in a given set differ by some integer multiple of

[1] 2 \pi i.

With either definition of \exp(z), Euler’s famous formula can be proven:

[2] \exp(\pi i) + 1 = 0.

But where’s the circle that gives rise to the \pi in [1] and [2]? The answer is easy to see if we establish another formula to which Euler’s name is also attached:

[3] \exp(i z) = \sin (z) + i \cos(z).

Thus complex numbers unify two of the most frequent natural phenomena: exponential growth and periodic motion. In the complex plane, the exponential is a circular function.

That’s amazing!

Here’s a second example appropriate for my story. Define the function on integers \text{factorial (n)} = n! in the usual way. Now ask whether there is a way to extend it to (some of) the complex plane, so that we can take the factorial of a complex number. There is, and as with \exp(z), there is only one way if we require that the resulting function be infinitely differentiable. The resulting function is (almost) called Gamma, written \Gamma. I say almost, because the function that we want has the following property:

[4] \Gamma (z - 1) = z!

Obviously, we’d like to stay away from negative values on the real line, where the meaning of (–5)! is not at all clear. In fact, if we stay in the half-plane where complex numbers have a positive real part, we can define \Gamma by an integral which agrees with the factorial function for positive integer values of z:

[5] \Gamma (z) = \int_0^\infty \exp(-t) t^{z - 1} dt .

If we evaluate \Gamma (\frac{1}{2}) we discover that the result is \sqrt{\pi} .

In other words,

[6] \pi = \Gamma(\frac{1}{2})^2 .

Pi are indeed square.

That’s amazing!

I suspect that the \pi arises because there is an exponential function in the definition of \Gamma, but in other problems involving \pi it’s harder to find where the \pi comes from. Euler’s Basel problem is a good case in point. There are many good proofs that

1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}

One proof uses trigonometric series, so you shouldn’t be surprised that \pi shows up there too.

\pi comes up in probability in Buffon’s needle problem because the needle is free to land with any angle from north.

Can you think of a place where \pi occurs, but you cannot find the circle?

George Lakoff and Rafael Núñez have written a controversial book that bolsters the argument that you won’t find any such examples: Where Mathematics Comes From. But Platonist that I am, I maintain that there might be such places.


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