As I’ve written previously, mathematics and technical communication (tech comm) both model reality. In math, numbers do not “exist” in the literal sense of the word. You can have 3 coins, but the concept of 3 does not occupy a physical point in time or space; it transcends it. Numbers, therefore, describe the quantities or properties of a person, place or thing but are not actual people, places or things.

Similarly, tech comm is a description of reality but is not reality itself. A guide explaining how to use a smartphone is not an smartphone but a representation of it. The ideas, lesson and concepts in the guide must be interpreted and understood by a human reader; therefore these things exist only in the reader’s mind.

Now, if mathematics and tech comm are attempts to describe reality, it follows that some of the basic principles of math should apply to tech comm.

Numbers are the building blocks of all mathematics. The 10 digits which form all numbers are math’s “alphabet”, however, not all numbers are equal; they fall into various groups.

Natural numbers are all whole positive numbers: 1, 2, 3 and so on. These are the practical, real-world numbers that we use each day when counting, ordering, adding, and so on. They are precise and complete because they exclude fractions or decimals. Any simple, clear and complete positively stated information corresponds to a natural number, for example: Sales increased 7% over last year.

Negative numbers are numbers less than 0. They were first envisioned by the Chinese over 2,000 years ago. There is a theory that the idea of duality in Chinese philosophy made it easier for this culture to develop the idea of a number less than zero.

Any negative statement corresponds to a negative number, for example:

Do not turn off your computer during the installation.

Fractions have at least two parts: the top number of the fraction or numerator and the lower portion or denominator. However, fractions can have more than two parts in the form of a complex fraction, for example (2/3)/( 5/7).

Complex modern content management systems (CMS) are actually composed of fractional pieces of information which are reused as required. For example, there may be many procedures which all refer to a specific part number. If you are using are using a typical Word processor to document these procedures and the part number changes, you’d have to manually search and replace every occurrence of this number. However, in a CMS, the part number is stored once in a database as a variable, and therefore only has to be changed once. All references to that part number are then automatically updated. Any piece of information can be a “informational fraction”, from a word, to a sentence, a paragraph, and even a page.

Irrational numbers have an infinitely long series of non-repeating digits after the decimal place. You can’t write them as a fraction or ratio. Examples include the square root of 2 (1.4142135…) and pi (π) which is equal to 3.14159265…. As you continue down the line of infinite digits, you get incrementally closer and closer to the true value of the number. However, when calculating values, you have to stop a certain point; you can’t simply go on forever. NASA scientists are able to keep the space station running using only 16 digits of pi. For calculating the fundamental constants of the universe, they need 32 digits.

Irrational communication is comprised of pieces of information which each add ever-decreasing value to the information. For example, let’s say you need to write a step that instructs users how to connect to a wi-fi network. The statement you develop is:

To connect to the wi-fi network, select the ABC_Network, then enter the following password: Pass1532.

For most people, this would suffice. However, what about novice users who don’t even know how to select a wi-fi network? We’d have to add another piece of information, underlined below:

To connect to the wi-fi network, on your device, under Settings, select Wi-Fi connections, select the ABC_Network, then enter the following password: Pass1532.

This seems complete, right? But what about people who are not sure what you mean by “device”? To address this, we add even more information:

To connect to the wi-fi network, on your SmartPhone, tablet, laptop or desktop, under Settings, select Wi-Fi connections, select the ABC_Network, then enter the following password: Pass1532.

But what about people who don’t know what a wi-fi network is? We add:

You can use our wi-fi network to connect to the Internet. To connect to the wi-fi network, on your SmartPhone, tablet, laptop or desktop, under Settings, select Wi-Fi connections, select the ABC_Network, then enter the following password: Pass1532.

And what about those poor souls who don’t know what the Internet is?

The internet is the world’s largest information network. It used to send and receive information, view news items, images, videos and sound, and to connect with others. You can use our wi-fi network to connect to the Internet. To connect to the wi-fi network, on your SmartPhone, tablet, laptop or desktop, under Settings, select Wi-Fi connections, select the ABC_Network, then enter the following password: Pass1532.

One could go adding information forever but I think you get the point. Each piece of new information, just like each additional digit in an irrational number, adds a bit more value to the original piece of information. How many “decimals” of information are required depends on the knowledge level of the average user. Too much information is as bad as not enough.

Now we come to one the most challenging types of numbers: imaginary. At some point, mathematicians asked: what is the square root of a negative number? There is no clear answer, because no number multiplied by itself produces a negative number. Two negative numbers multiplied together produce a positive number. To resolve this, mathematicians invented imaginary numbers, written with the letter i. For example, the square root of -9 is 3i.

The essence of imaginary numbers is:

• two numbers are combined together
• combining numbers normally creates a larger number but in this case actually creates a smaller one, therefore,
• an inherent contradiction is created

The informational equivalent of an imaginary number is a statement added to another statement that creates a conflict and therefore lowers the value of both statements.

For example:

1. If you over 18 years old, complete Form A.
2. If you are less than 25 years old, complete Form B.

The second statement contradicts the first, and thereby negatively impacts both statements. It is the equivalent of the mathematical i, in this case, the i standing for incomprehensible, impossible, inexcusable and, quite possibly, insane. This is actually a common problem, especially with complex policies, procedures and regulations that are riddled with contradictions.

Finally, an exponent is a number that dramatically increases the value of another number, for example, 3³ which equals 3 x 3 x 3 or 27. Conversely, a square root is a number that when multiplied by itself creates a larger number, for example, 10 is the square root of 100. In either case, we are changing a small number very rapidly to much larger one, or vice versa.

If there’s one thing that information experts agree on, it’s that the amount of information in the world has grown exponentially. How much? A quick math lesson is in order. An exabyte is one quintillion (1018) bytes, which is one billion gigabytes or one thousand million billion bytes, a byte being equivalent to about one letter. One exabyte is up to 3,000 times the size of all the content in the Library of Congress. Between the start of history and 2003, five exabytes of information in total were created. We now create five exabytes every two days. Big data, indeed.

The single informational device that has contributed to the ability to access this near-infinite amount of information is a textual object that is absurdly simple yet staggeringly complex: the hyperlink. For with a single hyperlink, a tiny piece of information directly connects to something much larger. (This small link, for example, links to something vast.) The destination of the hyperlink is the exponent of the hyperlink itself; the hyperlink, therefore, is the root of the much larger piece of information that it points to.

It’s no coincidence that the word mathematics literally means “to learn”. The primary goal of tech comm is that the user learns something, whether it is a concept or a task. The connections between mathematics and tech comm are, as with math itself, measured, complex, and infinite.

Here is one of the world’s most difficult mathematical problems:

For the equation: aⁿ + bⁿ = cⁿ, where a, b and c are whole numbers, n must equal 2. In other words, this equation only works if n =2.

For example, the following numbers fit this equation:
3² + 4² = 5² and 5² + 12² = 13². If n equals 3 or any other number, you won’t find any solutions to this equation.

This problem is known as Fermat’s Last Theorem, named after the French mathematician Pierre de Fermat, who lived during the 1600s. While annotating a book about mathematics, Fermat claimed to have found a solution. He wrote: “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” Too bad he wasn’t using a Word processor with its ability to add notes of unlimited size.

The problem remained unsolved for over 350 years until a British mathematician named Andrew Wiles finally conquered it in a monumental 200 page proof. How he solved it is a fascinating adventure into the strange and mysterious world of higher mathematics.

Wiles’ solution involved two very strange mathematical shapes: elliptic curves and modular forms. Elliptic curves resemble doughnuts, whereas modular forms don’t resemble anything and are therefore much more difficult to describe, but here goes:

A modular form is an incredibly complex, highly symmetrical form with many dimensions. It is impossible to draw one because it only exists as a conceptual form.

Elliptic curves and modular forms are very different from each other. However, the solution to the theorem involved proving that these two shapes, are, in fact, the same. When the idea that these two forms might be identical was initially proposed, it was a radical concept. It was like saying that an elephant is a banana, which is, quite simply, bananas.

However in 1995, Wiles proved these two forms were indeed identical. In doing so, he solved Fermat’s Last Theorem. How proving that these two forms were the same also solves Fermat’s Last Theorem is beyond the scope of this article. (For a full explanation, read the PBS transcript from the Nova documentary, The Proof.)

Mathematics and technical communication both attempt to model reality, and both use informational objects to do so. The primary object (or shape) that a technical communicator develops is the information repository, which is comprised of:

1. Topics (such as overviews or procedures) that answer specific questions.
2. Containers and sub-containers for the various topics (such as other topics, pages, chapters or other sections).
3. A function enabling the user to search the topics (an index, TOC, or content search function).
4. An environment that contains all the topics and the search function (for example, a PDF, help system, or website).

Users deal with another shape: informational queries, which are comprised of:

1. The generation of specific questions, such as “what is this thing?”, “how do I perform this task?”, “how do I resolve this problem or error?”
2. The process of determining where to find the answers.
3. Locating the relevant information repository.
4. Searching the information repository.
5. Locating the topic that they hope will answer their question.
6. Understanding the answer to their question, that is, the contents of the relevant topic.
7. Successfully resolving their question, for example, by understanding a concept, completing a task or resolving a problem or error.

Both of these shapes require all of their respective components in order to be considered complete shapes. For example, an informational query is incomplete if the user can only complete the first six steps. They may find and understand the relevant help topic, but if they cannot complete it (due to an error in the topic, the product or both), then the informational query is incomplete.

Just as elliptic curves and modular forms, two radically different shapes, were proven to be the same, both information repositories and informational queries are the same. This is because each shape is a reverse-engineered version of the other.

When a technical communicator creates an information repository, they are attempting to recreate the steps that a user will follow in an informational query.  Communicators try to anticipate as many of the questions that a user will have, then work backwards to create a resposity that will the answer the user’s question.

We can take the steps of an informational query, change their order (mostly by reversing them), and then structure them from the perspective of the technical communicator:

1. Consider all the potential questions a user could have.
2. Create topics that successfully resolve these questions.
3. Ensure the topics are written so that the user will understand them.
4. Index the topics so that they are searchable.
5. Create a search system that will enable the user to find the relevant topic.
6. Place the information repository in a location where the user can access it.
7. Make it obvious to the user where the information repository is located.

Conversely, we can reverse engineer an information repository from the user’s perspective:

A user needs to:

1. Locate the environment containing the relevant document that will answer their question.
2. Search the topics for the answer.
3. View the various topics that might contain the answer.
4. Find the topic that answers the question.

One shape is but a mirror-image of the other.

The commonality goes even further, for all end users are potential communicators, and all communicators should ideally “be” the end user. The greatest documentation is created when end users actually communicate directly with the technical communicator, and when the technical communicator imagines themselves to be the end user, with all of their worries, concerns and, most of all, questions.

In fact, I have developed a proof that verifies that the number of end users in the world is identical to the number of technical communicators.

Unfortunately, this blog is too small too contain it.

Game theory is a specialized field of mathematics that analyzes choices and results in strategic situations, or games, as the players try to maximize their success. It can be applied to practically any situation where one is making a choice for personal benefit, for example:

• choosing a restaurant
• hiring a worker
• buying and selling stocks
• deciding who to marry
• playing poker (or any other game)

Game theory has been applied to such diverse fields as: economics, evolutionary biology, engineering, political science, psychology, philosophy and business management. Google even uses it to maximize the advertising revenue generated from their AdWords. All of these areas require making the best decisions possible in order to create the maximum benefit.

Game theory takes into account the fact that humans are essentially self-centered, that is, that we tend to act in a way that we think will be best for us. Even when we appear to be selfless, we’re still acting in our own self-interest. For example, giving to charity gives us the benefit of feeling that we’re doing a good thing, a benefit that we’re willing to pay for.

The Nash equilibrium (also know as governing dynamics) is a set of game theory strategies. It states that the individuals in any situation will benefit the most if they do not only what is best for them individually, but also what is best for the entire group.

This equilibrium was developed by the mathematician John Nash, who was profiled in the wondrous film, A Beautiful Mind. The film gives a rather sexist but graphic example of his theory:

A group of men and women are at a bar. Each of the men wants to pair off with each woman. However, one of the women, a blond, is more attractive than the others. The question is: should each man go for a less attractive woman, or try for the blond?

The Nash equilibrium implies that no man should try to pick up the blond. Odds are they will all be rebuffed. If the men then try to go after the other women, they’ll most likely be rejected because each women will know that they were the man’s second choice. The best strategy, therefore, is for each man to try to pursue a woman other than the blond.

To sum up: the Nash equilibrium states that in any situation involving trade-offs, the maximum benefit is achieved if everyone is making the best decision they can taking into account the decisions of the others.

The Nash equilibrium has been applied to an amazing variety of areas including:

• arms races
• technical standards
• bank runs
• currency fluctuations
• traffic flow
• auctions

Nash’s theorem was so ground-breaking that in 1994 he won the Nobel prize in economics for his theorem. (There is no Nobel prize for mathematics.)

The philosophy of this equilibrium can be applied to information development on two different levels: within the architecture of information objects and to the information development process.

Documentation Objects

All documents contain sets of objects that can be thought of as “players” in a game. These objects include:

• topics
• cross-references
• TOC entries
• index entries
• graphics

In addition, documents themselves form objects in a documentation set, as part of a group of related documents.

Conflicts will arise if there are two or more objects within each of these areas that are difficult to distinguish. Examples include:

• topic with similar names, such as Overview and Introduction
• index entries that begin with the term removing and others that begin with the term deleting
• two graphics that describe the same thing but in a slightly different way
• using the same term to describe different things
• guides with similar names, for example an Administration Guide and a Technical Guide

Conflicts such as these create confusion for the end user, because they have no way of knowing which is the “correct” version. This creates an unnecessary game-like situation for the user, as they struggle to pick the winning object.

To avoid this, all documentation needs to be carefully reviewed and purged of all conflict. The end result is a series of objects that play nicely together. What is best for each individual documentation object is also best for the group of objects; the very essence of governing dynamics.

Multiple Authors

The Nash equilibrium can be applied even more directly to the information development process in a multi-author environment. Many organizations have content management systems in which multiple authors can create, edit and manage their documentation.

While such systems can create a more balanced workload, the potential for conflict is enormous. Even if the system only allows one editor at a time (a standard feature of any content management system), it’s still easy for writers to get into editing conflicts in which whoever updates last “wins”.

This is not a technical issue but a management issue. Writers must understand they are not competing against each other but against incomplete and inaccurate documentation. The end user wants relevant information, and is not interested the writers’ turf wars. As the Nash equilibrium implies, writers working on common documentation sets need to know that what is best for the group is also what is best for each of them.

There are probably few areas in life to which governing dynamics could not be applied. Therefore, the idea that “life is but a game” is far more true than we can possibly imagine.