Deliverable Overview
You'll write one explainer, 600 to 900 words long, for a reader with no mathematical background. Picture a friend who has asked how something they use every day, such as paying online, scanning a barcode, or searching the web, actually works. Your explainer is your answer.
It has five required components:
- The technology and the problem it solves
- At least two branches, chosen from number theory, algebra, and linear algebra, each with the specific idea it contributes
- One small worked calculation
- One paragraph on what the explanation simplifies or leaves out
- Two or more sources you consulted
Citations are welcome throughout, not only in the source list. A simple format works: author, year, title, and where to find it, such as "Brin and Page, 1998, 'The Anatomy of a Large-Scale Hypertextual Web Search Engine,' free PDF from Stanford." The source list doesn't count toward the word limit.
Detailed Requirements
The Technology and Its Problem
Name one technology you use and state, in your first paragraph, the problem it solves. Make the problem concrete. "Online payments need security" is too broad. "When you send a card number to a shop you've never dealt with, anyone along the way could read it, so the number has to be scrambled in a way only the shop can undo" is a problem a reader can hold onto.
Two or More Branches and What Each Contributes
Show at least two of the three branches at work, and for each one name the specific idea it contributes. A branch named without an idea doesn't count. Some pairings to consider:
- Number theory studies whole numbers: primes, divisibility, and remainders. Modular arithmetic, or clock arithmetic, keeps only the remainder after division by a fixed number. RSA encryption raises a number to a power and keeps the remainder (Rivest, Shamir, and Adleman 1978, sec. V). The ISBN check digit uses remainders on division by 10 (International ISBN Agency 2017, sec. 5.5).
- Algebra studies structures, such as groups. A group is a set with a way of combining its members in which every combination can be undone and one member, the "do nothing" element, changes nothing. The remainders that share no factor with a fixed number form a group under multiplication. The identity due to Euler and Fermat, which shows RSA decryption undoes encryption, is a fact about such a group (Rivest, Shamir, and Adleman 1978, sec. VI). Error-correcting codes, which add structured redundancy so that a damaged message can be repaired, are another place to look (Hamming 1950).
- Linear algebra studies vectors and matrices. A vector is a list of numbers; a matrix is a grid of numbers that turns one vector into another. PageRank treats the web's links as a matrix and looks for scores that the matrix leaves unchanged (Brin and Page 1998, sec. 2.1). Graphics software and machine learning also rely heavily on matrices.
You can pick a technology not on this list. The test is the same: which idea, from which branch, does which job.
One Small Worked Calculation
Work one calculation by hand that shows the idea in action, using numbers small enough that your reader could check every step. Show each step and then check the result, for example by decrypting what you encrypted or confirming a check digit.
Good candidates include a check digit for a real ISBN, a toy RSA encryption with primes under 100, a small matrix applied to a few points, or a ranking of three pages that link to one another. Small numbers are an honest choice, not a shortcut. The 1978 RSA paper illustrates its method with the primes 47 and 59, while recommending primes of about 100 digits for real use (Rivest, Shamir, and Adleman 1978, secs. VII–VIII).
What the Explanation Leaves Out
Write one paragraph, at least, on what your explainer simplifies or omits. Separate three things: what is proved, what is assumed, and what real systems add. For RSA, decryption working is proved, while security rests on the unproved belief that factoring large numbers is hard. For a check digit, say which errors it catches and which it misses. For a ranking method, say what other signals a real system combines with it; the 1998 PageRank paper already combines link scores with word position, font, and link text (Brin and Page 1998, sec. 4.3).
Your Sources
List at least two sources you consulted. Strong choices are an original paper, a standards document, a textbook, or an explainer from a university or standards body. Check your account against them, and make no claim they don't support.
Success Criteria
Your explainer meets the requirements when:
- The technology and its problem are stated in the first paragraph
- At least two branches appear, each with the specific idea it contributes
- One calculation is worked with small numbers and checked
- Every mathematical term is defined on first use
- A paragraph states what the explanation simplifies or leaves out
- No claim goes beyond what your sources support
- At least two sources are listed
Step-by-Step Guidance
Step 1: Choose a technology and state its problem. Pick something you use and can describe from the outside. Write one sentence naming the problem it solves. If you can't write that sentence, choose another technology.
Step 2: Identify the branches and the idea from each. For each branch you plan to use, write one line: the branch, the idea, and the job it does. For an ISBN, number theory supplies remainders on division by 10, and the job is catching typing errors. Aim for two or three lines.
Step 3: Work one small calculation. Choose numbers small enough to check by hand. For an ISBN check digit, multiply the first twelve digits alternately by 1 and 3, add the products, and find the remainder on division by 10. The check digit is 10 minus that remainder, or 0 if the remainder is 0. The manual's own example, 978-92-95055-12-?, gives a weighted sum of 126, a remainder of 6, and a check digit of 4 (International ISBN Agency 2017, app. 1). Then check your result a second way.
Step 4: Find two sources and check your account. Look for the original paper, a standards document, or a reputable explainer. Read the part that covers your idea and compare it with your notes. Where they disagree, trust the source and revise.
Step 5: Write for a reader with no mathematical background. Define each term the first time it appears, in words, before any symbol. Use short paragraphs. Keep the calculation in its own paragraph or short numbered list so a reader can follow it line by line.
Step 6: Add the paragraph on what you left out. Reread your draft and ask what a specialist would object to. Name what is proved and what is assumed, and what a real system adds. This paragraph is where you protect your reader from overclaiming.
You may draw on notes or practice work of your own.
Resources
Common Pitfalls
- Describing the technology without the mathematics. A tour of what happens when you press "Pay" isn't an explainer. Each stage you describe should point to an idea and the job it does.
- Naming a branch without saying what it contributes. "This uses linear algebra" tells your reader nothing. "Each page's score is a weighted sum of the scores of pages linking to it" tells them something.
- A calculation too large to check by hand. Real RSA keys run to hundreds of digits. Your reader needs to follow every step, so use small numbers and say that real systems use far larger ones.
- Overclaiming. Calling encryption "mathematically unbreakable" is wrong. Its correctness is proved; its security rests on the belief that certain problems, such as factoring large numbers, are hard.
- Treating a company's marketing as a source. A product page tells you what a company wants you to believe, not how the system works. Use papers, standards, and textbooks.
References
- Brin, Sergey, and Lawrence Page. 1998. "The Anatomy of a Large-Scale Hypertextual Web Search Engine." Computer Networks and ISDN Systems 30 (1–7): 107–117.
- Hamming, R. W. 1950. "Error Detecting and Error Correcting Codes." Bell System Technical Journal 29 (2): 147–160.
- Hefferon, Jim. n.d. Linear Algebra. 4th ed. Colchester, VT: Saint Michael's College. GNU Free Documentation License or CC BY-SA 2.5.
- International ISBN Agency. 2017. ISBN Users' Manual. International ed., 7th ed. London: International ISBN Agency.
- Rivest, Ronald L., Adi Shamir, and Leonard M. Adleman. 1978. "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems." Communications of the ACM 21 (2): 120–126. doi:10.1145/359340.359342.