Thursday, December 31, 2015

Multiplication is not completely reducible to addition, even for whole numbers

Mathematician Keith Devlin has written several articles about the relationship between multiplication and addition (1, 2, 3, 4, 5). Drawing on a variety of sources, including pure mathematics and empirical research about learning, Devlin argues that since multiplication isn't repeated addition, students shouldn't be taught that it is. I think that's basically right, though in practice it is complicated, as Devlin acknowledges. 

In this sidebar, I would like to mathematize the basic question. "Multiplication is repeated addition" might initially sound like a mathematical claim, something straightforwardly either true or false, but as it stands, this statement is too vague to admit of either proof or disproof. Some precise definitions are necessary before we can consider the question from the standpoint of mathematics.

Everybody agrees that repeated addition is useless as a way to understand multiplication of real numbers, as in cases like π × e. Where I think I differ from most people is that I do not even grant that multiplication reduces to repeated addition for the case of whole numbers. But that claim is also vague, so let's get started. You will let me know if anything below is incorrect or needs fixing.

Definition 1. A whole-number operation is a function that maps every pair of non-negative integers to some non-negative integer, which is called the result of the operation, or the resulting value. Since we only consider whole numbers, for brevity we simply call a whole-number operation an operation. The result of an operation B on two numbers m and n can be denoted mBn.

Examples: 

1) Addition (+) and multiplication (×) are operations. 

2) Let F be defined by mFn = 7 × n. Then F is an operation, in particular a noncommutative operation. For example, 4F6 = 42, but 6F4 = 28. 

3) Let G be defined by mGn = |m − n|. Then G is an operation, in particular a nonassociative operation. For example, 1G(1G2) = 0, but (1G1)G2 = 2. 

Definition 2. An addition expression is a string of symbols consisting of k ≥ 2 whole numbers separated from one another by k − 1 addition symbols. 

Examples. 

4) "2 + 3" is an addition expression.  

5) "2 + 3 + 9" is an addition expression. 

6) " + 1" is not an addition expression. 

7) "1" is not an addition expression. 

8) "6 × 7" is not an addition expression. 


Remark 1. "1" and "6 × 7" are algebraic expressions, but not addition expressions. 

Remark 2. An addition expression can be said to have a value in the obvious way, e.g., the value of "2 + 3 + 9" is 14. 

Definition 3. An operation C is completely reducible to addition if, for all whole numbers m and n, there exists an addition expression that equals mCn and contains no numbers besides m or n

Examples. 

9) The operation mFn = 7 × n is completely reducible to addition. Here is a particular instance: the addition expression 4 + 4 + 4 + 4 + 4 + 4 + 4 equals 5F4 and contains no numbers besides 5 or 4. 

Remark 3. One might have thought that only commutative operations are completely reducible to addition; Example 9 shows that this is not the case.

10) The operation mGn = |m − n| is not completely reducible to addition. To see this, consider that 1G1 = 0, whereas it follows immediately from the definition of an addition expression that an addition expression containing only 1s has value at least 2.

Theorem. Multiplication is not completely reducible to addition. 

Proof. As noted in Example 10, an addition expression containing only 1s has value at least 2. So an addition expression containing only 1s cannot equal 1 × 1 in value. ∎

The theorem states the obvious: 1 × 1 isn't equal in value to any well formed expression consisting of 1s and addition symbols. According to the sense of Definition 3, this means that multiplication is not completely reducible to addition. 

This being obvious, it must be that when people talk of reducing whole-number multiplication to addition, they have something else in mind besides Definition 3.


Wikipedia provides the following quasi-definition of multiplication as repeated addition:

m × n = n + n + ... + n   (m addends). 

I'm not sure the Wikipedia authors have thought about what happens to this prescription when m = n = 1. In this case, it says

1 × 1 = 1   (1 addend). 

The equation is true, of course, but to our point today I can't help noticing that there's no addition in it! Addition is a binary operation. If you are truly adding, then there's never just 1 addend. Whatever the Wikipedia formula reduces multiplication to, it isn't addition. 

Maybe the Wikipedia definition ought to say something like this instead?

m × n = 0 + n + n + ... + n   (m instances of n)

For m = n = 1, this becomes

1 × 1 = 0 + 1   (1 instance of 1)

which is a true equation, and, well, if it contains a number other than 1, then at least it is an addition expression. But now what happens when we take m = 0? Since, according to the prescription, we mustn't write any instances of 5, the result is an ill-formed expression:

0 × 5 = 0 + 

Maybe we could patch things up by agreeing that the phrase "no instances of 5" means "insert the number 0." But wait—isn't that just multiplying?

I'll leave off trying to patch up Wikipedia's formula (feel free to take up that project here) and just list some representative cases that prevent us from interpreting multiplication as repeated addition:

1 × 1
1/5 × 3/4
−1/5 × 3/4
π × e
xy

Hmm. Kind of a lot. It makes me wonder what the value is of defending the idea. 


In my own work, when I have needed to describe the relationship between multiplication and addition, I have pointed first to the distributive property: 

a × (b + c) = a × b + a × c.

Observe that both operations appear on both sides of this identity—it is not a prescription for eliminating one operation in favor of another. Nothing in the axioms is inviting us to undertake that project. 

Indeed the distributive property is the only field axiom that refers to both operations, which means that it captures the entirety of the relationship between the two operations. For that reason, to describe the relationship between multiplication and addition using formulations other than the distributive property itself is to invite error. But is there any true and valuable statement that juxtaposes multiplication with repeated addition? I'll try one:  
Repeated addition is a special-case algorithm for calculating whole-number products, m × n, when m > 1 or n > 1. 
Proving that the algorithm works involves the distributive property, of course:

Proof: By assumption, either m > 1 or n > 1. First suppose n > 1. Then, and only then, can n be written as a sum of 1s: = 1 + 1 + ... + 1 (n addends). So by the distributive property, m × = m × (1 + 1 + ... + 1) = m × 1 + m × 1 + ... + m × 1. Since 1 is the multiplicative identity, we have m × n = m + m + ... + m (n addends). Likewise, if (and only if) m > 1, m can be written as a sum of 1s: m = 1 + 1 + ... + 1 (m addends). So m × n = (1 + 1 + ... + 1) × n = 1 × n + 1 × n + ... + 1 × n = n + n + ... + n (m addends). Thus, if (and only if) n > 1, the product m × n may be calculated by adding n addends m, and if (and only if) m > 1, then the product m × n may be calculated by adding m addends n. ∎


Repeated addition is an algorithm, not an operation. This is one reason among many why multiplication and repeated addition are best thought of as two different things, even for whole numbers. 

Friday, December 25, 2015

Answers to Holiday Challenge

Using all lowercase letters, and using Times New Roman ten-point font with no special formatting...
  • What is the widest seven-letter word you can type?
  • What is the narrowest seven-letter word you can type?
  • Find a four-letter word that is wider than some seven-letter word.

Thanks for sending in so many great answers! As for myself, I sat out the challenge this year—though I did program a computer to try it. Most of the computer's attempts agreed with readers' entries, with some notable exceptions.

I measured candidate words by typing them into Adobe Illustrator and then viewing the type objects at maximum magnification. So without further ado...



Wide



The semitransparent blue bars are positioned in such a way that they bookend mugwump. So mommuck just barely pokes into the blue region, and hammaum falls just barely short of it.

These words were familiar to me with the exception of mommuck, which doesn't appear in any of my dictionaries. In Okracoke Island dialect, it means to harass: see this Baltimore Sun article. The widest standard word, mugwump, means a fence-sitter in politics, or a self-important leader. This word also has an interesting history. Hammaum is an archaic alternate spelling of hammam (Turkish bath).

As a reader observed via email, it would have been nice thematically if maximum or mammoth had turned out to be widest!

A couple of tongue-in-cheek responses that I received were mammony, an invented word that means wealthy (its width lies between that of mommuck and that of mugwump) and memmove, a Unix command (between hammaum and whammed).


Narrow

(Some of these words might be equally wide...I couldn't always distinguish them using the methods I was using.)


Give the computer credit for titlist—I'm a little surprised that no reader submitted this word. The reader favorite instead was illicit. (Maybe the golf ball brand TITLEIST has elbowed titlist out of our collective brains?)

Some of these words are really jargon: my dictionary tells me that illitic is the adjectival form of illite, a kind of mineral clay, and tillite is a stone made of consolidated till. And littlie is a slang term or colloquialism for a small child (chiefly British?).

Interestingly, illitic and illicit are anagrams, yet they differ in width. It seems illicit pays a price for ending with t, because the letterform for t has a little tail sticking out. Evidently, the problem of extremizing word width doesn't reduce to the consideration of individual letter widths. This must have something to do with the way lines of type are assembled. Thinking that it might to be so, I wrote my computer program holistically at the word level: the computer first renders a given word as a vector graphic, then rasterizes the graphic, next discards white pixels, and finally computes the difference between the maximum and minimum column indices among the remaining pixels. This difference is the approximate width of the word.


Four-letter words wider than some seven-letter words

Readers supplied a number of answers to this question. One interesting case is work: it is wider than titlist, but by such a small margin that it might be a candidate for the narrowest four-letter word that is wider than some seven-letter word.

One could ask additional questions along these lines. Is there a three-letter word that is wider than some seven-letter word? (My kids think so...when I put the question to them, they said, 'How about mmm?' As in, 'yummy!' We didn't find mmm in any of our dictionaries, but maybe it's an interjection?) Is there a four-letter word that is wider than some eight-letter word? How about a six-letter word that is wider than some 11-letter word?


Last but not least!


This year's prize, a copy of Word Games 4, goes to reader Melissa. Enjoy the puzzles in this year's volume! And a big thanks to everybody who tried the Holiday Challenge. My best wishes to you for a happy and healthy 2016.







Wednesday, December 16, 2015

Holiday Challenge

Last December, my readers and I played a memorable game of hangman. Here's the 2015 holiday challenge:

Using all lowercase letters, and using Times New Roman ten-point font with no special formatting...
  • What is the widest seven-letter word you can type?
  • What is the narrowest seven-letter word you can type?
  • Find a four-letter word that is wider than some seven-letter word.

Email your answers to zimblogzimblog@gmail.com.

Anybody who tries the challenge this year will be entered into a drawing to win a copy of my latest puzzle book, Word Games 4I'll announce the winner after Christmas!
















N.B., the words I like to work with are the “standard” word game variety: this means we avoid proper nouns, foreign words, slang, contractions, hyphenated words, or acronyms. Also, the width of a word is defined as the horizontal separation between its leftmost and rightmost extremes.

Tuesday, December 15, 2015

In Case You Need a Cheap Stocking-Stuffer This Season

A few weeks ago I made this funny toy for my kids...they are still messing around with it!




To make this toy, cut up a cardboard toilet-paper tube into four rings...snip each ring...then tape each ring back together leaving a bit of a gap remaining (tape from the top and bottom so that both faces are slippery). The rings can now turn on the outside of a second toilet-paper tube.

My goal was for the resulting sentences to be funny and/or nonsensical, but I didn't plan carefully what I would write. If you have a great "content model" for this kind of toy, feel free to put it in the comments!

Tuesday, December 8, 2015

The Eagle Have Landed




This image shows an article that was published the other day in The New Republic. I noticed a grammatical error (circled in red) in the sentence just underneath the article title. Errors like this usually get fixed before too long, but this prominent blurb has been up for a while now.

Since I have a few posts up about grammar, I should clarify that whenever I comment about grammar, I'm referring to written language, not speech. That's because speech is usually less formal than writing. Also, speaking is a very personal and revealing sort of performance. For both of these reasons, and probably others, correcting an adult's grammar during a conversation would be an extremely rude thing to do, almost without exception. So when it comes to grammar, my subject is writing—and in particular, professional writing. Like, say, if you worked at a magazine.

Sometimes words mean more than what they say—other times, they mean less. Sometimes they mean something else entirely. Words are used not only to communicate, but to intimidate. Rather than a thesis, a string of words might convey an emotion, an identity, a desire, or some other unspeakable message.

I use language that way too, but mostly in my writing I try to use words denotatively. I have to, because I generally consider writing to be the art of producing propositions that are true of what they describe. So you can imagine how my mind jams when I feed it a sentence like 'The search have led to a new theory' in which the number of searches is simultaneously one and more than one.

Over twenty years ago, Steven Pinker wrote a good article in The New Republic about 'the fallacies of the language mavens.' There, Pinker explained that some of the mavens' favorite grammar rules are really just historical artifacts of recent vintage. Pinker also pointed out some weaknesses in the arguments of famous language writers, like William Safire. Pinker is especially convincing when it comes to certain specific grammar rules, such as the one about split infinitives. But I couldn't help thinking that Pinker's attack on grammar rules might have been more powerful if in his article he had actually broken some of those rules. Well, maybe he tried. Maybe The New Republic had more copy editors in 1994 than it has today.

Pinker, who is a member of the Usage Panel for my dictionary of choice, recently wrote a book about writing called The Sense of Style: A Thinking Person's Guide to Writing in the 21st Century. I haven't read the book, but I'll bet it's a worthwhile read. Here are reviews of it from the New York Times and the New Yorker. Here too is a list of grammar "rules" that Pinker invites us to do away with. I note that subject-verb number agreement isn't among them.

Cutting a Triangle in Fourths, Cont'd

Consider an equilateral triangle in the xy-plane, centered at the origin.

1) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's area?

2) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's perimeter?

(See my previous post for animations.)

In short, the answer to problem (1) is "no." This can be proved from scratch or deduced from the information here.

As for problem (2), again the answer is no. This is not hard to see, if we begin with the premise that the only perimeter-bisecting lines passing through the center of the triangle are the three obvious ones that connect some vertex to the midpoint of its opposite side. More about this premise in a moment, but for now this fact is enough to solve the problem at hand. For suppose the triangle could be rotated so that each quadrant contains one fourth of its perimeter. In that case, the y-axis would bisect the perimeter (since the quarter in x > 0, y > 0 and the quarter in x > 0, y < 0 would add to half), and likewise the x-axis would also bisect the perimeter. There would exist perpendicular perimeter bisectors passing through the center. But none of the lines connecting vertices to midpoints of opposite sides are perpendicular to one another. In other words, there do not exist perpendicular perimeter bisectors passing through the center, and it follows that the triangle cannot be rotated in such a way that each quadrant contains one fourth of the triangle's perimeter.

Now as to that premise: I wrote up a couple of pages of discussion here about the fact that the only perimeter bisectors of an equilateral triangle are the obvious ones. (For amusement's sake, I approach the problem calculus-style.)


I was pleased to learn that a colleague of mine had pursued these problems in the margin of his in-flight magazine! All the mathematical travelers out there will be thanking the Wellendorff ring company for designing an ad with plenty of white space.

By the way, after a conversation with my colleague, I can describe the thought process reflected in these marginalia: what is being shown is that at any time when one quadrant has a quarter of the area, the quadrant immediately "behind" it will demonstrably not have a quarter of the area. Nice approach!

If you liked these problems, some related problems come to mind:

3) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's area.

4) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's perimeter.

5) If possible, place an equilateral triangle in the xy-plane in such a way that each quadrant has one fourth of the triangle's area and one fourth of the triangle's perimeter.

For further reading on these subjects, try "Lines Simultaneously Bisecting the Perimeter and Area of a Triangle," by Paul Yiu. Professor Yiu is the editor of Forum Geometricorum, the journal that published my Pythagoras-free derivation of the identity sin2 θ + cos2 θ = 1.

Thursday, December 3, 2015

Cutting A Triangle In Fourths

Consider an equilateral triangle in the xy-plane, centered at the origin.

1) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's area?

2) By suitably rotating the triangle about its center, is it possible for each quadrant to contain one fourth of the triangle's perimeter?

The answer to the area problem can be deduced from the information here. Just for fun, I also made an animation. If it doesn't show as embedded here, you can watch it on YouTube.





Endless loop GIF version:





Tuesday, December 1, 2015

Guess My Google

In each case below, guess the single word that I typed into Google image search to yield the results shown.

In some cases, the images shown are top results; in other cases, the images were chosen from further down the page.

You can check your answers by mousing over an image and determining the file name of the image file.




1.



2.



 3.



4.




5.



6.



7.



8.



9.


10.