Math, Unconventionally

Okay, boys and girls, here are some warm-up problems for today’s lesson…

1. What is the product of all single-digit prime factors of 143?
2. What is the value of 6 / 2 (1 + 2)?
3. When is a DECADE not equal to 10 years?

The correct solution to each of these problems relies on a mathematical convention.

A convention is an action that is deemed acceptable by other members of a group. There are plenty of conventions — most social, but many mathematical — by which you probably already abide.

• If, while passing through a door in a public building, you notice a person less than k feet behind you who will be passing through the same door momentarily, you should continue to hold the door open at least until that person reaches the door or, to be most proper, until she passes through it. (While we’re on the subject, I’d be happy to hear your opinions for the value of k. I was sure that k < 25, but I recently got a dirty look from a woman for whom I did not hold the door, and she was at least 40 feet away.)
• You should use X’s in place of a stick figure’s eyes to indicate unconsciousness or death. (See image below.) • You should drive on the right side of the road — unless you’re in Britain, Australia, Suriname or Guyana, where the right side is the wrong side. Incidentally, about 2/3 of the countries in the world follow the right-hand rule, accounting for nearly 3/4 of all traffic.
• You should assume that numbers are represented in base 10 (unless otherwise specified).
• You should use the plus sign (+) to indicate addition.

The order of operations is another accepted mathematical convention. Bon Crowder of Math Four says that it’s like driving on one side of the road or the other. “Doesn’t really matter which one, just as long as everyone else involved agrees to play by the same rules.”

But the following problem, which has been traversing the cyber-circuit recently, indicates that perhaps not all of us follow the same convention:

What is the value of 6 / 2 (1 + 2)?

There is an implied multiplication symbol just before the parentheses. In the order of operations, division and multiplication have equal precedence, so the value of the expression can be calculated as follows:

6 / 2 × (1 + 2)
6 / 2 × 3
3 × 3
9

But several references suggest that implied multiplication takes precedence over explicit multiplication. Perhaps you agree with this? If so, you’re not alone. Many have argued that the value should be found this way:

6 / 2 (1 + 2)
6 / 2 (3)
6 / 6
1

If you’d like to get involved in this argument, then feel free to join the discussion at Spiked Math.

Similarly, there is not universal consensus for the definitions of many math terms. One example is whole number. The James and James Mathematical Dictionary included three definitions for whole number:

• The non-negative integers 0, 1, 2, 3, …
• The positive integers 1, 2, 3, 4, …
• All positive and negative integers …, -3, -2, -1, 0, 1, 2, 3, …

The first definition is the one I learned in school, but apparently it’s not used in every school.

A similar thing occurs for the definition of proper divisor. A proper subset of a set is any subset that is not the original set itself. It would seem appropriate, then, that a proper factor would be any positive integer factor other than the number itself. But this definition is only used in some cases. For instance, when discussing perfect numbers, this definition is convenient: a number is perfect if the sum of its proper factors is equal to the number itself (e.g., the proper factors of 6 are 1, 2, and 3, and since 6 = 1 + 2 + 3, then 6 is a perfect number.)

Sometimes, however, the definition indicates that a proper factor of n should exclude both 1 and n. This definition is convenient when describing prime numbers; a positive integer is said to be prime if it has no proper divisors.

Which one is correct? Depends who you ask.

Finally, if you enjoy conventions, you might enjoy attending the annual convention of the Barbershop Harmony Society, which will occur July 1‑8, 2012, in Portland, Oregon. Mathy folks are sure to love it, as it is bound to be a harmonic function.

(No, in fact, this entire post wasn’t written as an elaborate set-up for that one joke. It just worked out that way.)

As to the problems at the start of this diatribe, here are the solutions. Sort of.

1. Since 143 = 11 × 13, it has no single-digit prime factors. But mathematical convention dictates that the empty product is 1, so the product of all single-digit prime factors of 143 is 1. Crazy, huh? (Incidentally, another mathematical convention is at play here; namely, that 1 is not considered a prime number. If 1 were a prime number, then the Fundamental Theorem of Arithmetic would fail, because integers would not have unique prime factorizations.)
2. See above. I agree with Spiked Math that the answer is 9, but some people still argue that the answer is 1.
3. When it’s written in hexadecimal: DECADE16 = 14600926. (And there’s another convention, although by no means is this one universal. Uppercase letters in normal font are used for the “numbers” in hexadecimal, whereas uppercase letters in italics are typically used to indicate sets or points in geometry. On the other hand, lowercase letters in italics are used to represent algebraic variables, such as x + y = 7, and they often are used to indicate a geometric length — for instance, many textbooks say that the side opposite angle C in a triangle has length c.)

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• 1. xander  |  January 18, 2012 at 11:19 am

While I agree that the “correct” answer to number (2) should be 9, I also think that it is an ambiguous question, and it is certainly not one that I would give to my algebra students (though I have been using it as an example for a while).

As you point out, implied multiplication may be seen to have higher precedence than division. This is not the general convention, but given the lack of a multiplication symbol and the fact that group symbols are generally given precedence (and that the 2 might be thought of as part of the group), it is easy to see where the confusion can arise.

There are a couple of things that can be done to reduce the ambiguity of the problem. I propose that each of the following is a better alternative (where the new expression may not be identical to the old, but where it is obvious how the new expression should be simplified):

$(6/2)(1+2)$
$\frac{6}{2}(1+2)$
$6\div 2 \times (1+2)$
$6/(2(1+2))$
$\frac{6}{2(1+2)}$

xander

• 2. venneblock  |  January 19, 2012 at 8:07 am

Fair point, Xander. I liken this to writing. There are often several grammatically correct ways to write a sentence, all of which sound pretty much the same. But typically, one of those forms makes it easier for the reader to understand and often removes ambiguity. I like your alternatives — though I wonder how many readers of this blog speak TeX.

Anyone know the quote by Donald Knuth about how TeX looks like Greek unless you speak Greek? It was in the intro to one of his books that I no longer own, and I couldn’t find the quote on the web.

• 3. Chris Smith  |  January 18, 2012 at 2:49 pm

Quite enjoyed that wee discussion (or rant). Keep up the good work!

• 4. venneblock  |  January 19, 2012 at 8:08 am

Rants are my specialty!