Link: Tricky Rational Exponents

Consider the following apparent paradox:

\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)

Of the seven equalities in this statement, exactly which of them are false? Give a specific number between (1) and (7). Join in the discussion where I posted this at StackExachange, if you like:


Seat Belt Enforcement

Yesterday in the Washington Post, libertarian police-abuse crusader Radley Balko wrote an opinion piece arguing against mandatory seat-belt laws. He opens:
The ACLU of Florida just released a report showing that in 2014, black motorists in the state were pulled over for seat belt violations at about twice the rate of white motorists... Differences in seat belt use don’t explain the disparity. Blacks in Florida are only slightly less likely to wear seat belts. The ACLU points to a 2014 study by the Florida Department of Transportation that found that 85.8 percent of blacks were observed to be wearing seat belts vs. 91.5 percent of whites. The only possible explanation for the disparity that doesn’t involve racial bias might be that it’s easier to spot seat-belt violations in urban areas than in more rural parts of the state... even if it did explain part or all of the disparity, it still means that blacks in Florida are disproportionately targeted.

Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.

Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book Rise of the Warrior Cop for more. And I'm pretty sensitive to issues of overly-punitive enforcement and fines that are repressively punishing to the poor; back in the 80's I used to routinely listen to Jerry Williams on WRKO radio in Boston, when he was crusading against the rise of seat-belt laws, and I found those arguments, at times, compelling.

But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.


Grading on a Continuum

Anecdote: I had a social-sciences teacher in high school who didn't understand that real numbers are a continuum.

On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).

Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.

And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting again to generate something like: D 60-69.55, C 69.56-79.55, B 79.56-89.55, A 89.56-100. And then of course noticed that there were still gaps between the intervals and went at it for a few more cycles.

I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that real numbers are dense? And had he never thought through his grading schema until this very moment?

I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class.

But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.



Whenever one learns a new mathematical operation, it is imperative also to learn the limitations under which the operation may be performed. Lack of this additional knowledge can lead to the employment of the new operation in a blindly formal manner in situations where the operation is not properly applicable, perhaps resulting in absurd and paradoxical conclusions. Instructors of mathematics see mistakes of this sort made by their students almost every day...
- Howard Eves, Great Moments in Mathematics, Lecture 32.


Public Shaming

A collection of Twitter messages from the many people who think that a $550 million Powerball payout will let you give $1 million to every person in the U.S.:


New Year, New Name

With the new year, I've rolled out a new name and domain for this site: MadMath.com. This is something I actually wanted to do at the outset, as it much better captures the "I'm mad about math!" double entendre that I was really going for. So I'm freakishly psyched that I get a chance to use the proper name at this time. Expect to see more frequent blogging at this address, and hopefully some other expansions to the site in the feature. Thanks for reading!