# Reading the Comics, January 4, 2020: Representations Edition

The start of the year brings me comic strips I can discuss in some detail. There are also some that just mention a mathematical topic, and don’t need more than a mention that the strip exists. I’ll get to those later.

Jonathan Lemon’s Rabbits Against Magic for the 2nd is another comic strip built on a very simple model of animal reproduction. We saw one late last year with a rat or mouse making similar calculations. Any calculation like this builds on some outright untrue premises, particularly in supposing that every rabbit that’s born survives, and that the animals breed as much as could do. It also builds on some reasonable simplifications. Things like an average litter size, or an average gestation period, or time it takes infants to start breeding. These sorts of exponential-growth calculations depend a lot on exactly what assumptions you make. I tried reproducing Lemon’s calculation. I didn’t hit 95 billion offspring. But I got near enough to say that Lemon’s right to footnote this as ‘true’. I wouldn’t call them “baby bunnies”, though; after all, some of these offspring are going to be nearly seven years old by the end of this span.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd justifies why “mathematicians are no longer allowed to [sic] sporting events” with mathematicians being difficult. Each of the signs is mean to convey the message “We’re #1”. The notations are just needlessly inaccessible, in that way nerds will do things.

$0.\bar{9}$ first. The bar over over a decimal like this means to repeat what is underneath the bar without limit. So this is the number represented by 0.99999… and this is another way to write the number 1. This sometimes makes people uncomfortable; the proof is to think what the difference is between 1 and the number represented by 0.999999 … . The difference is smaller than any positive number. It’s certainly not negative. So the difference is zero. So the two numbers have to be the same number.

$0^0$ is the controversial one here. The trouble is that there are two standard rules that clash here. One is the rule that any real number raised to the zeroth power is 1. The other is the rule that zero raised to any positive real number is 0. We don’t ask about zero raised to a negative number. These seem to clash. That we only know zero raised to positive real numbers is 0 seems to break the tie, and justify concluding the number-to-the-zero-power rule should win out. This is probably what Weinersmith, or Weinersmith’s mathematician, was thinking. If you forced me to say what I think $0^0$ should be, and didn’t let me refuse to commit to a value, I’d probably pick “1” too. But.

The expression $x^x$ exists for real-valued numbers x, and that’s fine. We can look at $\lim_{x \rightarrow 0 } x^x$ and that number’s 1. But what if x is a complex-valued number? If that’s the case, then this limit isn’t defined. And mathematicians need to work with complex-valued numbers a lot. It would be daft to say “real-valued $0^0$ is 1, but complex-valued $0^0$ isn’t anything”. So we avoid the obvious daftness and normally defer to saying $0^0$ is undefined.

The last expression is $e^{\frac{\pi}{2}} \imath^{\imath}$. This $\imath$ is that famous base of imaginary numbers, one of those numbers for which $\imath^2 = -1$. Complex-valued numbers can be multiplied and divided and raised to powers just like real-valued numbers can. And, remarkably — it surprised me — the number $\imath^{\imath}$ is equal to $e^{-\frac{\pi}{2}}$. That’s the reciprocal of $e^{\frac{\pi}{2}}$.

There are a couple of ways to show this. A straightforward method uses the famous Euler formula, that $e^{\imath x} = \cos(x) + \imath\sin(x)$. This implies that $e^{\imath \frac{\pi}{2}} = \imath$. So $\imath^{\imath}$ has to equal $(e^{\imath \frac{\pi}{2}})^{\imath}$. That’s equal to $e^{\imath^2 \frac{\pi}{2}})$, or $e^{- \frac{\pi}{2}})$. If you find it weird that an imaginary number raised to an imaginary number gives you a real number — it’s a touch less than 0.208 — then, well, you see how weird even the simple things can be.

Gary Larson’s The Far Side for the 4th references Abraham Lincoln’s famous use of “four score and seven” to represent 87. There have been many ways to give names to numbers. As we’ve gotten comfortable with decimalization, though, most of them have faded away. I think only dozens and half-dozens remain in common use; if it weren’t for Lincoln’s style surely nobody today would remember “score” as a way to represent twenty. It probably avoids ambiguities that would otherwise plague words like “hundred”, but it does limit one’s prose style. The talk about carrying the one and taking away three is flavor. There’s nothing in turning eighty-seven into four-score-and-seven that needs this sort of arithmetic.

I hope later this week to list the comic strips which just mentioned some mathematical topic. That essay, and next week’s review of whatever this week is mathematical, should appear at this link. Thanks for reading.