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What’s the intersection of the set of mathematicians and the set of popes?

Hint: a man who started life with one name but later adopted the one he is today remembered as.

Some infinities (and egos) are bigger than others

Here’s a tale of a rational (or irrational?) legal battle from the 1990s re: Cantor’s diagonal argument.

Cantor’s diagonal argument from 1891 was truly revolutionary: an ingenious way to demonstrate that no matter what proposed list of all real numbers (or, say, just those between $0$ and $1$) is put forth, it’s easy to find a number which is definitely missing from the list.1

In a nutshell, Cantor was the first to show that some infinities are bigger than others.

Cantor’s diagonalisation argument for the reals is watertight, and has proved to be a model of elegance and simplicity in the century plus that has passed since it first appeared.

That didn’t stop engineer William Dilworth publishing A correction in set theory, in which he refutes Cantor’s argument, in the Transactions of the Wisconsin Academy of Sciences in 1974.

  1. One has to pay close attention to realise that the same proof doesn’t also establish that the rationals are uncountable, bearing in mind that the Cantor pairing function shows that the rationals most certainly are countable. See []

A student is artistic and loves poetry. Is it more likely she’s studying Chinese or Business?

Let’s suppose that:

  1. $60\%$ of students who study Chinese are artistic and love poetry.
  2. $20\%$ of students who study Business are artistic and love poetry, and
  3. Only about $1\%$ of students study Chinese, whereas about $15\%$ of students study Business.

Thus, out of every $1000$ students, there are $10$ studying Chinese, of whom $6$ are artistic and love poetry, and also there are $150$ studying Business, of whom $30$ are artistic and love poetry.

So if a student is artistic and loves poetry, it’s $5$ times more likely she’s studying Business than Chinese.

So much for preconceptions (and “correlation”).

Knit your mother’s sweater

Here is a clever display of the prime factorisation of the numbers 1-200 on a sweater, from knitter Sondra Eklund.

Each prime is represented (as a square) by its own colour, and luckily there’s an infinite number of both. Composites are represented by squares composed of collections of smaller squares or rectangles of appropriate colours.

She has arranged the natural numbers in columns of width ten. Interesting geometric and visual patterns emerge, and on the other side she’s knitted a version with eight to a column, which makes it easier to work in Octal.

As Sondra says, “One of the cool things about this sweater is that it works in any language and on any planet!!!”

Thanks to Ivars Peterson (on Twitter at‏ @mathtourist) for the pointer.

The Odds Gods smile on birthday/card matches

The classic birthday problem asks how many people are required to ensure a greater than 50% chance of having at least one birthday match, meaning that two or more people share a birthday. The surprisingly small answer, assuming that all birthdays are equally likely and ignoring leap years like 2012, is 23 people.

What’s the (implicit) equation for “equation”?

See for yourself with this inverse graphing calculator.