### The Empty Set is Nothing to Worry About

Today’s article is about nothing: nothing at all, as encapsulated in the number zero and the empty set. It took humanity millennia to move beyond the counting numbers. Zero emerged in several civilizations, first as a place-holder to denote a space or gap between digits, and later as a true number, which could be manipulated like any other. [see TM143, or search for “thatsmaths” at irishtimes.com].

A selection of images of zero (google images).

### Grandi’s Series: Divergent but Summable

Is the Light On or Off?

Suppose a light is switched on for a half-minute, off for a quarter minute, on for one eighth of a minute and so on until precisely one minute has elapsed. Is the light on or off at the end of this (infinite) process? Representing the two states “on” and “off” by ${1}$ and ${0}$, the sequence of states over the first minute is ${\{ 1, 0, 1, 0, 1, 0, \dots \}}$. But how do we ascertain the final state from this sequence? This question is sometimes known as Thomson’s Lamp Puzzle.

### Trigonometric Comfort Blankets on Hilltops

On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country  [see TM142, or search for “thatsmaths” at irishtimes.com].

Trig pillar on summit of Croaghan Moira, Wicklow [Image from https://mountainviews.ie/%5D.

### Numbers with Nines

What proportion of all numbers less than a given size N have a 9 in their decimal expansion? A naive argument would be that, since 9 is one of ten distinct digits, the answer must be about 10%. But this is not “remotely close” to the true answer.

Counting the Nines

It is a simple counting exercise to determine the proportion of numbers less than 10 or 100 or 1000 for which at least one of the digits is a 9.

For N < 10 there is just one such number. For N < 100 there are 10 whose first digit is 9 and 9 others, making 19 in all. For N < 1000 there are 100 whose first digit is 9 and 9 by 19 others, making 271 in all. Here is a table of the number of numbers with at least one 9 digit:

A Better Way

The above way of counting the 9s is a very awkward way of solving the problem. It is better to turn it around and ask what proportion of numbers less than a given limit have no 9s. Now it is simple to use a probabilistic argument.

Any given digit has a one-in-ten chance of being a 9. We consider all numbers less than 10N, which have N digits (counting leading 0s). Assuming the digits are chosen randomly, the chances of any particular digit not being a 9 are 9/10. The digits are independent so, for N digits, the chances of having no 9s are (9/10)N.

For a randomly chosen number less than one million (with six digits) the probability that there are no 9s is (9/10)6 = 0.531441. For numbers less than a billion, the chance decreases to 0.38742, well less than 50%.

Clearly, the probability (9/10)N becomes smaller, approaching zero as N increases. In that sense, “almost every number” has a 9.

### Optical Refinements at the Parthenon

The Parthenon is a masterpiece of symmetry and proportion. This temple to the Goddess Athena was built with pure white marble quarried at Pentelikon, about 20km from Athens. It was erected without mortar or cement, the stones being carved to great accuracy and locked together by iron clamps. The building and sculptures were completed in just 15 years, between 447 and 432 BC. [TM141 or search for “thatsmaths” at irishtimes.com].

### “Dividends and Divisors Ever Diminishing”

Next Saturday is Bloomsday, the anniversary of the date on which the action of Ulysses took place. Mathematical themes occur occasionally throughout Ulysses, most notably in the penultimate episode, Ithaca, where the exchanges between Leopold Bloom and Stephen Dedalus frequently touch on weighty scientific matters. [Last week’s ThatsMaths post]

Joyce in Zurich: did he meet Zermelo?