Every day, crucial business and political decisions are made on the basis of numerical data. Only rarely do the key decision makers produce that data; rather they rely on others, not only to produce it, but to present it to them. Yet how many quants – the data producers – know how to present data effectively? To put it another way, how many of them know how to tell a story using numbers?

This is a surprisingly ancient question. It was Aristotle who first introduced a clear distinction to help make sense of it. He distinguished between two varieties of infinity. One of them he called a potential infinity: this is the type of infinity that characterises an unending Universe or an unending list, for example the natural numbers 1,2,3,4,5,…, which go on forever. These are lists or expanses that have no end or boundary: you can never reach the end of all numbers by listing them, or the end of an unending universe by travelling in a spaceship. Aristotle was quite happy about these potential infinities, he recognised that they existed and they didn’t create any great scandal in his way of thinking about the Universe.

Aristotle distinguished potential infinities from what he called actual infinities. These would be something you could measure, something local, for example the density of a solid, or the brightness of a light, or the temperature of an object, becoming infinite at a particular place or time. You would be able to encounter this infinity locally in the Universe. Aristotle banned actual infinities: he said they couldn’t exist. This was bound up with his other belief, that there couldn’t be a perfect vacuum in nature. If there could, he believed you would be able to push and accelerate an object to infinite speed because it would encounter no resistance. […]

But in the world of mathematics things changed towards the end of the 19th century when the mathematician Georg Cantor developed a more subtle way of defining mathematical infinities. Cantor recognised that there was a smallest type of infinity: the unending list of natural numbers 1,2,3,4,5, … . He called this a countable infinity. […] This idea had some funny consequences. For example, the list of all even numbers is also a countable infinity. Intuitively you might think there are only half as many even numbers as natural numbers because that would be true for a finite list. But when the list becomes unending that is no longer true.

There are several trends that might suggest a diminishing role for mathematics in engineering work. First, there is the rise of software engineering as a separate discipline. It just doesn’t take as much math to write an operating system as it does to design a printed circuit board. Programming is rigidly structured and, at the same time, an evolving art form—neither of which is especially amenable to mathematical analysis.

Some scientists think that subway networks are an emergent phenomenon of large cities; each network is the product of hundreds of rational but uncoordinated decisions that take place over many years. And whereas small cities rarely have subway networks, 25 percent of medium-sized cities (with populations between one million and two million) do have them. And all the world’s megacities—those with populations of 10 million or more—have subway systems.

It’s famously tough getting through the Google interview process. But now we can reveal just how strenuous are the mental acrobatics demanded from prospective employees. Job-seekers can expect to face open-ended riddles, seemingly impossible mathematical challenges and mind-boggling estimation puzzles. (…)

1. You are shrunk to the height of a 2p coin and thrown into a blender. Your mass is reduced so that your density is the same as usual. The blades start moving in 60 seconds. What do you do? (…)

3. Design an evacuation plan for San Francisco. (…)

5. Imagine a country where all the parents want to have a boy. Every family keeps having children until they have a boy; then they stop. What is the proportion of boys to girls in this country? (…)

6. Use a programming language to describe a chicken. (…)

7. What is the most beautiful equation you have ever seen? (…) Most would agree this is a lame answer:
E = MC2
It’s like a politician saying his favorite movie is Titanic.
You want Einstein? A better reply is:
G = 8πT (…)

8. You want to make sure that Bob has your phone number. You can’t ask him directly. Instead you have to write a message to him on a card and hand it to Eve, who will act as a go-between. Eve will give the card to Bob and he will hand his message to Eve, who will hand it to you. You don’t want Eve to learn your phone number. What do you ask Bob? (…)

11. How much would you charge to wash all the windows in Seattle? (…)

Reliable and unbiased random numbers are needed for a range of applications spanning from numerical modeling to cryptographic communications. While there are algorithms that can generate pseudo random numbers, they can never be perfectly random nor indeterministic.

Researchers at the ANU are generating true random numbers from a physical quantum source. We do this by splitting a beam of light into two beams and then measuring the power in each beam. Because light is quantised, the light intensity in each beam fluctuates about the mean. Those fluctuations, due ultimately to the quantum vacuum, can be converted into a source of random numbers. Every number is randomly generated in real time and cannot be predicted beforehand.

{ We show that if a car stops at a stop sign, an observer, e.g., a police officer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satisfied: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign. | Dmitri Krioukov/arXiv | PDF }

Math can be a fun, logic puzzle for some people. But for others, doing math is a headache-inducing experience. Scientists at the Stanford University School of Medicine have recently shown that people who experience math anxiety may have brains that are wired a little differently from those who don’t, and this difference in brain activity may be what’s making people sweat over equations.

Benford’s Law, also known as the rule of first-digits, is a rule that says in data sets borne from real-life (perhaps sales of coffee or payments to a vendor), the number 1 should be the first digit in a series approximately 30% of the time, instead of 11% as would happen had a random number between one and nine been generated.

The rule was first developed by Simon Newcomb, who noticed that in his logarithm books the first pages showed much greater signs of use than those pages at the end. Later the physicist Frank Benford collected some 20,000 observations to test the theory, which he too stumbled upon.

Benford found that the first-digits of a variety of things in nature, like elemental atomic weights, the areas of rivers, and the numbers that appeared on front pages of newspapers, started with a one more often than any other digit.

The reason for that proof is the percentage difference between consecutive single-digit numbers. Say a firm is valued at $1 billion. For the first digit to become a two (or to reach a market cap of $2 billion), the value of the firm will need to increase by 100%. However, once it reaches that $2 billion mark, it only needs to increase by 50% to get to $3 billion. That difference continues to decline as the value increases.