2018 Abel Laureate Robert Langlands

The Norwegian Academy of Science and Letters kindly provided me with a scholarship to attend the Abel Prize week in Oslo earlier this year where I interviewed the 2018 Abel Laureate Robert Langlands.

In the first of a series of videos documenting my experience, Robert describes how he came to do Mathematics at university…

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How can you show geometrically that 3 < π < 4?

Approximating Pi was a favourite pastime of many ancient mathematicians, none more so than Archimedes. Using his polygon approximation method we can get whole number bounds of 3 and 4 for the universal constant, with only high-school level geometry.

This is the latest question in the I Love Mathematics series where I answer the questions sent in and voted for by YOU. To vote for the next question that you want answered next remember to ‘like’ my Facebook page here.

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Equations Stripped: Normal Distribution

Stripping back the most important equations in maths so that everyone can understand…

The Normal Distribution is one of the most important in the world of probability, modelling everything from height and weight to salaries and number of offspring. It is used by advertisers to better target their products and by pharmaceutical companies to test the success of new drugs. It seems to fit almost any set of data, which is what makes it SO incredibly important…

You can watch all of the Equations Stripped series here.

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JFM China Symposia: Beijing

Video highlights from the third and final stop of the JFM China Symposia in Beijing. We were hosted by Tsinghua University with further speakers from Peking University, Xidian University, Beihang University and the Chinese Academy of Sciences.

Ke-Qing Xia describes how water in the ocean travels the entire globe over the course of 1000 years

 

Colm Caulfield explains how to the shape of a hanging chain is related to turbulence

 

Charles Meneveau discusses wind energy and its future as the current cheapest form of energy in the US

 

Photo: Christian Steiness

 

JFM China Symposia: Hangzhou

I’m in China this week documenting the JFM Symposia ‘from fundamentals to applied fluid mechanics’ in the three cities of Shenzhen, Hangzhou and Beijing. Check out the CUP website for daily blog entries as well as some of my favourite video highlights from the scientific talks in Hangzhou below.

Detlef Lohse describes how a good scientist must be patient like a good bird-watcher as demonstrated by his experiments with exploding ice droplets

Hang Ding discusses falling droplets and shows a video of one hitting a mosquito

Quan Zhou presents some amazing visuals of Rayleigh-Taylor turbulence 

Tom Rocks Maths S02 E01

Tom Rocks Maths is back on Oxide – Oxford University’s student radio station – for a second season. The old favourites return with the weekly puzzle, Funbers and Equations Stripped. Plus, the new Millennium Problems segment where I tell you everything that you need to know about the seven greatest unsolved problems in the world of maths, each worth a cool $1 million. And not to forget the usual selection of awesome music from artists such as Rise Against, Panic at the Disco, Thirty Seconds to Mars – and for one week only – Taylor Swift. This is maths, but not as you know it…

JFM China Symposia: Shenzhen

I’m in China this week documenting the JFM Symposia ‘from fundamentals to applied fluid mechanics’ in the three cities of Shenzhen, Hangzhou and Beijing. I’ll be writing daily blog entries on the CUP website as well as posting some of my favourite video highlights from the scientific talks, starting with the first symposium in Shenzhen.

Detlef Lohse explains the evaporation of a drop of Ouzo (a traditional Greek alcohol)

Colm Caulfield describes the two types of mixing present in the ocean (including a fantastic visualisation of KH instability)

Anderson Shum demonstrates how a fluid can behave as a ‘dancing ribbon’

Can you pee on the moon?

Question

If, in some miraculous way, one were able to pee standing on the surface of the Moon, what kind of arc would it create?

Answer

Dr Chris Messenger from the University of Glasgow was on hand to help me with Michael’s question…

  • The moon’s gravity is 16% of that on Earth, which means the pee will travel in a straighter arc and about 2.5 times further
  • In a uniform gravitational field objects travel in a parabolic arc – sort of a ‘u-shape’
  • On the moon, the atmosphere is so thin that the pee would follow a very accurate parabola, as can be seen with the dust thrown up by the lunar rover
  • The low atmospheric pressure on the moon would immediately boil the pee which would then fall to the surface as steam
  • Despite the low temperature of the moon (as low as -170 degrees Celsius), the pressure reduces the boiling point of water so dramatically that your pee would boil way below body temperature of 37 degrees Celsius, which is why it immediately turns to steam
  • The freezing temperature of water on the moon also occurs in the same range as the boiling point, which means that the steam molecules will then freeze into yellow ice crystals

You can listen to the full version of Question of the Week with the Naked Scientists here.

Take me to your chalkboard

Is alien maths different from ours? And if it is, will they be able to understand the messages that we are sending into space? My summer intern Joe Double speaks to philosopher Professor Adrian Moore from BBC Radio 4’s ‘a history of the infinite’ to find out…

Can ants feel pain?

Question

Carol asks: Can ants feel pain?

Answer

I went crawling around for the answer with York University’s Eleanor Drinkwater…

  • Ants can sense that they’ve been harmed and react but this is different to actually feeling pain
  • Nociception is the sensory nervous system informing the brain that you’ve been hurt, whereas pain is an unpleasant sensation with a negative emotional response
  • One can occur without the other eg. when playing sports you often don’t realise that you are injured until afterwards, or people who have lost limbs experience phantom limb pain
  • Robots can also be programmed to experience nociception without experiencing pain, for example in the video game The Sims characters will jump around if they’re burnt by fire
  • We currently know very little about insect expressions of pain, but we do know that the pain expression systems are different to those in mammals, meaning that insects are likely to experience pain in a different way to humans
  • In short, the jury is still out, so best to be nice to any ants that you may come across!

Part of the Naked Scientists Question of the Week series – you can listen to the full version here.

Complex Numbers – they don’t have to be complex!

The idea of complex numbers stems from a question that bugged mathematicians for thousands of years: what is the square root of -1? That is, which number do you multiply by itself to get -1?

Such a simple question has blossomed into a vast mathematical theory, for the simple reason that the answer isn’t real! It can’t be 1, as 1 * 1 = 1; it can’t be -1, as -1 * -1 = 1; whichever number you multiply by itself, you can’t get a negative number. Up until the 16th century, almost everyone ignored this issue; perhaps they were afraid of the implications it could bring. But then, gradually, people began to realise that there was a whole new world of mathematics waiting to be discovered if they faced up to the question.

In order to explain this apparent gap in maths, the idea of an ‘imaginary’ number was introduced. The prolific Swiss mathematician Leonhard Euler first used the letter i to represent the square root of -1, and as with most of his ideas, it stuck. Now i isn’t something that you’ll see in everyday life in relation to physical quantities, such as money. If you’re lucky enough to have money in your bank account, then you’ll see a positive number on your bank statement. If, as is the case for most students, you currently owe money to the bank (for example, if you have an overdraft), then your statement will display a negative number. However, because i is an ‘imaginary’ unit, it is neither ‘positive’ nor ‘negative’ in this sense, and so it won’t crop up in these situations.

Helpfully, you can add, subtract, multiply and divide using i in the same way as with any other numbers. By doing so, we expand the idea of imaginary numbers to the idea of complex numbers.

Take two real numbers a and b – these are the type that we’re used to dealing with.

They could be positive, negative, whole numbers, fractions, whatever.

A complex number is then formed by taking the number a + b * i. Let’s call this number z.

We say that a is the real part of z, and b is the imaginary part of z.

Any number that you can make in this way is a complex number.

For example, let a = -3 and b = 2; then -3 + 2*i, which we write as -3 + 2i, is a complex number.

As we saw before, complex numbers don’t actually pop up in ‘real-life’ situations. So why do we care about them? The reason is that complex numbers have some very neat properties that allow them to be used in all sorts of mathematical contexts. So even though you may not see the number i in everyday life, it’s very likely that there are complex numbers involved behind the scenes wherever you look. Let’s have a quick glance at some of these properties.

The key observation is that the square of i is -1, that is, i * i = -1.

We can use this fact to multiply complex numbers together.

Let’s look at a concrete example: multiply 2 + 2i by 4 – 3i.

We use the grid method for multiplying out brackets:

  4 -3i
2 2 * 4 = 8 2 * -3i = -6i
+2i 4 * 2i = 8i 2i * -3i = -6 * i * i = -6 * -1 = 6

Adding the results together, we get (2 + 2i)(4 – 3i) = 8 + 6 – 6i + 8i = 14 + 2i.

Therefore, multiplying two complex numbers has given us another complex number!

This is true in general, and it turns out to be very handy. In fact, Carl Friedrich Gauss proved a very famous result – known as the Fundamental Theorem of Algebra because it’s so important – that effectively tells us that the solutions to all equations can be written as complex numbers. This is extremely useful because we know that we don’t have to go any ‘deeper’ into numbers; once you’ve got your head around complex numbers, you can proudly declare that you’ve mastered them all!

Because of this fundamental theorem, our little friend i pops up all over the place in physics, engineering, computer science, and of course, in all sorts of areas of maths. While it may only be imaginary, its applications can be very real, from air traffic control, to animating characters in films. It plays a really important role in much of theoretical mathematics, which in turn is used in almost every scientific discipline. And to think, all of this stemmed from an innocent-looking question about -1; what were they so scared of?!

Kai Laddiman

Funbers 21

Funbers has reached the age of adulthood as the series turns 21 today! Twenty-one is a popular number in gambling, sports and politics, as well as being the number of shots fired in a ceremonial gun salute. To find out why you’ll have to listen to the latest episode below…

You can listen to all of the Funbers episodes from BBC Radio Cambridgeshire and BBC Radio Oxford here.

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