One of the reasons we love maths is that at its core it doesn't care about who you are, where you are from, the language you speak, the colour of your skin or your gender. Mathematics is a universal language. We feel this as people doing mathematics. But as we attend conferences, read journals and interview the many wonderful mathematicians we meet, we are reminded that as women we are in a minority.
Maryam Mirzkhani, the first female Fields medallist
But currently in the UK only 6% of maths professors are women, despite women making up over 44% of maths undergraduates (see this paper from the London Mathematical Society). The reasons for this are not well understood but the problem seems to be exacerbated by the small number of women itself: fewer women in the community mean they are overlooked when names are sought for speakers or prizes and the relatively few women are disproportionally asked to sit on committees and participate in other non-research activities at the detriment of their research time. And of course this can be compounded by the broken career patterns and other challenges that stem from child-rearing and family responsibilities.
The under-representation of women at the highest levels of maths has been recognised by learned societies, universities and government resulting in many initiatives, such as the Athena Swan Charter in the UK. Thankfully we now rarely hear of experiences of direct sexism from female mathematicians. Instead for many women, particularly young female mathematicians, there remains an uneasiness about being one of a very few women in a maths department. They have spoken of being "the odd one out" or feeling like "the other" in their work places. More role models – high profile women in the mathematical community – are needed both for younger and established female mathematicians.
So we are thrilled that the brilliant mathematician Maryam Mirzakhani has been awarded the Fields Medal at the ICM 2014 in Seoul, Korea – the first women to be so recognised. Mirzakhani's work is on investigating mathematical surfaces and the geometric structures they can have. She has been recognised for her "rare combination of superb technical ability, bold ambition, far-reaching vision, and deep curiosity" which led to "striking and highly original contributions to geometry and dynamical systems".
Now that the IMU has finally recognised a woman for her mathematical achievements with a Fields medal, we hope that more female mathematicians will be nominated and recognised in the future. It seems fitting that Mirzakhani received her prize from a female head of state, President Park Geun-hye of South Korea, in a ceremony led by Ingrid Debauchies, the president of the International Mathematics Union, and mc-ed by Seon-Hee Lin, a professor from Seoul National University. These four women made up a third of the guests on the stage at the opening ceremony – we hope the recognition of women in mathematics stays at at least this level in the future.
The prizewinners and guests on stage at the opening ceremony of ICM 2014
We have set up our temporary headquarters in Seoul, South Korea! We are here to attend the International Congress of Mathematicians, a huge maths conference that happens every four years. We are particularly excited because we'll witness the the award of the Fields Medal, one of the highest honours in maths. The Fields medal is awarded every four years by the International Mathematical Union to up to four mathematicians for "outstanding mathematical achievement for existing work and for the promise of future achievement." (There is an age limit of 40 though, so one half of Plus is sadly already out of the running.)
We will be reporting live on the award of the medals, as well as a range of other prizes that will be handed out, and no doubt meet many fascinating mathematicians to interview. Stay tuned!
Most people are aware of the fact that computers work using strings of
0s and 1s. But how do you write numbers using only those two symbols?
To see how, let's first remind ourselves of how the ordinary decimal way
of writing numbers works. Let's take the number 4302 as an example. The
digit 4 in this number doesn't stand for the number 4, rather it stands
for 4000, or 4 x 1000. Similarly, 3 doesn't stand for 3 but for 300 = 3
x 100, 0 stands for 0 x 10, and 2 stands for 2 x 1. So 4302 means
4 x 1000 + 3 x 1000 + 0 x 10 + 2 x 1.
Similarly, 7396 stands for
7 x 1000 + 3 x 100 + 9 x 10 + 6 x 1.
What do the numbers 1000, 100, 10 and 1, which appear in these
expressions, have in common? They are all powers of 10:
1000 = 103
100 = 102
10 = 101
1 = 100.
To write a number in decimal notation, you first write it as a sum of
consecutive powers of 10 (with the largest power on the left) and then
pull out the coefficients of these powers. We can do the same with
powers of 2 rather than 10. For example, the binary number 110 stands
1 x 22 + 1 x 21 + 0 x 20 = 4 + 2 +0 = 6
(written in decimal).
And the binary number 10001 stands for
1 x 24 + 0 x 23 + 0 x 22 + 0 x 21
+ 1 x 20
= 16 + 0 + 0 + 0 + 1 = 17 (written in decimal).
You can convince yourself that a binary number only consists of the
digits 0 or 1: when you write a number as a sum of consecutive powers of
2, no other coefficients are necessary.
This sorts out the natural numbers, but what about numbers that have a
fractional part? To write a number between 0 and 1 in binary, you play
the same game using powers of 1/2 instead of powers of 2. For example,
0.75 = 1/2 + 1/4 = 1 x 1/21+ 1 x 1/22.
In binary, 0.75 is written as 0.11. The binary number 0.1001 stands for
the decimal number
1 x 1/21+ 0 x 1/22 + 0 x 1/23+ 1 x 1/24
= 1/2 + 1/16 =0.5625.
You can find out more about the positional way of writing numbers here and about the use of 0s and 1s in logical operations here.
A perfect number is a natural number whose divisors add up to the number itself. The number 6 is a perfect example: the divisors of 6 are 1, 2 and 3 (we exclude 6 itself, that is, we only consider proper divisors) and
1+2+3 = 6.
If a non-perfect number were an animal, it might look something like this.
Hooray! People have known about perfect numbers for millennia and have always been fascinated by them. Saint Augustine (354–430) thought that the perfection of the number 6 is the reason why god chose to create the world in 6 days, taking a rest on the 7th. The Greek Nicomachus of Gerasa (60-120) thought that perfect numbers produce virtue, just measure, propriety and beauty. Numbers that are not perfect, for example numbers whose proper divisors add up to more than the number itself, Nichomachus found very disturbing. He accused them of producing excess, superfluity, exaggerations and abuse, and of being like animals with "ten mouths, or nine lips, and provided with three lines of teeth; or with a hundred arms, or having too many fingers on one of its hands."
If you play around with numbers for a while you will see why people have always been so fond of perfect numbers: they are very rare. The next one after 6 is 28, then it's 496, and for the fourth perfect number we have to go all the way up to 8128. Throughout antiquity, and until well into the middle ages, those four were the only perfect numbers that were known. Today we still only know of 48 of them, even though there are fast computers to help us find them. The largest so far, discovered in January 2013, has over 34 million digits.
Will we ever find another one? We can't be sure — mathematicians believe that there are infinitely many perfect numbers, so the supply will never run out, but nobody has been able to prove this. It's one of the great mysteries of mathematics. You can find out more in Number mysteries.