- Hello, my name's Marcus du Sautoy
and I'm a Professor of Mathematics
at the University of Oxford and also the Simonyi Professor
for the Public Understanding of Science.
When I was at school
our mathematics teacher told us a story.
It was a story of the young Carl Friedrich Gauss.
He was nine or 10 years old at the time.
And his maths teacher decided to challenge the class
with a mathematical problem.
He said, I want you to add up the numbers
from one to a hundred.
Now I think that maths teacher probably thought
he was going to get a a nice bit of rest
whilst the students went through the numbers
one at a time adding them together,
but before he'd even finished asking the question
the young Gauss had written that down,
a number on his slate board
and slammed it down in front of the teacher.
The teacher thought the student was being impudent
but when he looked at the number,
there was the correct answer.
He asked, "Gauss how did you get that so quickly?"
Gauss replied, "Well, there's a shortcut,
you see all my fellow students are starting
at the beginning adding one plus two plus three plus four
and it's gonna take them ages
to get all the way up to a hundred.
And I bet they're gonna make loads of mistakes
but I thought there's a clever shortcut here
because I can combine the beginning
and the end of the story.
So one plus a hundred is hundred and one
two plus 99 is also 101
three plus ninety eight, a hundred and one.
So I realized that there were 50 pairs
of numbers adding up to 101.
So the answer was 50 times 101, 5,050."
The beautiful thing about this shortcut
is even if the teacher said,
"Okay, well I'm gonna give you
an even more difficult problem.
I want you to add up the numbers
from one to a million,"
Gauss can still use the same trick.
You can step back, combine the beginning
and end of the journey.
And again, get these pairs all adding up to the same thing
and get the answer very quickly.
My mathematics teacher, I think I was about 12 or 13
at the time said, "That's what mathematics is all about.
Mathematics, it's the art of the shortcut."
And my lazy teenage self began to think.
I like the look of this subject.
And so over the years at school
he began to teach us all these different, amazing shortcuts
that mathematicians have come up with
over the 2000 years that we've been doing mathematics.
Some of the ones I really like the pattern shortcuts.
And in fact I sometimes call mathematics the science
of patterns because if you can spot a pattern
in the data that you're looking at you can read that data
into the future and make predictions.
So you don't have to wait for the future
to become the present because those patterns give you
a short cut to look and just see what's gonna happen next.
If you think about even the way we write numbers
as a kind of shortcut, the ancient Egyptians,
they had hieroglyphics for numbers,
but they had to invent a new picture each time
they got a new number multiplied by 10.
They didn't understand the value of the place value system.
It was the Babylonians across the road who realized
that if you use the position of a number,
it was a shortcut to making numbers very efficiently.
And the Arabs, they came up with amazing shortcut, algebra,
a kind of language to be able to find patterns in numbers.
So rather than writing down lots of examples
one equation would succinctly tell you exactly
what was going on inside these numbers.
And geometry too, if you think about it,
sometimes what looks like the shortest distance
between two points isn't actually the shortest.
If you look on a map and you try and predict
where an airplane will fly from London to New York,
it isn't the straight line you'd naturally draw
but a curve line that goes over Greenland.
So sometimes the shortest path
is not the most obvious one in geometries,
and light actually uses this
because light is very lazy like me as a teenager
and likes to find the shortest way to its destination.
And so we've discovered in our universe
that light will sometimes bend round an object
because it can go faster that way.
I think one of the greatest shortcuts
that mathematicians ever came up with is the calculus
Newton-Leibniz idea of a technique
which means you can look at a complex situation
and find for example the fastest way to a solution
or the way to maximize your profits or minimize energy.
Calculus in a way is the shortcut that we've discovered
for the things that nature quite often does naturally.
And a diagram I found that actually sometimes
you don't want to do is in language
but sometimes drawing a picture is a great way
to find a shortcut to understanding a problem.
If you think about the London underground map
that people use every day here in London,
it's actually a diagram of the way
that the underground stations are connected.
It isn't a geometric map
but it's a much more useful diagram.
It throws away information about the distances
which are not important,
just shows how things are connected together.
In our modern age we're kind of on sort of data
that we have to kind of negotiate.
but statistics has given us a shortcut
to actually understand.
We don't have to look at all the data,
just a small sample of it can give you great insight
into what's actually going on in the numbers.
And what about chance and probability and risk,
trying to predict what's gonna happen next?
If you're in a casino for example,
well we've come up with this shortcut of probability.
It doesn't tell you how a coin is gonna land
on the next throw but over the long term
it'll tell you kind of what's going to happen.
And you can use that shortcut to make sure
that you will lose as little money as possible in the casino
and our modern age we are actually having to negotiate
lots of different sorts of networks.
And we discovered that some networks
have incredible shortcuts get from one place to the other.
Certain networks called small world networks
mean that if you take two points in the network
there might be just six lines
which you'll get you from one point to the other.
And people might have heard of this idea
of the six degrees of separation,
choose two people in the world
you'll find that there are six friendships
that will get you from one person to the other.
But there are some problems in mathematics
where there don't seem to be any shortcuts.
And the incredible thing about mathematics,
it can sometimes look at itself
and say when there isn't a solution to a problem.
So we have a problem called the traveling salesman problem
where a traveling salesman has to look at a network
of cities and roads between them
and look at the distances and try and find the shortest path
to visit all of the cities.
Turns out it doesn't seem to be a smarter way
than trying all the different possibilities
and finding the one which is the shortest.
We'd love a kind of clever shortcut
and algorithm to find the shortest part
but we believe that this is a sort of problem
that there isn't a shortcut,
but trying to prove there isn't a shortcut
is actually a really difficult problem.
Anyway, mathematicians have worked for 2000 years
trying to come up with all of these extraordinary shortcuts.
But the amazing thing today is that we can use
all of these amazing tunnels that they've built
in order to find our way to our goal
as efficiently and fast as possible.
If you'd like to find out more about the art of the shortcut
then I've written a new book called
"Thinking Better The Art of the Shortcut,"
which documents these amazing shortcuts
that mathematicians have come up with
over the last 2000 years.
And you can also check out the lecture I gave
at the Royal Institution.