Maths in a minute |
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words. From symmetry to Euclid's axioms, and from binary numbers to the prosecutor's fallacy, learn some maths without too much effort.
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Ramanujan: Dream of the possible
A hundred years ago Srinivasa Ramanujan was elected Fellow of the Royal Society. Here is a look at the maths that gained him the title.
Inspired by the spirit of Ramanujan
Are you an emerging engineer, mathematician or scientist who doesn't have the support of a school, university, or other research organisation? Then the Spirit of Ramanujan project might be able to help.
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A favourite from the archive
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How the leopard got its spots
Alan Turing is famous as a WWII code breaker and computing pioneer. But he also came up with a beautifully elegant theory of how the patterns we see on animal coats and skin arise.
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Maths in a minute: Equal temperatures
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At every given point in time there are two points on the equator that have the same temperature.
How do we know this? Well, here's a proof. Let's look at the equatorial plane which slices through the Earth at the equator. The equator is a circle which lies in that plane, and we can choose a coordinate system on the plane so that the point (0,0) lies at the centre of the equator. For each point x on the equatorial circle there is a point -x, which lies diametrically opposite x.
Now each point x on the equator comes with a temperature t(x). We can assume that the function t, which allocates a temperature to each point, is continuous. That's because temperature doesn't suddenly jump up or down as you move around on the Earth.
Now consider the function
f(x) = t(x)-t(-x).
It is also continuous.
If this function is equal to 0 for some point x, then we are done because if
f(x) = t(x)-t(-x)=0
then
t(x)= t(-x)
so the temperature at x is the same as the temperature at -x.
If f(x) isn't equal to 0 anywhere, then let's assume (without loss of generality) that there is a point x at which f(x)>0, so
f(x) = t(x)-t(-x)>0.
This implies that
f(-x) = t(-x)-t(x)=-f(x)<0.
There is a result, called the intermediate value theorem, which says that if a continuous function is greater than 0 at some point of its domain and less than 0 at another, then it must equal 0 at some point in between the two. Thus, since f(-x)<0 and f(x)>0, there must be a point y on the circle such that f(y)=0. So
f(y) = t(y)-t(-y)=0
which means that
t(y)=t(-y).
So the temperature at the point y is the same as the temperature at the point -y.
The result actually holds for any circle on the Earth, not just the equator. In fact, the result is the one-dimensional case of the Borsuk-Ulam theorem, which says that for any continuous function t from the circle to the real numbers there is a point x such that t(x)=t(-x).
The more general version of the Borsuk-Ulam theorem says that for any continuous function t from the n sphere to n-tuples of real numbers there is a point x such that t(x)=t(-x).
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