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Magic
numbers |
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Numerology
by the experts |
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If a mathematician had a favourite number what would that number be? |
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We
asked three mathematicians to tell us about their favourite
number and why they consider it special. |
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Tom
Korner's favourite number was zero |
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 The
centre of our universe
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Is
zero a number? |
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Surely
a number is how many you've got, if you haven't got any then
it's not a number. |
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Tom
Korner claims that mathematicians can build a whole universe
from nothing. They start with nothing, then consider a bag containing
nothing which is an object. They then consider a bag containing
the bag with nothing in it and nothing, and that is a different
object. In this way they can build up an infinity of objects
which they can use to do mathematics on. This is why for mathematician's
zero is the foundation of everything. |
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"I
don't see how any mathematician could reply anything but zero."
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The
value of nothing |
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What
was life like without zero? |
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Before
zero people used abacuses and Roman Numerals to display numbers.
Without the discovery of zero, life today could have been very
complicated. |
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The
origination of zero goes back to some completely unknown Indian
in the 8th century who realised the empty column on the abacus
could be represented by some sort of symbol, a zero. This gave
you a different representation for numbers, what we call a place-holding
method. For example One hundred is one, zero, zero. |
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Without
the zero we would be in a world of roman numerals where 100
men can go off to fight or a man can own 50 ships but where
calculations are very complex and would require specialists
to do them. With the introduction of zero it is possible for
every man to do his own calculations. |
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John
Casti's favourite number was Omega |
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The
invisible number |
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How
can a number exist when we can't see it? |
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When
the number is Omega, a number that cannot be reproduced by computation. |
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According
to Dr
John Casti,
to describe the number Omega, one has to think about the process
of carrying out a computation. |
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All
computing machines today are descendants of a theoretical computing
machine that Alan Turing developed known as a Turing machine.
One of the central questions of computation is if you have this
theoretical Turing machine what numbers can you actually compute? |
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In particular are there numbers that can never be computed ?
Numbers for which we can not find a program or formula which
would spit the digits out one after another. |
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What
Turing found out was almost every number is uncomputable. In
the whole history of the human race every number that has ever
been calculated or written down amounts to only a small subset
of the sum of all possible numbers. |
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Almost
all numbers have an infinite number of digits and they do not
repeat in any periodic way. They are irrational numbers. |
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There
are algorithms which if you let them run will push out every
single digit of PI. But there are other numbers, for which there
are no such programs. Because there are no programs or algorithms
which will reproduce them we can not see those numbers but we
can prove mathematically that they exist. |
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We
can produce the first few digits of Omega but we will never
be able to see every digit of the number because each digit
of the number becomes harder and harder to calculate and you
will never get to the end of this number. |
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What
are the properties of Omega?
i)It is uncomputable.
ii)It can never be compressed into a formula. It's digits are
related to the solution of a particular type of algebraic equation
which in turn is
equivalent to Turing's famous computing machine. |
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"If
it were possible to actually calculate this number omega, calculate
its digits, you could answer any possible question in mathematics."
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