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THE MOST ELEGANT EQUATION |
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THE MOST ELEGANT EQUATION
After a bit of Newtonianism, let us look at the beauty of
complex numbers for a while. Complex numbers are
the most mystical numbers in science. No one knows
where they exist, even though they are used more
frequently than simple numbers. They are very useful
in quantum mechanics. Complex number is the sum
of a real and an imaginary number. Why a sum of
two kinds of numbers? What do we understand by
saying this? Basically real number and imaginary
number belong to two different number worlds. They
are not related to each other in any way as the
basis vectors along x and y coordinates are not
related to each other. The representation of real
numbers is given by the symbol ‘1(unity)’ which
we do not write generally. The representation of
complex numbers is given by the symbol ‘i’. So,
change your view towards number system from now.
They are deeper than what they seem to be. For
complex numbers, the laws of mathematical operations
are same as those for normal numbers (real numbers).
‘The square root of minus 1 is i (read iota). A complex
number is, in general, written as a + ib. You might
be familiar with some basic operations of complex
numbers. Here, we will see the formulations of
Abraham De Moivre and Leonhard Euler.
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The relation eiθ =
cosθ + i sinθ is called de Moivre’s relation.
This can be easily proved. Expand eiθ using
standard form ula for exponential function (ex =
1+ x + x2/2! + x3/3! …). Separate
the terms containing iota from those not containing iota.
Identify the series with cosine and sine series.
This relation is very useful in representing complex
numbers in a plane called the complex plane.
It is just like the familiar x-y plane, but you have
complex numbers on y-axis. To represent the complex
number a+ib, you take a point on plane whose x-coordinate
is a and y-coordinate is b. If I use polar coordinates,
then the value of ‘a’ is rcosθ and that
of ‘b’ is rsinθ. So, the value of this complex
number is (rcosθ + irsinθ). Take r common
and from there, applying de-Moivre’s theorem, you
find the value to be reiθ, where θ is
the angle used in polar coordinates. Leonhard Euler
did a great work. He simply put θ = p and
obtained:
eip = -1
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De Moivre |
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What’s so great about this relation? Well firstly, it is less
than 2 cm long. There is a much better reason for
its elegance. It relates two very far away objects
worlds - the worlds of real and imaginary. If you
take the logarithm of this equation, you get the
only relation in the cosmos, which gives the value
of square root of minus one in terms of real numbers
only. One more point, consider the complex plane.
Y-axis represents the imaginary part. If you know
about vectors, you will see that the basis vector
on y-axis is ‘i’. So iota represents the
basis vector for y-axis. And ‘1(unity)’ represents
basis vector on x-axis. How can these two orthogonal
vector-like things be related? Through this great
relation of this greatest mathematician (or one
of the greatest), it is possible to relate two
orthogonal dimensions. |
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Leonhard Euler |
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Complex
numbers are useful in mechanics. The de Moivre’s theorem
helps us to solve many equations related to waves.
We will see how we can solve a wave equation for a
quantum particle in the later sections. The use of
complex numbers was also incorporated in relativity
when it was thought that the 4-dimensional space-time
was a flat space-time, time axis being imaginary one.
But, later, this idea was discarded and another form
of flat space-time was taken.
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So,
here, we saw a new form of numbers which is not
like other number forms. It does not represent
any common physical quantity, but is more important
than any real number. How far can we go? Let us
see in the next section ‘Hypercomplexity’.
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