Page 9 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes If we simply abbreviate the rational number (n, 1) by n, there is absolutely no danger of confusion:
2 + 3 = 5 stands for (2, 1) + (3, 1) = (5, 1). The equation 3x = 8 that started this all may then be
interpreted as shorthand for the equation (3, 1) (u, v) = (8,1), and one easily verifies that x = (u, v)
= (8, 3) is a solution. Now, if someone runs at you in the night and hands you a note with 5
written on it, you do not know whether this is simply the integer 5 or whether it is shorthand for
the rational number (5, 1). What we see is that it really doesn’t matter. What we have ”really”
done is expanded the collection of integers to the collection of rational numbers. In other words,
we can think of the set of all rational numbers as including the integers–they are simply the
rationals with second coordinate 1.
One last observation about rational numbers. It is, as everyone must know, traditional to write
n n
the ordered pair (n, m) as . Thus, n stands simply for the rational number , etc.
m 1
Now why have we spent this time on something everyone learned in the second grade? Because
this is almost a paradigm for what we do in constructing or defining the so-called complex
numbers. Watch.

Euclid showed us there is no rational solution to the equation x = 2. We were thus led to
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defining even more new numbers, the so-called real numbers, which, of course, include the
rationals. This is hard, and you likely did not see it done in elementary school, but we shall
assume you know all about it and move along to the equation x = –1. Now we define complex
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numbers. These are simply ordered pairs (x, y) of real numbers, just as the rationals are ordered
pairs of integers. Two complex numbers are equal only when there are actually the same–that is
(x, y) = (u, v) precisely when x = u and y = v. We define the sum and product of two complex
numbers:
(x, y) + (u, v) = (x + u, y + v)

and
(x, y) (u, v) = (xu – yv, xv + yu)
As always, subtraction and division are the inverses of these operations.
Now let’s consider the arithmetic of the complex numbers with second coordinate 0:
(x, 0) + (u, 0) = (x + u, 0),

and
(x, 0) (u, 0) = (xu, 0).
Note that what happens is completely analogous to what happens with rationals with second
coordinate 1. We simply use x as an abbreviation for (x, 0) and there is no danger of confusion:
x + u is short-hand for (x, 0) + (u, 0) = (x + u, 0) and xu is short-hand for (x, 0) (u, 0). We see that our
new complex numbers include a copy of the real numbers, just as the rational numbers include
a copy of the integers.

Next, notice that x(u, v) = (u, v)x = (x, 0) (u, v) = (xu, xv). Now any complex number
z = (x, y) may be written
z = (x, y) = (x, 0) + (0, y)

= x + y(0, 1)
When we let  = (0,1), then we have
z = (x, y) = x + y

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