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Complex Numbers
Michael Fowler
2/15/07
Real Numbers
Let us think of the ordinary numbers as set out on a line which goes to infinity in both positive
and negative directions.
We could start by taking a stretch of the line near the origin (that is, the
point representing the number zero) and putting in the integers as follows:
Next, we could add in rational numbers, such as ½ ,
23/11, etc., then the irrationals like
2 ,
then numbers like
π
, and so on, so any number you can think of has its place on this line.
Now let’s take a slightly different point of view, and think of the numbers as represented by a
vector
from the origin to that number, so 1 is
and, for example,
–2 is represented by:
Note that if a number is multiplied by –1, the corresponding vector is turned through 180
degrees.
In pictures,
The “vector” 2 is turned through
, or 180 degrees, when you multiply it by –1.
What are the square roots of 4?
Well, 2, obviously, but also –2, because multiplying the backwards pointing vector –2 by –2 not
only doubles its length, but also turns it through 180 degrees, so it is now pointing in the positive
direction.
We seem to have invented a hard way of stating that multiplying two negatives gives a
positive, but thinking in terms of turning vectors through 180 degrees will pay off soon.
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Solving Quadratic Equations
In solving the standard quadratic equation
ax
2
+
bx
+
c
= 0
we find the solution to be:
2
4
2
bb
a
c
x
a
−±
−
=
.
The problem with this is that sometimes the expression inside the square root is negative.
What
does that signify?
For some problems in physics, it means there is no solution.
For example, if I
throw a ball directly upwards at 10 meters per sec, and ask when will it reach a height of 20
meters, taking
g
= 10 m per sec
2
, the solution of the quadratic equation for the time
t
has a
negative number inside the square root, and that means that the ball doesn’t get to 20 meters, so
the question didn’t really make sense.
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 Fall '07
 MichaelFowler
 Complex Numbers, Cos

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