Any science fiction fan, or movie buff for that matter, will be familiar with the concept of hyperspace wormholes. Watch any Star Trek movie and you’ll see the Enterprise zip through two points in space faster than the time takes to conventionally travel in a straight line. The possibility of wormholes elevated to another level when Einstein published his general theory of relativity. In this section, we’ll see that mathematicians had this idea in mind even before the physics community.
MATHEMATICAL WORMHOLE.2
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MATHEMATICAL WORMHOLE.1
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ADDITION AND MULTIPLICATION
THE COMPLEX PLANE
VIDEO LECTURE part 2
MAIN CONCEPTS
For complex valued functions, 
, we have entered the wormhole. Get the Printer Friendly VersionCOMMENTS
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CONCEPT
In high school calculus course, you’ll come across the differential arc length
 along the curve  given by
This isn’t too hard to derive. Simply notice that The arc length, s, from
Here is where our mathematician hero comes in. American mathematician Edward Kasner in 1914 showed how if the function Let’s set out the situation formally. We are all riding in the Enterprise and need to travel from point P at
Alternatively, we could take what everyone calls the shortest distance between two points, that is the chord between P and Q and travel through the distance c. Clearly, we all know that c < s. Upon closer inspection, we can also write
which should be no surprise as the two points gets closer, they merge into each other and so their ratio is 1. Or is it? Well, it certainly is for real-valued functions. Let’s just take a tour with a complex function. While complex-valued functions can’t be drawn on a curve - draw for me Now we present the genius of Edward Kasner. From
As
The reasoning is that
Again, as
given that Finally, let us compare the two different routes we can choose to travel along.
My goodness! The arc length distance is almost 6% shorter than the straight line. Time to tell your friends that the old adage of “the shortest distance between two points is a straight line connecting the two” is wrong. This means that by talking the complex-valued function path to travel to point Q, we have actually reached there before someone who would have taken the straightline path. The Enterprise has entered the wormhole. All only possible in the realm of complex-valued functions of course.All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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, 
and 
forms a right-angle for small values and then apply Pythagoreans theorem.
to 
is simply
is complex valued, we have just created wormhole in space!
in space to point Q at 
along distance s. Our route of travel is marked out by the graph 
as shown below.
and I’ll give you a hundred bucks – that does not stop us from performing the usual algebraic manipulation. Please read carefully that we are using the function 
and not 
which can be drawn. This time, its impossible to have the value 
on the y axis.
, the chord length from 
to 
is 
, we approximate c as 
given that 
is close to zero.
goes to zero much faster than does 
.
?
, we approximate s as
is close to zero.
