As
a child sitting in math class I always wondered how any of the stuff we learned
would actually be used. Like why on earth do we need to study the quadratic
formula? Well if you are like the vast majority of people, and work at the
local hardware store or work a regular desk job, you never will. But I assure
you, the quadratic formula and math in general are very important to how the
world works. They are just used in ways that are hidden to most of us, so we
rarely see it. This series of blogs will be focused on showing the ways math is
used in the modern world, most of it in ways none of us ever new about. Despite
being a math blog, I will try and keep all of the math basic and as simple as
possible, so we can focus on the real content of how it is used.
Chapter 1:
Taylor Series and Modern Computing
If you have ever taken Calculus
II you will remember learning about Taylor Series. If you haven’t don’t worry,
they are rather simple and I will explain them in a second. They are one of the
things Calculus students generally hate learning because they are hard to fully
understand and most teachers don’t explain how they are applied in real life.
As such many students simply learn what they need to in order to pass the test,
and forgot all about them the second they know they have passed. Although
Taylor series are very important in the modern world, and we use them every
day, most of us just don’t ever realize it.
For those of you who never learned
about them (or forgot once you saw that B), Taylor Series are a representation of a function as an infinite of
terms that are calculated from the values of the function's derivatives at a single point. More simply, Taylor series
are a way of approximating a complex function as a simpler one. It isn’t a 100%
accurate, but even a simple Taylor Series can approximate some very complex
rational function to an astounding accuracy. To do this one simply repeated follows
the Taylor formula and follows derivative laws until they have achieved the
level of accuracy that is acceptable to them.
The
Taylor Series formula:
Example of
Taylor Approximation for sin(x):
This approximation for sin(x) only
goes to the fourth order and is very simple, but even so is very accurate. For
example, when calculating an x value less than 1 and greater than -1, the error in approximation is less
than 0.000003, which is very small and accurate enough for most
applications. But I am not here to lecture you on calculus and this brings me
to my real point. While sin(x) is a simple function, these simple Taylor Series
are much easier to calculate once you start getting into complex functions.
Especially for computers with limited CPU and RAM capabilities, they can use
these Taylor Series to calculate values of complex functions within an
acceptable accuracy level all while using much less memory and don’t need to
store giant banks of trig function values. All most all computer calculator
programs do this, from your basic four button calculator all the way up to some
super computers. Because of the way computers “think” it is much easier for
them to calculate in this way (especially for non-mathematical programming
languages such as C+). For example, the function arctan(x) can be very
complicated to calculate when dealing with higher numbers, so a computer would
simply solve the Taylor Series instead, and they can solve it with only basic mathematical
operators.
So
when you are working on homework and enter a function into your IPhone, it is
actually solving something much different than what you entered and technically
giving you a value that isn’t truly accurate. Now is this something that you
will use every day? No. In fact, unless you’re a computer science major you may
never use it since this is built into the basics of any mathematical programing.
But that isn’t the point of learning, and now you know something that 99% of
people don’t. Plus you now know how computers and the world work a little better;
you can now finally say that calculus II wasn’t a total waste.
If you are interested in learning more about
Taylor Series, Paul’s Online math does a great job going over them:
More info on how they are used in the computer
science field:
Sources:
"Taylor Series." Wikipedia. Wikimedia Foundation, 06 June 2014. Web. 06 June
2014. <http://en.wikipedia.org/wiki/Taylor_series>.
"Mathematical Foundations of
Computer Science 2011." Google Books.
N.p., n.d. Web. 06 June 2014.
<http://books.google.com/books?id=C1K-Z6BIDZsC&pg=PA176&lpg=PA176&dq=taylor+series+computer+science&source=bl&ots=PnwdEU8FIi&sig=IbZsRoj4eNekcXDXHnVhbxmnZfg&hl=en&sa=X&ei=42ORU8L7K8vTsATO4YCADw&ved=0CH8Q6AEwCDgK#v=onepage&q=taylor%20series%20computer%20science&f=false>.
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