What are continued fractions for?
I learned about continued fractions from some YouTube videos I watched about a year ago, and I know how to write numbers as continued fractions. However, I am still in the dark as to why continued fraction notation is useful. Are there some calculations it makes easier? If so, how?
I learned about continued fractions from some YouTube videos I watched about a year ago, and I know how to write numbers as continued fractions. However, I am still in the dark as to why continued fraction notation is useful. Are there some calculations it makes easier? If so, how?
5 Answers
As far as I'm concerned, they're not really that important. But then again, I have no exposure to graduatelevel concepts, and they may be useful there.
Explanation:
Nevertheless, for math nerds like me, they present some very interesting problems and manifest some very beautiful patterns. For instance, the continued (infinite) fraction:
Is very famous because it simplifies to
But check this out. This part in red:
Is really just equal to
See? It's just the same infinite repeating fraction, so we replace it with
Now we solve, through these steps:
The
which is the number
In fact, you can also express such math celebrities as
See below.
Explanation:
It is a powerful analysis tool. Among other, It can be used to reveal the nature of numbers. If rational, irrational or even transcendent!
Here's a link to an advanced article (if you are interested) with a summary of continued fraction applications to "dynamical systems", which has its origin in the "nbody" system of how n masses interact under mutual gravitational attraction (using Newton's laws).
The key reason they were developed as a concept in theory of numbers is that in a sense they give the "best" approximation of a real number with a rational number of "small" denominator.
Specifically, if
A few thoughts...
Explanation:

Take an A4 sheet of paper (210mm x 297mm).

Cut off the largest possible square (210mm x 210mm) from one end, leaving a remaining rectangular piece 87mm x 210mm.

Cut off two 87mm x 87mm squares, leaving a 36mm x 87mm rectangle.

Cut off two 36mm x 36mm squares, leaving a 15mm x 36mm rectangle.

Cut off two 15mm x 15mm squares, leaving a 6mm x 15mm rectangle.

Cut off two 6mm x 6mm squares, leaving a 3mm x 6mm rectangle.

Cut the rectangle into two 3mm x 3mm squares.
Why do we get one square, then two squares 5 times?
The ratio of the width of an A4 sheet to its length is:
#1 : 1+1/(2+1/(2+1/(2+1/(2+1/2))))#
Notice the unmentioned steps above, namely rotating the remaining rectangle through a quarter turn  effectively inverting the ratio of width and length  corresponding to the reciprocals in the continued fraction.
The proportions of an A4 sheet of paper are chosen to be
What we see here is some relationship between geometric constructions, square roots and continued fractions.
In fact, we have:

Any rational number has a terminating continued fraction.

Any number of the form
#a+sqrt(b)# where#a# and#b# are rational has a recurring continued fraction and vice versa.
Continued fractions allow us to find efficient rational approximations to square roots, namely approximations
There are interesting generalised continued fractions for numbers related to