Question

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?

Answer 1

As far as I'm concerned, they're not really that important. But then again, I have no exposure to graduate-level 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:
`1+1/(1+1/(1+...)`

Is very famous because it simplifies to `phi`, or the Golden Ratio (`1.618...`). To see why this is, we will employ a very clever trick. We begin by setting the fraction equal to `x`, as we would do with any unknown entity:
`x=1+1/(1+1/(1+...)`

But check this out. This part in red:
`1+1/color(red)((1+1/(1+...)))`

Is really just equal to `x`! Let's take it out of context and add a few more terms to see why:
`color(red)(1+1/(1+1/(1+1/(1+...))`

See? It's just the same infinite repeating fraction, so we replace it with `x`:
`x=1+1/color(red)((1+1/(1+...))`
`->x=1+1/color(red)(x)`

Now we solve, through these steps:
`x^2=x+1`
`->x^2-x-1=0`
`->x=((-1)^2+-sqrt((-1)^2-4(1)(-1)))/((2)(1))`
`->x=(1+-sqrt(5))/2`

The `(1-sqrt(5))/2` solution is a negative number, and obviously the infinite fraction is not negative; therefore, our solution is
`(1+sqrt(5))/2`
which is the number `phi` in radical form.

In fact, you can also express such math celebrities as `pi` and `e` as infinite fractions, and discover some cool patterns like the Fibonacci sequence in them too. As for continued fractions in general, you can use them to approximate mathematical constants but you would be punishing yourself when other techniques (e.g. Taylor series-derived expansions) are way easier. I'm not really sure how else they're useful; infinite fractions are, to the extent of my knowledge, the most useful type.

Answer 2

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!

Answer 3

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 "n-body" system of how n masses interact under mutual gravitational attraction (using Newton's laws).

Link: Number Theory and Dynamical Systems

Answer 4

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 `x in RR` and `p/q` is a convergent of the development of `x` in continued fraction, any other rational number `(p')/(q')` can approximate `x` better only if `q' > q`.

See: http://sites.millersville.edu/bikenaga/number-theory/approximation-by-rationals/approximation-by-rationals.html

Answer 5

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 `1 : sqrt(2)` to the nearest millimetre. In fact the same is true of an A0 sheet, plus the area is made as close as possible to `1 m^2`.

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 `p/q` to `sqrt(n)` such that `abs(p^2-q^2n) = 1` (see https://socratic.org/s/aJmwNJJA).

There are interesting generalised continued fractions for numbers related to `e` and `pi`, which have implications for proving their irrationality, etc.