The General form of a Continued Fraction
where a, b, c, d, e, etc are all whole numbers. If P/Q is less than 1, then the first number, a, will be 0.
The second form takes lots of vertical space on the page. The third form is the same mathematically, but all those brackets are confusing to read! The final form is used in some books as a convenient shorthand as it is both easy to read and takes little space on the page.
There is an even simpler notation (the list notation) that we introduce in the next section.
Note that usually all the numbers in the continued fraction will be positive although alternative forms are possible where negative whole numbers are allowed, but not on this page.
The fractional form that we have derived is called the continued fraction.
There is no need to draw the rectangles-as-squares pictures each time, unless you want to, because we can merely look at the numbers. If the fraction is less than 1, we use its reciprocal and then we can split it
into a whole-number part plus another fraction which will be less than 1 and repeat. We stop when the fraction has a numerator or a denominator of 1.
Take for instance, 7/30. It is already less than 1 so we start off by writing it as and then we apply the method of the last paragraph: Either of the last two lines is a valid continued fraction form for 7/30 and we can, of course, just omit the "0 +" part.
The List Notation for Continued Fractions
We can write down any continued fraction such as just as a list of the numbers a; b, c. The first number, a. is special as it is the whole number part of the value. The rest is written as a list with comma separators (,) like this: None of the values will be zero, except possibly the first - the one before the semicolon (;) - if the value of the fraction is less than 1.
The FIRST number in the list
For the continued fraction examples above, we can now write them as: