Squares

Suppose we have a rectangle which is 45 units by 16. We shall use this to express 45/16 as a continued fraction since at present 45/16 is just a simple fraction.

Looking at the rectangle the other way, its sides are in the ratio 16/45. We shall use this change of view when expressing 45/16 as a continued fraction. 45/16 is 2 lots of 16, with 13 left over, or, in terms of ordinary fractions:

In terms of the picture, we have just cut off squares from the rectangle until we have another rectangular bit remaining. There are 2 squares (of side 16) and a 13 by 16 rectangle left over.

Now, suppose we do the same with the 13-by-16 rectangle, viewing it the other way round, so it is 16 by 13 (so we are expressing 16/13 as a whole number part plus a fraction left over). In terms of the mathematical notation we have:

Repeating what we did above but on 16/13 now, we see that there is just 1 square to cut off of side 16, with a 3-by-13 rectangle left over, expressing 13/3 as a whole-number-plus-fraction:

45 16

16/13

Notice how we have continued to use fractions and how the maths ties up with the picture. Now we do the same thing on the left-over 3-by-13 rectangle, but looking at it as a 13-by-3 rectangle. There will be 4 squares (of side 3) and a rectangle 1-by-3 left over:

45 16

13/3

Now we have ended up with an exact number of squares in a rectangle, with nothing left over so we cannot split it down any more.

45 16

In the rectangle of the rectangle, we can relate the geometry to the arithmetic as follows: we see 2 orange squares (16 by 16), 1 brown square (13 by 13), 4 red squares (3 by 3) leaving a blue rectangle of size 1 by 3 (or you can think of this as 3 blue squares of size 1 by 1): the numbers are 2, 1, 4 and 3, as seen in the continued fraction above.

The General form of a Continued Fraction

We can do the same to any fraction, P/Q (P and Q are whole, positive numbers) expressing it in the form of a continued fraction as follows:

where ag, a^, a2, etc are all whole numbers. If P/Q is less than 1, then ag will be 0. 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 7/30 = 0 + 1/(30/7)

and then we apply the method of the last paragraph:

7/30

= 0 + 1/(4 +

2/7)

= 0 + 1/

4 +

1/

(7/2)

)

= 0 + 1/

4 +

1/

(3 +

1/2))

= 0 + 1/

4 +

1/

(3 +

1/(1 + 1/1

Either of the last two lines is a valid continued fraction form for 7/30.

Continued Fractions - An introduction

The List Notation for Continued Fractions

We can write down any continued fraction such as

P/Q = a0 + 1/(a-, + 1/( a2 + 1/(a3 + ...))) just as a list of the a's:

The first number is the whole number part of the fraction, so we separate from the other coefficients by using a semi-colon (;) after it.

For the continued fractions used above, we can now write them as:

If the first number in the list is 0, then the numerical value is less one. For instance, one half is:

Also, there is a simple way to find the reciprocal of a continued fraction, for instance 16/45, since its list form is 0 + 1/(45/16), so we have:

If its list form begins with a zero already, as in 1/2 = [0,2], then its reciprocal is found by removing the 0 from the start of the list:

1. Express the following as continued fractions

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