In the USA we often measure areas in acres.
One acre is defined to be 43,560 square feet.
A football field ( 120 yd X 160 ft ) has an area of 57,600 ft2.
So, one acre is exactly 121 / 160 of a football field.
If you are a person who needs to measure things
in terms of acres
but you find it easier to
visualize football fields, then you might
want to
use the fact that one acre is
exactly 121 / 160 of a football field.
Since 121 / 160
≈
120 / 160 = 3 / 4,
we can more easily remember
that one acre is about three-fourths of a football field.
But if we express 121 / 160 as a
continued fraction
,
we can use its convergents to
find that a much better approximation
than 3 / 4 is 31 / 41.
| So when comparing areas, we should remember that | ||
| 1) an acre is less than a football field | ||
| 2) an acre is roughly 3 / 4 of a football field | ||
| 3) an acre is almost exactly 31 / 41 of a football field | ||
Here is a very brief look at continued fractions.
| When written as a continued fraction, | |||
| 121 / 160 is 0 + 1 / w | |||
| where w is 1 + 1 / x | |||
| where x is 3 + 1 / y | |||
| where y is 9 + 1 / z | |||
| where z is 1 + 1 / 3 | |||
The ( messy ) result looks something like this :
| 121 / 160 = | 0 + |
1
|
If you've never heard of continued fractions before then you probably wonder why anyone would want to write a simple fraction in such a complicated way.
One application: continued fractions can sometimes be used to find "good" estimates of numbers, where the estimate is simpler than the number being estimated.
In our example, the numbers 0 1 3 9 1 3 are called the partial quotients of the continued fraction.
< 0, 1, 3, 9, 1, 3 > = < q0, q1, q2, q3, q4, q5 >
Many applications of continued fractions begin by putting the partial quotients into a table whose construction starts out like this:
|
Next we copy q0 into the empty slot below it.
|
For our example, we have
|
Now fill in the middle row,
moving from left to right, as follows:
Into the 1st empty slot of the middle row we will put
( q0 * q1) + 1
= ( 0 * 1) + 1 = 1
Into the next middle row slot we will put
( the_value_just_computed * q2) +
q0 =
( 1 * 3 ) + 0 = 3.
Continue filling slots from left to right using this rule
( also see Fig 1 ) :
After computing the number that you entered into slot j,
you next compute the number to be put into slot j+1 by
multiplying the number in slot j by the
number in the slot ( diagonally ) above and to the right
of slot j slot and then adding to that product the number
found in slot j-1.
The bottom row is then filled out in a similar way.
See Fig 1 where we show how the "v" and
"w" cells are computed.
For our example, here is the final result:
|
Except for the 0 and 1 in the 1st two columns, choose any number in the bottom row, and let it be the denominator of a fraction whose numerator is the number in the cell above it. This fraction can be regarded as an estimate of the final fraction ( 121 / 160 ) in the bottom row. As you move from left to right, the estimates get better. Thus, 1/1 is a worse estimate than 3/4, which is worse than 28/37, ....
Other than 121/160 itself, the best estimate of 120/160 is 31 / 41.
Tables like the one above have some interesting properties. For example, choose any two contiguous cells in the bottom row and regard those cells plus the two above as forming a 2X2 matrix. The determinant of this matrix will be either 1 or -1. In fact, the leftmost matrix has a determinant of +1. The next matrix will have a determinant of -1, the next will be +1, and so on, always alternating between 1 and -1.
If you want to learn more about continued fractions,
there are many books available.
One source that is old but still worth looking at is
chapter 7 of
An Introduction to the Theory of Numbers
by Ivan Niven and Herbert Zuckerman
© 1960 John Wiley & Sons, Inc.