Approximate the value of π using a rational
representation of the quantity `pi`

.

The mathematical quantity *π* is
not a rational number, but the quantity `pi`

that
approximates it *is* a rational number since all
floating-point numbers are rational.

Find the rational representation of `pi`

.

The resulting expression is a string. You also can use `rats(pi)`

to
get the same answer.

Use `rat`

to see the continued fractional
expansion of `pi`

.

The resulting string is an approximation by continued fractional
expansion. If you consider the first two terms of the expansion, you
get the approximation $$3+\frac{1}{7}=\frac{22}{7}$$,
which only agrees with `pi`

to 2 decimals.

However, if you consider all three terms printed by `rat`

,
you can recover the value `355/113`

, which agrees
with `pi`

to 6 decimals.

$$3+\frac{1}{7+\frac{1}{16}}=\frac{355}{113}\text{\hspace{0.17em}}.$$

Specify a tolerance for additional accuracy in the approximation.

R =
3 + 1/(7 + 1/(16 + 1/(-294)))

The resulting approximation, `104348/33215`

,
agrees with `pi`

to 9 decimals.

Create a 4-by-4 matrix.

X =
1.0000 0.5000 0.3333 0.2500
0.5000 0.3333 0.2500 0.2000
0.3333 0.2500 0.2000 0.1667
0.2500 0.2000 0.1667 0.1429

Express the elements of `X`

as ratios
of small integers using `rat`

.

N =
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
D =
1 2 3 4
2 3 4 5
3 4 5 6
4 5 6 7

The two matrices, `N`

and `D`

,
approximate `X`

with `N./D`

.

View the elements of `X`

as ratios using ```
format
rat
```

.

X =
1 1/2 1/3 1/4
1/2 1/3 1/4 1/5
1/3 1/4 1/5 1/6
1/4 1/5 1/6 1/7

In this form, it is clear that `N`

contains
the numerators of each fraction and `D`

contains
the denominators.