Chapter 1.2, Problem 70PPSE

Solution

i.

To state: The value of $\pi$ using a calculator.

The resultant value is $\pi =3.1415926535897932384626433832795$ . It is not an exact representation of $\pi$ .

Given information:

The given statement says to find the value of $\pi$ .

Explanation:

It is known that $\pi$ is an irrational number.

Use a calculator to find the value of $\pi$ ,

  $\pi =3.1415926535897932384626433832795$

It can be seen that the resulted value of $\pi$ after decimal is showing non-terminating and non-recurring numbers.

The exact representation of $\pi$ cannot be obtained because irrational number gives us non-terminating and non-recurring decimals in decimal representation.

ii.

To state: The comparison of decimal representation of $\pi$ and the value $\frac{22}{7}$ which is often considered equivalent to $\pi$ .

The two values are not exactly same but they are very close to each other.

Given information:

The given represented value of $\pi$ as $\frac{22}{7}$ .

Explanation:

It is known that is an rational number.

Use a calculator to find the value of $\frac{22}{7}$ ,

  $\frac{22}{7}=3.142857142857142857142857142857$

It can be seen that the resulted value of $\frac{22}{7}$ after decimal is showing recurring numbers.

It can be observe that the decimal value of Pi obtained using a calculator and its representative $\frac{22}{7}$ are same to two places after decimal.

Therefore, the value of $\pi$ is very close to the calculated value of $\frac{22}{7}$ but it is exactly same.