Question

What is the percent error in using 3.14 as an approximation for \[\pi \] (which is 3.14159265358979323846...) ?
A. \[0.5\% \]
B. \[0.05\% \]
C. \[0.005\% \]
D. None of these

Answer 1

Hint: It is known that \[\pi \] is a transcendental number, it is not the root of any polynomial with rational coefficients. The transcendence of \[\pi \] implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Error Percent of anything is calculated by \[\dfrac{{\left| {\left. {Approximate - Exact} \right|} \right.}}{{\left| {\left. {Exact} \right|} \right.}} \times 100\] So place the values in this formula and see what result you can obtain.

Complete step-by-step answer:
The number \[\pi \] is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter \[\pi \] since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.
Here in this question the approximate value is given as 3.14 and it is also known that \[\pi \] has an never ending value; So let us take the value of \[\pi \] as 3.141592 and we will try to find the percent error in this;
\[\begin{array}{l}
\therefore \% Error = \dfrac{{\left| {\left. {Approximate - Exact} \right|} \right.}}{{\left| {\left. {Exact} \right|} \right.}} \times 100\\
 \Rightarrow \% Error = \dfrac{{\left| {\left. {3.14 - 3.141592} \right|} \right.}}{{\left| {\left. {3.141592} \right|} \right.}} \times 100\\
 \Rightarrow \% Error = \dfrac{{\left| {\left. { - 0.001592} \right|} \right.}}{{\left. {\left| {3.141592} \right.} \right|}} \times 100\\
 \Rightarrow \% Error = 0.05\%
\end{array}\]
So from here it is clear that option B is the correct option and 0.05% is the correct answer.

So, the correct answer is “Option B”.

Note: We cannot find the error percentage by taking the exact value of \[\pi \] as the exact value is yet not defined so we increase the digits after decimal and then find it accordingly. Being an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.