Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π=c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Answers
Step-by-step explanation:
Given :-
is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π=c/d.
To find :-
π=c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Solution :-
Let the diameter of the circle be d units
Let the Circumference of the circle be C units
Given that
π is the ratio of the circumference of a circle to its diameter
=> π = C : d
(or)
=> π = C/d
=> it is in the form of p/q
=> it is a rational number.
But it is not a rational number.
It is an irrational number.
Reason -1:-
The circumference of the circle and the diameter of the circle are immeasurable.There is no measure to measure the values of C and d accurately. So we regarded as π is an irrational number.
Reason -2:-
The decimal expansion of π is non terminating and non recurring so It is not a rational number .So it is an irrational number.
Points to know about π :
- Archimedes was a Greek generous who first calculated the digits in the decimal expansion of π .
- He found that the value of π lies between 3.140845 and 3.142857
- 3.140845 < π < 3.142857
- Aryabhatta was the great Indian Mathematician who found the value of π correct to 4 decimal places i.e.π= 3.1416.
- The present super fast computers has calculated π value upto 1.24 trillion decimal places.
- π=3.141592653589....,The decimal expansion of π is non terminating and non recurring decimal,it is an irrational number.
- We use π=22/7 but π≠22/7 .The value of 22/7 is closely nearer to π.
- We celebrate every year π day on March 14th on the behalf of The Einstein's birthday.
- The first three numbers of the value of π=3.14 is represents March 14th (coincident).
Answer:-
π is an irrational number.