Recall, π is defined as the ratio of the circumference(say c) of a circle to its diameter(say d). That is, . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Properties of Circle

Question

Recall, $π$ is defined as the ratio of the circumference(say c) of a circle to its diameter(say d). That is, $π = \frac{c}{d}$ . This seems to contradict the fact that $π$ is irrational. How will you resolve this contradiction?

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Here,

π = \frac{22}{7}

Now, it is in the form of

\frac{p}{q}

, which is a rational number, but this is an approximate value of

π

If you divide 22 by 7, the quotient (3.14⋯ ) is a non-terminating number i.e. it is irrational.

Approximate fractions include (in order of increasing accuracy)

\frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \dots

If we divide anyone of these we get, the quotient (3.14⋯ ) is a non-terminating number. i.e. it is irrational.

So,

There is no contradiction as either c or d irrational and

hence

π

is a irrational number.