The value of \( \pi \) is defined as the ratio of the circumference \( c \) of a circle to its diameter \( d \): \[ \pi = \frac{c}{d} \] This is a fundamental definition in geometry and holds true for all circles, regardless of their size.
While \( \pi \) is defined as a ratio, it is an irrational number, which means that it cannot be expressed as a simple fraction of two integers. This implies that its decimal expansion is non-terminating and non-repeating: \[ \pi \approx 3.14159265358979\ldots \]
The contradiction arises when we try to think of \( \pi \) as a "simple ratio" of integers, but its irrationality means that no exact fraction exists that can represent \( \pi \). The **irrationality** of \( \pi \) does not contradict the definition of \( \pi \) because the ratio \( \frac{c}{d} \) is a general relationship, while \( \pi \)'s value as an irrational number comes from the fact that the circumference and diameter of a circle are related in such a way that no exact rational ratio can describe it.
The definition of \( \pi \) as \( \frac{c}{d} \) holds universally for all circles, and while \( \pi \) is irrational, this does not imply that the ratio is undefined. It simply means that the ratio cannot be expressed exactly as a fraction of two integers, but the relationship itself still exists and is fundamental in geometry.
For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.