Difference between revisions of "Pi"
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<strong>Theorem:</strong> The value of $\pi$ is independent of which circle it is defined for. | <strong>Theorem:</strong> The value of $\pi$ is independent of which circle it is defined for. | ||
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| + | <strong>Proof:</strong> █ | ||
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| + | <strong>Theorem:</strong> The real number $\pi$ is an [[irrational number]]. | ||
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 15:49, 7 April 2015
A circle in Euclidean plane geometry is defined to be the set of points equidistant from a center point. The length around a circle is called its circumference and the length a line from the circle through the center is called a diameter of the circle. All diameters have the same length by definition of the circle. Let $A$ be a circle. The number $\pi$ is defined to be the ratio $\dfrac{C}{D}$ where $C$ is the circumference of $A$ and $D$ the diameter of $A$. It requires proof to show that the value obtained from the circle $A$, call this $\pi_A$, is the same number one obtains from another circle $B$, the value $\pi_B$.
Properties
Theorem: The value of $\pi$ is independent of which circle it is defined for.
Proof: █
Theorem: The real number $\pi$ is an irrational number.
Proof: █
References
Proof that $\pi$ is constant for all circles without using limits
Proof that $\pi$ exists (video)
