Difference between revisions of "Pi"
(Created page with "=References= [http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits Proof that $\pi$ is constant for all circles...") |
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| + | 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= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Theorem:</strong> The value of $\pi$ is independent of which circle it is defined for. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
=References= | =References= | ||
| − | [http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits Proof that $\pi$ is constant for all circles without using limits] | + | [http://math.stackexchange.com/questions/3198/proof-that-pi-is-constant-the-same-for-all-circles-without-using-limits Proof that $\pi$ is constant for all circles without using limits]<br /> |
| + | |||
[http://www.oocities.org/cf/ilanpi/pi-exists.html Proof that $\pi$ exists] | [http://www.oocities.org/cf/ilanpi/pi-exists.html Proof that $\pi$ exists] | ||
Revision as of 15:45, 4 October 2014
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: █
References
Proof that $\pi$ is constant for all circles without using limits
