Discussion: Is value of $\pi = 4$?

so what is the "real definition" of a circle?

i think the original solution from wikipedia is too ambigous, i couldn't find why the circumference of the circle is 2*r*pi=3.14...

if we really use the method that i provide from the link how are we able to get an infinite series of pi and find the circumference of an ellipse?

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    $\begingroup$ Is en.wikipedia.org/wiki/Circle not clear? To quote: "a circle is the set of points in a plane that are a given distance from a given point, the centre". $\endgroup$ – lhf Oct 17 '11 at 21:21

The standard notion of a circle is a set of all points in a plane equidistant from some fixed point in that plane (its center).

By distance we mean standard Euclidean distance. So a circle in $\mathbb{R}^2$ is a set $\{ (x,y) \in \mathbb{R}^2 \,|\ \mathrm{distance}((x,y),(a,b))=r \}$ for some fixed point $(a,b) \in \mathbb{R}^2$ (the center of the circle) and some fixed (positive) distance $r \in \mathbb{R}$, $r>0$ (the radius).

Euclidean distance is given by the formula: $\mathrm{distance}((x,y),(a,b)) = \sqrt{(x-a)^2+(y-b)^2}$. Thus squaring both sides of the equation "$\mathrm{distance}((x,y),(a,b))=r$" gives us

$$\{ (x,y) \in \mathbb{R}^2 \,|\ (x-a)^2+(y-b)^2=r^2 \}$$

This is the circle with radius $r$ and center $(a,b)$.

If we wish to discuss a circle in $\mathbb{R}^n$, then we need to specify the plane in which it lies. The circle with center ${\bf c}$ and radius $r>0$ which lies in the plane $({\bf x} - {\bf p}) {\bf \cdot} {\bf n}=0$ (the plane through the point ${\bf p}$ with normal vector ${\bf n}$) is given by $\{ {\bf x} \in \mathbb{R}^n \,|\ ({\bf x}-{\bf p}){\bf \cdot}{\bf n}=0 \mbox{ and } |{\bf x}-{\bf c}|=r \}$. Again $|{\bf x}-{\bf c}|$ is the standard Euclidean norm.

If you mess with the definition of distance, then you can still call the corresponding set a "circle", but it's not a circle is the standard "classical" sense.


That discussion had nothing to do with the definition of "circle", and everything to do with the definitions of "length" and "limit". But at any rate, the definition of a circle (in the plane, $\mathbb{R}^2$) is a subset of the form $$\{(x,y)\in\mathbb{R}^2\mid (x-h)^2+(y-k)^2=r\}$$ where $r\in\mathbb{R}$, $r>0$ and $(h,k)\in\mathbb{R}^2$ is any point.

The circumference of a circle of radius $r$ is $2\pi r$ essentially by the definition of $\pi$ (so unless your circle has radius $\frac{1}{2}$, it doesn't have circumference $\pi$...). See here or here for an explanation of why this actually defines a single number $\pi$ for all circles.


Here's a totally rigorous definition.

Definition. For all $C \subseteq \mathbb{R}^2,$ we call $C$ a circle iff there exists $r>0$ and $c \in \mathbb{R}^2$ such that the following holds. $$\forall x \in \mathbb{R}^2(x \in C \leftrightarrow d(x,c)=r)$$


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