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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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Is 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". – lhf Oct 17 '11 at 21:21
up vote 8 down vote accepted

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.

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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.

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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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