section_path: Fundamentals/Definition
section_name: Definition
paragraph_text: Definition
  
  π is commonly defined as the ratio of a circle 's circumference C to its diameter d: π = C d {\displaystyle \pi ={\frac {C}{d}}}
  
  The ratio C d {\textstyle {\frac {C}{d}}} is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio C d {\textstyle {\frac {C}{d}}}. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C d {\textstyle \pi ={\frac {C}{d}}}.
  
  Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits —a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x 2 + y 2 = 1 {\textstyle x^{2}+y^{2}=1}, as the integral: π = ∫ − 1 1 d x 1 − x 2. {\displaystyle \pi =\int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}.}
  
  An integral such as this was proposed as a definition of π by Karl Weierstrass, who defined it directly as an integral in 1841.
  
  Integration is no longer commonly used in a first analytical definition because, as Remmert 2012 explains, differential calculus typically precedes integral calculus in the university curriculum, so it is desirable to have a definition of π that does not rely on the latter. One such definition, due to Richard Baltzer and popularized by Edmund Landau, is the following: π is twice the smallest positive number at which the cosine function equals 0. π is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential equation.
  
  In a similar spirit, π can be defined using properties of the complex exponential, exp z, of a complex variable z. Like the cosine, the complex exponential can be defined in one of several ways. The set of complex numbers at which exp z is equal to one is then an (imaginary) arithmetic progression of the form: { …, − 2 π i, 0, 2 π i, 4 π i, … } = { 2 π k i ∣ k ∈ Z } {\displaystyle \{\dots,-2\pi i,0,2\pi i,4\pi i,\dots \}=\{2\pi ki\mid k\in \mathbb {Z} \}} and there is a unique positive real number π with this property.
  
  A variation on the same idea, making use of sophisticated mathematical concepts of topology and algebra, is the following theorem: there is a unique (up to automorphism) continuous isomorphism from the group R / Z of real numbers under addition modulo integers (the circle group), onto the multiplicative group of complex numbers of absolute value one. The number π is then defined as half the magnitude of the derivative of this homomorphism.