Application of Pi
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Application of Pi()
Monil shahRohan shah
is commonly defined as the ratio of circles circumference C to its diameter d.
Geometry and trigonometryMonte Carlo methodsComplex numbers and analysisNumber theory and Riemann zeta functionProbability and statistics
Outside mathematics:Describing physical phenomenaMemorizing digitsIn popular culture
Uses of pi
1) Electrical engineers used pi to solve problems for electricalapplications2) Statisticians use pi to track population dynamics3) Medicine benefits from pi when studying the structure of the eye.4) Biochemists see pi when trying to understand the structure/function of DNA5) Clock designers use pi when designing pendulums for clocks6) Aircraft designers use it to calculate areas of the skin of the aircraft
Circumference is never exactWe can never truly measure the circumference or the area of a circle because we can never truly know the value of pi. Pi is an irrational number, meaning its digits go on forever in a seemingly random sequence
Analytical Geometry and CalculusDuring the 17th century, analytic geometry and calculus were developed. They had a immediate effect on Pi. Pi was freed from the circle! An ellipse has a formula for its area which involves Pi (a fact known by the Greeks); but this is also true of the sphere, cycloid arc, hypocycloid, the witch, and many other curves.
Geometry and trigonometry appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve .
The circumference of a circle with radius r is 2r.The area of a circle with radius r is r2.The volume of a sphere with radius r is 4/3r3.The surface area of a sphere with radius r is 4r2.The formulae above are special cases of the surface area Sn(r) and volume Vn(r) of an n-dimensional sphere.
appears indefinite integralsthat describe circumference, area, or volume of shapes generated by circles. For example, an integral that specifies half the area of a circle of radius one is given by
In that integral the function 1-x2 represents the top half of a circle (the square root is a consequence of the Pythagorean theorem), and the integral 1 -1 computes the area between that half of a circle and the x axis.
Sine and cosine functions repeat with period 2.
The trigonometric functions rely on angles, and mathematicians generally use radians as units of measurement. plays an important role in angles measured in radians, which are defined so that a complete circle spans an angle of 2 radians.The angle measure of 180 is equal to radians, and 1 = /180 radians.
Common trigonometric functions have periods that are multiples of ; for example, sine and cosine have period 2,so for any angle and any Integer k,
Monte Carlo methods: based on random trials, can be used to approximate .
Monte Carlo methods, which evaluate the results of multiple random trials, can be used to create approximations of . Buffon's needle is one such technique: If a needle of length is dropped n times on a surface on which parallel lines are drawn t units apart, and if x of those times it comes to rest crossing a line (x > 0), then one may approximate based on the counts:
Complex numbers and analysisThe association between imaginary powers of the number e and points on the unit circle centered at the origin in the complex plane given by Euler's formula.Any complex number, say z, can be expressed using a pair of real numbers. In the polar coordinate system, one number (radius or r) is used to represent z's distance from the origin of the complex plane and the other (angle or ) to represent a counter-clockwise rotation from the positive real line as follows:
where i is the imaginary unit satisfying i2 = 1.
The frequent appearance of in complex analysis can be related to the behavior of theexponential function of a complex variable, described by Euler's formula:
where the constant e is the base of the natural logarithm. This formula establishes a correspondence between imaginary powers of e and points on the unit circle centered at the origin of the complex plane. Setting = in Euler's formula results in Euler's identity, celebrated by mathematicians because it contains the five most important mathematical constants:
There are n different complex numbers z satisfying zn = 1, and these are called the "n-th roots of unity". They are given by this formula:
Probability and statisticsA graph of the Gaussian function
(x) = ex2. The colored region between the function and the x-axis has area .
The fields of probability and statistics frequently use the normal distribution as a simple model for complex phenomena; for example, scientists generally assume that the observational error in most experiments follows a normal distribution. is found in the Gaussian function (which is the probability density function of the normal distribution) with mean and standard deviation :
The area under the graph of the normal distribution curve is given by the Gaussian integral
while the related integral for the Cauchy distribution is
Outside mathematics Describing physical phenomena: Although not a physical constant, appears routinely in equations describing fundamental principles of the universe, often because of 's relationship to the circle and tospherical coordinate systems.
A simple formula from the field of classical mechanics gives the approximate period T of a simple pendulum of length L, swinging with a small amplitude (g is the earth's gravitational acceleration):
One of the key formulae of quantum mechanics is Heisenberg's uncertainty principle, which shows that the uncertainty in the measurement of a particle's position (x) and momentum (p) cannot both be arbitrarily small at the same time (where h is Planck's constant):
In the domain of cosmology, appears in Einstein's field equation, a fundamental formula which forms the basis of the general theory of relativity and describes the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy:
where Rik is the Ricci curvature tensor, R is the scalar curvature, gik is the metric tensor, is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tik is the stressenergy tensor.
Coulomb's law: from the discipline of electromagnetism, describes the electric field between two electric charges (q1 and q2) separated by distance r (with 0 representing thevacuum permittivity of free space):
is present in some structural engineering formulae such as the buckling formula derived by Euler, which gives the maximum axial load F that a long, slender column of lengthL, modulus of elasticity E, and area moment of inertia I can carry without buckling:
The field of fluid dynamics contains in Stokes' law, which approximates the frictional force F exerted on small, spherical objects of radius R, moving with velocity v in a luid withdynamic viscosity :
The Fourier transform, defined below, is a mathematical operation that expresses time as a function of frequency, known as its frequency spectrum. It has many applications inphysics and engineeing, particularly in signal processing.