How is pi derived

The contour lines give the rise or fall in height vertically between the two points. Using a ruler and the scale of the map you can find the horizontal distance between the points but make sure it is in the same units as the horizontal distance! Dividing one by the other gives the ratio measuring the steepness of the road between the two points.

But they look the same slope? You can see that they "measure" very different slopes the green line and the black line are clearly different slopes now. What do you think a slope of "1 in 1" means in the two interpretations? So we had better be clear about what we mean by slope of a line in mathematics!!

The first interpretation is called the sine of the angle of the slope where we divide the change in height by the distance along the road hypotenuse. The second interpretation is called the tangent of the angle of the slope where we divide the change in height by the horizontal distance. So in mathematics, as on road-signs, we measure the slope by a a ratio which is just a number.

The higher the number, the steeper the slope. A perfectly "flat" road will have slope 0 in both interpretations. Uphill roads will have a positive steepness and downhill roads will be negative in both interpretations. Note that with the other interpretation using the sine of the angle of 1 in 1 is a rise of 1 metre for every metre along the road.

This would mean a vertical road a cliff-face which is not at all the same thing as a tangent of 1! Similarly, in mathematics, a slope of -1 would be a hill going downwards at 45 degrees.

In maths, lines can have slopes much steeper than roads designed for vehicles, so our slopes can be anything up to vertical both upwards and downwards. Such a line would have a slope of "infinity". The tangent of an angle So we can relate the angle of the slope to the ratio of the two sides of the right-angled triangle. This ratio is called the tangent of the angle.

This would mean using the tangent function "backwards" which in mathematics is called finding the inverse function of the tangent. The inverse function is called the arctangent function , denoted atan or arctan. Because of this confusion, some mathematical authors prefer the atan or arctan notation, which is what we shall use on this page. So arctan t takes a slope t a tangent number and returns the angle of a straight line with that slope. Actually, it is not so much a formula as a series, since it goes on for ever.

So we could ask if it will it ever compute an actual value an angle if there are always terms to come? Provided that t is less than 1 in size then the terms will get smaller and smaller as the powers of t get higher and higher. Borwein et al. Further sums are given in Ramanujan ,. Beeler et al. Equation 78 is derived from a modular identity of order 58, although a first derivation was not presented prior to Borwein and Borwein The above series both give.

Such series exist because of the rationality of various modular invariants. A class number field involves th degree algebraic integers of the constants , , and. Of all series consisting of only integer terms, the one gives the most numeric digits in the shortest period of time corresponds to the largest class number 1 discriminant of and was formulated by the Chudnovsky brothers The appearing here is the same one appearing in the fact that the Ramanujan constant is very nearly an integer.

Similarly, the factor comes from the j -function identity for. The series is given by. Borwein and Borwein ; Beck and Trott; Bailey et al. This series gives 14 digits accurately per term. The same equation in another form was given by the Chudnovsky brothers and is used by the Wolfram Language to calculate Vardi ; Wolfram Research ,.

The best formula for class number 2 largest discriminant is. Borwein and Borwein This series adds about 25 digits for each additional term. The fastest converging series for class number 3 corresponds to and gives digits per term. The fastest converging class number 4 series corresponds to and is. This gives 50 digits per term. Borwein and Borwein have developed a general algorithm for generating such series for arbitrary class number. A complete listing of Ramanujan's series for found in his second and third notebooks is given by Berndt , pp.

These equations were first proved by Borwein and Borwein a, pp. Borwein and Borwein b, , proved other equations of this type, and Chudnovsky and Chudnovsky found similar equations for other transcendental constants Bailey et al. Adamchik, V. Monthly , , Backhouse, N. Pancake Functions and Approximations to. Bailey, D. Experimental Mathematics in Action. Beck, G. Beckmann, P. A History of Pi, 3rd ed. New York: Dorset Press, Beeler, M. Item in Beeler, M. Berndt, B. Ramanujan's Notebooks, Part IV.

New York: Springer-Verlag, Beukers, F. Blatner, D. The Joy of Pi. New York: Walker, Boros, G. Cambridge, England: Cambridge University Press, Borwein, J. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Experimentation in Mathematics: Computational Paths to Discovery. New York: Wiley, a. Andrews, B. Berndt, and R. New York: Academic Press, pp. Monthly 96 , , Castellanos, D. Part I. It is also a transcendental number , a concept that exceeds the scope of this post but is interesting in and of itself.

While you ruminate over the irrational nature of pi while enjoying your favorite slice of pie, take a moment to consider what makes pie, well, pie as opposed to cake. Memorizing as many digits of pi as possible has become an obsession for many.

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