0_pi_day

7/27/2019 0_pi_day

http://slidepdf.com/reader/full/0piday 1/8

Pi (3.14) Day (3/14) and Einstein's Birthday

= 3.14159265…

Where have you seen ?

7/27/2019 0_pi_day


Area of a Circle A=r 2

Circumference of a Circle C=d or C=2r

Area of a Sector A=r 2

360

m, where m is the measure of the central angle

S = 2 r 2 + 2 rh S = r 2 + rl S =4 r 2

V = r 2h V =

3

1 r

2h V =

3

4 r

3

7/27/2019 0_pi_day


Interesting Facts

01ie

Euler’s Formula

...77553311

...88664422

2

...4

1

3

1

2

1

1

1

6 2222

2

...11

1

9

1

7

1

5

1

3

11

4

Gregory-Leibniz series

The following TI83 program evaluates the above series times 4 for the number of

terms entered by the user and displays the result every 100 terms.

PI

4 N

0 P

1 DClrHomeDisp "ENTER NO > 100:"

Input I

round(I / 100, 0) I

For(J, 1, I)

For(K, 1, 100)

D + 2 D

1 * N N N / D + P P

End

Disp (K 1) J

Disp P

End

7/27/2019 0_pi_day


Early Estimates of Pi

Archimedes estimated as 3.1418 using a 96-sided regular polygon to

approximate a circle. By 1600, was estimated to 35 decimal places.

Computers made it possible to estimate to thousands of places.

7/27/2019 0_pi_day


My Estimate for Using Ideas from Calculus

Sum of rectangular areas is

approximately ¼ of area of circle.

As width of rectangles approaches

zero (gets smaller), sum approaches¼ area of circle.

A = r 2

= 12

=

= 4 * area of ¼ of circle

7/27/2019 0_pi_day


The Buffon Needle Problem

The Buffon Needle Problem proves that the

probability of a needle of length L crossingone of many parallel lines a distance of D

apart is D

L

2. (Buffon discovered this through

Calculus.)

To simplify things, we will let L = D. In other words, the length, L, of our needle

is the same as the distance, D, between the parallel lines. So when L = D, the

probability of the needle crossing one of the parallel lines is 2 .

The Monte Carlo Method

The Monte Carlo Method is a method of approximating values by doing a large

number of random experiments.

Example: What is the area of region A?

1. Select n points at random inside the square.

2. If r points fall in region A, then the area of

region A is2

sn

r .

7/27/2019 0_pi_day


Estimating using the Monte Carlo Method and the Buffon Needle Solution

Using the Monte Carlo Method, we will estimate the probability of a needle

landing on a line using the ratio drops

hits.

According to Buffon, the probability is

2.

The two are equal:

2

drops

hits.

Solving for we have:hitsdrops 2 .

Our Experiment

1. Drop the dowel rod on the floor 100 times.

2. Count the number of drops which result in the dowel rod crossing a parallel

line on the floor (parallel in one direction only). These are called “hits.”

3. Estimate using the ratio

hits

drops2

A great computer simulation of our experiment:http://www.mste.uiuc.edu/reese/buffon/bufjava.html

7/27/2019 0_pi_day


Digits of Pi(groups of 10; 100 per line)

Pi 3.

1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899

8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502

8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165

2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817

4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094

3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724

8912279381 8301194912

(500 Digits)

9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523

8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901

2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977

4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026

4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303

5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787

6611195909 2164201989

(1,000 Digits)

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