3,1 4 1 5 9 2 6 5 3 5 8 9 79
the 3 2 3 8 4 6 2 643 3 83 27 9 5 0 2 8 8 4 1 9 71 6 9 ‌
A Learning path in mathematics • Introducing a fun approach to the knowledge of maths facts • Обучаваме ученици и те след време запознават други с математически факти • We educate students and later they inform others • От 2009 година в ПМГ“Нанчо Попович“ празнуваме ден на ПИ • Since 2009, in PMG "Nancho Popovich" celebrate PI day • Предлагаме презентации, галерия фотоси и филм за Архимед • We offer presentations, gallery photos and movie Archimedes
π
? C π= 2r
contents
π
The history of the number Interesting facts about the number Computer programs for the calculation of What students from our school know about the number
π
π
π
The history of the number π The task to calculate the square of a circle
Pi and Archimedes
The history of pi after Archimedes
The task to calculate the square of a circle To draw a square with the same area of a circle one of the three outstanding problems of ancient times.
This problem is related to the number pi - one of the most important constants in modern mathematics..
Experimental solution of the problem to calculate the square of a circle in Ancient Egypt a circle and a square were covered with beans. The number of beans in the two figures is the same, if the side of the square is 8/9 of the diameter of the circle.
8/9 d
8/9d
d
•
How would you describe the solution today: 2
64 2 8 S = d = d 81 9 ⇓
πd 64 2 S= = d 4 81 ⇓ 256 π= = 3,1605 81 2
•3,1605 is the Egyptian’s approximation of pi. It has been used in the construction of the pyramids.
•In
the geometry of Ancient Babylon the circumference of a circle was calculated by tripling its diameter.
Pi and Archimedes
π
Who is Archimedes? The most famous Greek mathematician and engineer; Lived in Syracuse in 287-212.BC; 287-212
In the essay "Measuring range" he set THE task for finding the ratio between the circumference and its diameter for the first time .
Results obtained by Archimedes
3
10 1 10 1 d <C <3 d ⇒ 3 <π <3 7 7 71 71
What is the method of Archimedes? He inscribed regular polygons with 6, 12, 24, 48 and 96 sides within a circle and around it, comparing their perimeters and determining lower and upper limits of .
Archimedes was aware of the infinity of the circle and the value of pi is approximate.
Archimedes and mathematics
The importance of Archimedes, is best expressed by another great mathematician - Leibniz: â&#x20AC;&#x153; If you carefully read the writings of Archimedes, you will cease to be amazed at all the latest discoveries in geometry."
History of pi after Archimedes Greek mathematician and astronomer Ptolemy (2nd century AD):π = 3
17 120
Dutch mathematician Andrian Mæcen (17th century): π = 3,1415929 Indian mathematician Bramagupta (5-6 century AD): Ariabhata
Indian mathematician Ariabhata (6. AD) π = 3,1416 , by listing and describing the polygon with 384 sides
French mathematician Francois Viet (16th century) π =
π = 10
2
1 1 1 1 1 1 . 1+ . 1+ 1+ ...... 2 2 2 2 2 2
Francois Viet
And more ...
The Dutch mathematician Hummel (16. century) calculates pi to the 16th character the German mathematician Ludolph Van Ceulen (16century.) calculates pi to the 35th character,the so called Ludolphine number; In the 18th century Euler introduced the name
Ď&#x20AC;
Euler
In the 18th century the method of Archimedes' calculation of pi was abandoned and was replaced by more accurate one, introducing pi as an infinite series;
In the late 18th century the German mathematician Lambert and the French mathematician Lezhandar found that pi is an irrational number. In 1883.the German mathematician Lindemann proved that pi is a trantscendental number and the task to find the square of a circle is unsolvable with a ruler and a compass.
Interesting facts about the number
Ď&#x20AC;
On 14.03 we celebrate not only the birth of one of the most famous scientists Albert Einstein, but the day of pi. At the Massachusetts Institute of Technology and the Museum of Science "Exploratorium" in San Francisco many people glorify the perfection of this constant.
Givenchy produced a perfume "Pi" Kate Bush sings the digits of pi in a song YouTube keeps track of people who can recite it from memory
And more ... What's the easiest way to remember the digits of the number? π “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics!” „Как е леко и бързо запомнено пи, всички знаят щом желаят!”
Pi is all around us
Programs for calculating Ď&#x20AC;
l b o r p
pro ble m
1 m e
results
2
1 Set up a program in C + + language for calculating pi using the formula:
4 4 4 4 π = 4 - + − + − ... 3 5 7 9
#include<iostream> using namespace std; int main() { long long j,n; long double i=1,p=0; cin>>n; for(j=1;j<=n;j++) { if(j%2==1)p+=4/i; else p-=4/i; i+=2; } cout.precision(40); cout<<p<<endl; system("pause"); return 0; }
n
problem 1
problem 2
1000
3,1405926538397928
3,1416035449129027
10000
3,1414926535900434
3,1415929980246386
100000
3,1415826535897935
3,1415926644818426
1000000
3,1415916535897935
3,1415926539342052
10000000
3,1415925535897933
3,1415926536006156
100000000
3.1415926435897941
3,1415926535914473
1000000000
3,1415926525897953
3,1415926535794609
3000000000
3,1415926532564651
3,1415926536705565
2 #include<iostream> #include<math.h> using namespace std; int main() { long double i,r=1,x,s=0,a,n,s1; cin>>n; a=r/n; x=a/2; for(i=1;i<=n;i++) {s1=a*sqrt(r*r-x*x); s+=s1; x+=a; } ' a=4*s; i cout.precision(40); cout<<a<<endl; system("pause"); return 0;}
Set up a program in C + + language for calculating pi using a geometric approach: Construct n rectangles (n = 5 in the drawing). For large values of n the sum of the areas of these rectangles is "closer" to the area of a quarter of a circle
S ≈ S i'' ⇒ S ABPQ ≈ S ABCD
What students from our school know about the number Ď&#x20AC;
Училище за Себеутвърждаване и Подготовка към Европейски Хоризонти
ви кани на
π = 3,14 ден
през март 3, ден
14 от 3 и 14 часа след обед в ПМГ”нанчо Попович”, Шумен
The
numerical value
Calculating Π is defined analytically by trigonometric functions..
The number π is approximately 22/7 or 3,14 .
The numerical value of π, to the 100th decimal place is
•
3,14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06 286 20899 86280 34825 34211 70679 ...
Interesting facts
Despite much analytical work in the calculations done by supercomputers no pattern has been found to end the sequence.
On December 31, 2009. the French programmer Fabrice Belar reached a precision of 2699999990000 digits in the decimal base .
For practical purposes the precision of π from 2 to 5 characters is sufficient for all calculations.
Приложения
Celebrating the number π
Pi Day was proposed by Larry Shaw in 1988
In 2004 on Pi Day Daniel Tamet recited 22,514 digits of π.
Pi Day was marked in Bulgaria, Greece and Turkey on March 14, 2010 at 3:14 am.
March 14 is the birthday of Albert Einstein.
2009 PI Day
3,14... PI Day - 2009
2012 PI Day
2012 PI Day
PI Day 2013
sources Книга за извънкласно четене по математика – А.А.Колосов По стъпките на Питагор – Шчепан Еленски Чрез развлечение към знание – Станислав Ковал http://bg.wikipedia.org/ http://www.higherpi.info/ http://arbuz.uz/x_pi.html http://www.bg-science.info http://mathworld.wolfram.com/PiIterations.html .
The authors thank Dimitar Dimitrov for the videos and Monica Staykova and Rosalina Miteva for the translation into English