3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196
4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609
What is pi ()? Who first used pi? How do you find its value? What is it for? How many digits is it?
By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it.
For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159, which is really pretty good, but if you want a better approximation you can use a computer to get it. Here's pi to many more digits: 3.14159265358979323846.
The area of a circle is pi times the square of the length of the radius, or "pi r squared":
A = pi*r^2
A very brief history of pi
Pi is a very old number. We know that the Egyptians and the Babylonians knew about the existence of the constant ratio pi, although they didn't know its value nearly as well as we do today. They had figured out that it was a little bigger than 3; the Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians had a somewhat worse approximation of 4*(8/9)^2 (about 3.160484), which is slightly less accurate and much harder to work with. For more, see A History of Pi by Petr Beckman (Dorset Press).
The modern symbol for pi [] was first used in our modern sense in 1706 by William Jones, who wrote:
There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) - 1/3(16/5^3 - 4/239^3) + ... = 3.14159... = (see A History of Mathematical Notation by Florian Cajori).
Pi (rather than some other Greek letter like Alpha or Omega) was chosen as the letter to represent the number 3.141592... because the letter [] in Greek, pronounced like our letter 'p', stands for 'perimeter'.
About Pi
Pi is an infinite decimal. Unlike numbers such as 3, 9.876, and 4.5, which have finitely many nonzero numbers to the right of the decimal place, pi has infinitely many numbers to the right of the decimal point.
If you write pi down in decimal form, the numbers to the right of the 0 never repeat in a pattern. Some infinite decimals do have patterns - for instance, the infinite decimal .3333333... has all 3's to the right of the decimal point, and in the number .123456789123456789123456789... the sequence 123456789 is repeated. However, although many mathematicians have tried to find it, no repeating pattern for pi has been discovered - in fact, in 1768 Johann Lambert proved that there cannot be any such repeating pattern.
As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) pi is irrational: it cannot be written as a fraction (the ratio of two integers).
Pi shows up in some unexpected places like probability, and the 'famous five' equation connecting the five most important numbers in mathematics, 0, 1, e, pi, and i: e^(i*pi) + 1 = 0.
Computers have calculated pi to many decimal places. Here are 50,000 of them and you can find many more from Roy Williams' Pi Page.
2006-09-15 13:06:49
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answer #2
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answered by Anjanette A 3
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obviously, 22/7 is just an approximation and will not give you a correct value beyond 3 decimal points
although you seem to have something against the wikipedia, it has a good description of one of the simpler algebraic techniques
there are also caluculus techniques and statistical techniques (and experimental ones where you draw a really big circle and accurately measure the circumference and diameter)
Calculating Ï
The formulae often given for calculating the digits of Ï have desirable mathematical properties, but are often hard to understand without a background in trigonometry and calculus. Nevertheless, it is possible to compute Ï using techniques involving only algebra and geometry.
For example, one common classroom activity for experimentally measuring the value of Ï involves drawing a large circle on graph paper, then measuring its approximate area by counting the number of cells inside the circle. Since the area of the circle is known to be
a = \pi r^2,\,\!
Ï can be derived using algebra:
\pi = a/r^2.\,\!
This process works mathematically as well as experimentally. If a circle with radius r is drawn with its center at the point (0,0), any point whose distance from the origin is less than r will fall inside the circle. The pythagorean theorem gives the distance from any point (x,y) to the center:
d=\sqrt{x^2+y^2}.
Mathematical "graph paper" is formed by imagining a 1x1 square centered around each point (x,y), where x and y are integers between -r and r. Squares whose center resides inside the circle can then be counted by testing whether, for each point (x,y),
\sqrt{x^2+y^2} < r.
The total number of points satisfying that condition thus approximates the area of the circle, which then can be used to calculate an approximation of Ï. Mathematically, this formula can be written:
\pi \approx \frac{1}{r^2} \sum_{x=-r}^{r} \; \sum_{y=-r}^{r} \Big(1\hbox{ if }\sqrt{x^2+y^2} < r,\; 0\hbox{ otherwise}\Big).
In other words, begin by choosing a value for r. Consider all points (x,y) in which both x and y are integers between -r and r. Starting at 0, add 1 for each point whose distance to the origin (0,0) is less than r. Divide the sum, representing the area of a circle of radius r, by r2 to find the approximation of Ï. Closer approximations can be produced by using larger values of r.
For example, if r is set to 2, then the points (-2,-2), (-2,-1), (-2,0), (-2,1), (-2,2), (-1,-2), (-1,-1), (-1,0), (-1,1), (-1,2), (0,-2), (0,-1), (0,0), (0,1), (0,2), (1,-2), (1,-1), (1,0), (1,1), (1,2), (2,-2), (2,-1), (2,0), (2,1), (2,2) are considered. The 9 points (-1,-1), (-1,0), (-1,1), (0,-1), (0,0), (0,1), (1,-1), (1,0), (1,1) are found to be inside the circle, so the approximate area is 9, and Ï is calculated to be approximately 2.25. Results for larger values of r are shown in the table below:
r area approximation of Ï
3 25 2.777778
4 45 2.8125
5 69 2.76
10 305 3.05
20 1245 3.1125
100 31397 3.1397
1000 3141521 3.141521
Similarly, the more complex approximations of Ï involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations.
before computers, calculating pi was rough work, but now, they are cranking out pi digits like crazy
2006-09-15 13:12:11
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answer #3
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answered by enginerd 6
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