Basic ideas and concepts:
- Why we assume that exists? In other words, why it is certain that the ratio of the circumference to its diameter is always the same for any given circle? Short answer: all circles are geometrically similar, and therefore the ratios of corresponding parts are proportional. For two circles, C_{1} and C_{2}, of radii r_{1} and r_{2} must be true that: C_{1}/r_{1} = C_{2}/r_{2} which is equivalent to C_{1}/d_{1} = C_{2}/d_{2}
Long answer: Inscribe a regular hexagon in a circle. All angles of the hexagon are equal (60), all sides also equal, therefore, the perimeter of the hexagon of radius 1 is 6 and the ratio of the perimeter to the diameter (2r) is 3. Since equal chords cut off equal arcs, the perimeter of the circle will always keep the same ratio to its radius no matter the size of the circle.
- Why the letter ? The letter stands for perimeter, the distance around a figure. The ratio perimeter/diameter was written –in Greek letters— p/d. In 1706 the Welsh Mathematician William Jones dropped the d and started using the symbol alone.
The symbol began to be extensively used after Leonhard Euler adopted it in 1737.
- A constant called ? Since ancient times was observed that the ratio of the circumference of a plane circle to its diameter –the larger distance across-- was constant. This is, no matter how big or small the circle is that constant has a value of approximately 3.
- Which is the exact value of ? Most people respond to the value of pi as 3.14; is a number whose decimal expansion neither ends nor repeats a block of digits, like rational numbers do. (Rational numbers can be expressed as ratio of integers). is considered an irrational number, one that cannot be expressed as a ratio of two integers. Today billions of digits of the expansion of pi have been calculated. These are the first fifty:
= 3.14159265358979323846264338327950288419716939937510... up to infinity.
- How can a number be called irrational? Indeed “irrational” in this context should not be understood as “beyond reason”. Ancient Greeks called “logos” a ratio of two integers. Logos in Greeks also means word, verb, expressible. Numbers that were demonstrated not to be expressible as the ratio of two integers, like the square root of two, were called “alogos”, meaning “inexpressible”. Again, the decimal expansion of , like all irrational numbers, neither terminates nor become periodic.
- Who proved the irrationality of ? Johann H. Lambert in 1761. His approach goes as follows: He demonstrated that if x is a rational number, tangent of x is irrational; on the other hand, whenever tan(x) is rational, x must be irrational. Since tan(/4) = 1, therefore is irrational. There are other more convincing proofs of the irrationality of , this one was the first and simple enough to be understood by everyone.
- Is a transcendental number or an algebraic number? 1873 Charles Hermite proved that the euler’s number e, is transcendental. This is, there is no equation of the form ae^{x} + be^{y} + ce^{z} +....= 0 where the coefficients a, b, c and the exponents x, y, z are rational numbers. Then in 1882, Ferdinand von Lindermann proved that is transcendental based in Hermite’s work. Lindermann proved that the exponent in the “e’ equation cannot be either algebraic numbers, therefore e^{ix }+ 1 = 0 is not satisfied when x is an algebraic number, therefore x must be transcendental. (The imaginary unit i is algebraic since √-1 = i).
Note: For more info on this topic of the transcendentally of numbers, search on Lindemann–Weierstrass theorem .
- Squaring the circle. Is this possible? Since is transcendental, it is impossible to construct a square of equal area of a given circle. This is, it is impossible to construct a square of side √ by using straightedge and compass and the ancient Greek attempted unsuccessfully.
In common language, the expression squaring the circle is used as a figure of speech for trying something obviously impossible.
- formulas: Nothing better that mathworld website as a reference for pi formulas:
http://mathworld.wolfram.com/PiFormulas.html
http://www.wolframalpha.com/input/?i=Representations+of+Pi
- Ramanujan, 1914:
- Continued fractions of at Mathworld.wolfram.com:
- 100 club: http://www.acc.umu.se/~olletg/pi/club_100.html
- everywhere: Buffon's Needle, An Analysis and Simulation http://mste.illinois.edu/reese/buffon/buffon.html
Additional Remarks:
1) As no one can write down pi or any other irrational number, we do not use irrational numbers but their rational approximations by truncating a few digits, say 3 or ten, but always a few of them. These approximations turn out to be good enough for all practical purposes.
2) There are infinitely many irrational numbers. This is a necessary clarification to the ambiguity of text books and teachers who demonstrate that all integers can be considered rationals, as well as all other fractions of the sort p/q where q is different from zero; then a few irrationals are meant to be mentioned: √2, , Euler number. Let’s add here: the square root of any non perfect square is an irrational number. The same is valid for the cubic roots of any non perfect cube. Summary: the nth root of a number ( x^{1/n}) is irrational unless x is the nth power of an integer. Most logarithms and values of trigonometric functions are also irrational numbers. There are also irrational numbers that cannot be found as solutions (roots) of integer polynomials. This kind of numbers are meant to transcend algebra, and consequently were called “transcendental” numbers. √2 is an algebraic number, the solution or root of the equation: x^{2}-2 = 0; rational numbers are algebraic numbers of degree one: 2x-1 = 0 the solution of the equation is ½. George Cantor demonstrated in 1874 that the set of irrational numbers is larger (uncountable) than the set of rational numbers (which is a countable set). Not all infinities are the same: there are more irrational numbers than rational numbers.
3) How can it be that irrational numbers and even rational numbers never end? Because space is continuous and extensively infinite in depth.
4) Irrationality of √2. Can this number be represented as the ratio of two integers?
Let's assume √2 is a rational number. Then we can write it √2 = p/q where
p, q are integers, q different form zero, and also p and q have no common divisors (p/q is reduced to its lowest terms).
By squaring both sides of the equation follows that,
2 = p^{2}/q^{2},
p^{2} = 2q^{2}
So p^{2} is a multiple of two and therefore p itself is a multiple of 2.
Then since p is a multiple of two, or even number, we can represent it as 2k, this is, p = 2k
Substitute p = 2k into:
p^{2} = 2q^{2}
And we have
(2k)^{2} = 2q^{2} which leads to 2k^{2}= q^{2}
Therefore q^{2} is even, q is even: contradiction! Both numbers p and q cannot be even or multiple of two since we established as a condition that this ratio p/q had been reduced to its lowest terms.
Proving that other square roots are irrational takes more or less the same steps than the √2.
5) Irrationality and logaritms: Is log2 rational?
log 2 (base 10) is equal to 10^{log2} = 2
Assuming log 2 is rational the log can be equated to the ratio of two integers p/q:
log 2 = p/q and since the logarithm can be expressed in the exponential form:
10^{p/q} = 2 then by raising both sides to the power of q we obtain:
10^{p }= 2^{q}
Which is impossible: no power of ten raised to an integer is equal to a power of two raised to another integer. Contradiction. Therefore, log 2 and similarly log 3 etc are irrational numbers. However, log_{2} 8 = 3 a rational number, simply because 8 is a power of 2. Only when the number whose log is sought is a power of the base of the log, logaritms are rational numbers. |