Instruction
1
π is an irrational number. This means that it cannot be represented as a fraction with integer numerator and denominator. Moreover, π is a transcendental number, i.e. it cannot serve as a solution to any algebraic equation. Thus, the exact value of the number π cannot be written. However, there are methods to calculate it with any required degree of accuracy.
2
The earliest approach used by the geometers of Greece and Egypt, they say that π is approximately equal to the square root of 10, or fraction 256/81. But these formulas give the value of π, equal to 3.16, and this is clearly not enough.
3
Archimedes and other mathematicians have calculated π using geometric complex and time-consuming procedure — measuring the perimeters of inscribed and circumscribed polygons. The obtained value is 3,1419.
4
Another approximate formula determines that π = √2 + √3. It gives a value for π, is approximately equal to 3,146.
5
With the development of differential calculus and other mathematical disciplines at the disposal of scientists a new tool — power series. Gottfried Wilhelm Leibniz in 1674 found that the infinite series
1 - 1/3 + 1/5 - 1/7 + 1/9... + (1/(2n+1)*(-1)^n
in the limit converges to a sum equal to π/4. To calculate this amount just, however, to achieve sufficient accuracy, it will take a lot of steps, because the series converges very slowly.
6
Later were found and other power series that can calculate PI faster than using the Leibniz series. For example, it is known that tg(π/6) = 1/√3 hence arctg(1/√3) = π/6.
The arc tangent function is expanded in a power series, and for the given values we get:
π = 2√3*(1 - (1/3)*(1/3) + (1/5)*(1/3)^2 - (1/7)*(1/3)^3... + 1/((2n + 1)*(-3)^n)...)
With this and other similar formulas for the number π was calculated with an accuracy of millions of decimal places.
7
For most practical calculations it is enough to know the number of π accurate to seven decimal places: 3,1415926. It can be easily remembered with the mnemonic phrase: "Three — fourteen — fifteen — ninety-two and six".