Hello! This is Presh Talwalkar.
Here are 5 facts that everyone should know about the mathematical constant pi.
Fact number 1: the definition.
Pi is the ratio of the circumference of a circle to its diameter.
That is, pi equals the circumference divided by the diameter.
What's special is this is true for every single circle,
whether the circle is very big or whether the circle is very small.
Pi will always be the ratio of the circumference to its diameter, for every single circle.
Fact number 2: pi is irrational.
Pi is equal to approximately 3.141592653
and so on; the decimal form continues on forever and it never repeats.
The fraction 22/7 is accurate to two decimal places.
And the fraction 355/113 is accurate to six decimal places.
But because pi is irrational,
there is no fraction A/B, where A and B are integers, that will be exactly equal to pi.
Pi is also transcendental, which means
pi will never be the root or the solution of a non-constant polynomial with integer coefficients.
Fact number 3...
Archimedes estimated pi more than two thousand years ago.
Archimedes considered a circle which had a diameter of 1.
The circumference of this circle will be equal to pi.
So Archimedes thought he could estimate the circumference on the circle
by considering drawing a hexagon inside the circle.
The perimeter of this hexagon will be clearly less than the circumference of the circle, which is pi.
Archimedes also considered drawing a hexagon around the circle.
Now the perimeter of this hexagon will be larger than the circumference of the circle,
so we can have been upper bound for pi.
Using trigonometry, Archimedes could calculate the perimeter of the hexagons and estimate pi.
He then ingeniously doubled the number of sides and considered the same argument for 12-gon.
He doubled the number of sides again,
and considered the argument for a 24-gon.
He could calculate these perimeters of the inscribed and circumscribed perimeters
to get a lower bound and an upper bound for pi.
He got all the way to the 96-gon,
in which he found that pi must be wedged between 223/71 and 22/7.
This gave him an approximation that pi is about 3.14 which was very good and often practical.
Fact number 4: pi can be expressed in many infinite series.
This is my favorite infinite series.
Pi over 4 is the alternating sum of odd fractions.
This is the Leibniz formula, which was also discovered by the Indian mathematician Madhava.
This formula was found by Euler.
And it's a beautiful relationship between the numbers and pi.
There are also many complicated infinite series
This is one the Indian mathematician Ramanujan would have come up with,
and these types of formulas are used in the Chudnovsky algorithm,
which is how pi is determined algorithmically on computers.
Fact 5. pi comes up in probability.
Here's a drawing of the normal curve or the bell curve which you've seen in statistics class to describe many different data.
The equation of the curve involves pi!
So even in probability, pi is a very important constant.
Consider Buffon's needle problem.
Imagine randomly dropping a needle of length L on a floor where the lines are 1 unit apart.
What's the probability the needle will intersect one of the lines?
The answer is two times the length of the needle divided by pi!
So unexpectedly pi comes up even in probability.
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