[Written by ChatGPT. Image credit]
Pi Day is celebrated on March 14 (3/14) because the date represents the first three digits of π (3.14). The celebration was founded in 1988 by physicist Larry Shaw at the Exploratorium in San Francisco. Shaw and his colleagues marked the occasion by marching in circles and eating fruit pies, a tradition that continues today.
Who Showed Pi is Non-Repeating and Infinite?
The fact that π is an irrational number (meaning its decimal expansion neither terminates nor repeats) was first rigorously proven in 1768 by Johann Lambert, a Swiss mathematician. Lambert used continued fractions to prove that π cannot be expressed as a ratio of two integers.
Later, in 1882, Ferdinand von Lindemann proved that π is not only irrational but also transcendental, meaning it is not a root of any nonzero polynomial equation with rational coefficients.
How Do We Know Pi is Irrational?
A number is irrational if it cannot be written as a fraction a/b, where a and b are integers.
One way to prove π is irrational involves a proof by contradiction using calculus. Here’s a rough outline of an argument originally credited to Lambert:
1. Assume π is rational, meaning π = a/b for integers a and b.
2. Construct a specific function involving sin(x) that has integer properties and shows contradiction when assuming π is rational.
3. The contradiction implies π cannot be rational.
A more advanced proof by Ivan Niven (1947) uses calculus to show that if π were rational, it would violate the fundamental properties of derivatives and integrals of certain trigonometric functions.
Thus, mathematicians know with certainty that π is irrational and has infinite non-repeating digits, a fact confirmed computationally today with trillions of digits.
[Written by DeepSeek]
Pi (π) is one of the most fascinating and important numbers in mathematics. Here are some interesting facts about it:
- Definition: Pi is the ratio of a circle’s circumference to its diameter. It is an irrational number, meaning it cannot be expressed as a simple fraction, and its decimal representation never ends or repeats.
- Irrational and Transcendental: Pi is not only irrational but also transcendental, which means it is not a root of any non-zero polynomial equation with rational coefficients. This implies that it cannot be expressed exactly using a finite number of integers, fractions, or their roots.
- Decimal Expansion: The decimal representation of pi has been calculated to over 50 trillion digits, though only a few dozen are needed for most practical calculations.
- Approximations: Throughout history, various cultures have used approximations for pi. For example, the ancient Egyptians used a value of around 3.1605, and the Babylonians used 3.125. The fraction 22/7 is a common approximation.
- Pi Day: Pi Day is celebrated on March 14 (3/14) around the world. It also happens to be Albert Einstein’s birthday.
- Infinite Series: Pi can be computed using various infinite series. One of the most famous is the Leibniz formula: π/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – …
- Buffon’s Needle: Pi appears in the solution to Buffon’s needle problem, a probability experiment that involves dropping needles on a lined surface to estimate pi.
- Pi in Nature: Pi appears in various natural phenomena, such as the distribution of rivers’ meanders, the shape of waves, and the structure of DNA.
- Pi in Popular Culture: Pi has been featured in numerous books, movies, and songs. The movie “Pi” (1998) and the novel “Life of Pi” by Yann Martel are notable examples.
- Pi Memorization: There are competitions and records for memorizing the digits of pi. The current world record is held by Rajveer Meena, who recited 70,000 decimal places of pi in 2020.
- Pi and the Universe: Pi is used in various equations that describe the universe, including those in general relativity, quantum mechanics, and cosmology.
- Pi Symbol: The symbol π was first used by Welsh mathematician William Jones in 1706 and was later popularized by Leonhard Euler.
These facts highlight the ubiquity and importance of pi in mathematics, science, and culture.
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