The number π, the ratio of a circle’s circumference to its diameter, has been known for almost 4000 years. The ancient Babylonians calculated the area of a circle by taking three times the square of its radius, which gave a value of π = 3. By the 17th century BC, they had estimated a closer approximation value of π = 3.125. The Greek mathematician Archimedes of Syracuse (287β212 BC) calculated the value of π by approximating the area of a circle using the Pythagorean Theorem, in which he showed that π was between 3 1/7 and 3 10/71. It was not until 1706 that this ratio was officially named π by the Welsh mathematician William Jones, for π is the first letter of the Greek name for periphery and for perimeter.
π is also used to calculate the area of a circle and the surface area of a sphere. No matter how big the circle or sphere, the value of π holds, thus we say it is a dimensionless number. As π cannot be expressed as a simple fraction of two integers, it is an irrational number. As there is no integer that equals a power of π, a root of π, or a multiple of π, it cannot be part of an algebraic expression. Thus we say it is a transcendental number.
π is found in numerous equations and formulae, such as Heisenbergβs uncertainty principle, which is at the heart of quantum mechanics, which states that it is impossible to measure the location of a subatomic particle (i.e. an electron) and its speed simultaneously. It is also found in Einsteinβs field equations explaining gravitation as a result of space-time being curved by the presence of matter and energy. Solutions for problems in number theory also involve π. For example, the probability that two numbers chosen at random are relatively prime numbers, meaning they share no positive common factor other than 1, or the probability that an integer does not contain any squared factors is 6/π2.
Electric power delivered to us in the form of alternating current (AC), an electrical current in which electric charge periodically reverses direction, is expressed in mathematical formulae involving π. π is also used to compute the shortest routes for airplanes, as they travel on what is known as a Great Circle, which has its center at the center of the earth, and its radius is the distance from this center to the earthβs surface. Consequently a Great Circle is the shortest distance between two points on the surface of the earth. When airplanes fly long distances, they traverse along these routes in order to save time and fuel.
Remarkably, the Swiss mathematician Leonard Euler also discovered the formula to illustrate how π and prime numbers are intimately related:
π/4 = (3/4)(5/4)(7/8)(11/12)(13/12)(17/16) β¦,
where all the odd prime numbers appear in order as numerators, and the denominators are each nearest to the corresponding denominator and a multiple of 4.
Whereas π has been known to mathematicians for thousands of years, e, another transcendental number, has only been recognized since the beginning of the 17th century.Β It was not until 1683 that the Swiss mathematician Jacob Bernoulli realized that a significant number laid between 2 and 3, when he attempted to solve the problem of calculating compound interest.Β This number, 2.71828β¦, was named e by Leonard Euler in a letter he wrote to the Russian mathematician Christian Goldbach in 1731.Β Because ex is the only function that is equal to its own derivative, or rate of change, which makes e the most suitable choice of base for doing calculus, it is the base of natural logarithms.Β The number e crops up in areas of mathematics dealing with growth and decay.
One way to appreciate the importance of e is to consider the behavior of money left on deposit and earning a compound interest, of which the interest is compounded once or more frequently per annum. Itβs easy to see that the money compounded quarterly would earn more interest than annually. However if the money were to be compounded an infinite number of times per annum, the earned interest would converge to the formula that is e^(annual interest rate).
Another most well known application of e involves dating ancient objects, using a method called carbon-14 dating. Every living organism contains carbon-12 and carbon-14. About 5730 years after an organism dies, the amount of carbon-14 within it is reduced to half of what it initially was; another 5730 years goes by and itβs reduced to a quarter. Since it is an exponential function, the formula for calculating this rate of decay involves e. Reciprocally, e is also used to compute bacteria population growth and historical human population growth.
Inspired by the transcendental number π, this jacket is adorned with patchwork motifs of a special pie chart, called the Polar Pie Chart. It was developed by the 19th-century English statistician Florence Nightingale to count the death rate and the causes of mortality in the army. Nightingale was a Victorian cultureβs icon who embodied the persona of “The Lady with the Lamp,” making ward rounds for wounded soldiers at the military hospital at Scutari, on the Asiatic side of the Bosporus, opposite Constantinople, during the Crimean War:
“Lo! in that house of misery
A lady with a lamp I see
Pass through the glimmering gloom,
And flit from room to room.”1
The Polar Pie Chart was later used to show wind speed and direction distribution at a particular location, which was called the Wind Rose, and is still used by meteorologists today.
1. Henry Wadsworth Longfellow, Santa Filomena.
