‘Beautiful Equations I’ was one of my favourite posts to write, so I decided to continue with this series and write a part 2.
Leonhard Euler was one of the most influential and prolific mathematicians in history, laying the foundations of an array of areas in mathematics for his successors to build upon. His output was immense; he published more than 500 books and papers during his lifetime and a further 400 appeared posthumously.
Euler’s Identity is often considered the most beautiful equation in mathematics as it combines five of the most fundamental mathematical constants:
- e: the base of natural logarithms
- i: the imaginary unit of complex numbers, equivalent to the square root of -1
- π: the ratio of a circle’s circumference to its diameter
- 1: the multiplicative identity
- 0: the additive identity
Feynman described it as “the most remarkable formula in mathematics”, and I must say that I completely agree.
So where does this identity come from?
If you’ve studied complex numbers you will know Euler’s relation:
Simply substitute the angle as π!
For more information on where Euler’s relation comes from, click here.
Boltzmann’s Entropy Formula
As a chemistry student, one of my favourite topics (apart from my beloved organic chemistry) is entropy. Put simply, entropy is the degree of disorder in the system. For a reaction to occur, entropy must always increase.
Boltzmann’s formula relates entropy (S) of an ideal gas and the number of ways that the atoms or molecules can be arranged (k log W). The more ways the particles can be arranged, the greater the disorder and therefore entropy of the system. K is Boltzmann’s constant and W is the number of microscopic elements of a system in a macroscopic system in a state of balance.
Edwin Schrödinger’s famous partial differential equation illustrates how subatomic particles change with time when under the influence of a force. Any particular atom or molecule is described by its wave function (represented by the Greek letter psi), which predicts the probability of where and when the particles appear.
However, physicists are still unsure on how to interpret this equation. Some believe that it’s just a useful calculation tool, but does not actually correspond to anything real, whilst others argue that it demonstrates the limit to the amount that we can learn about the universe, as we can only learn about a particle once it’s measured.
Schrödinger believed that the wavefunction represented a real, physical object and rejected the interpretation that a particle only collapses when it’s measured. In fact, his famous cat experiment actually intended to demonstrate the weakness of this interpretation.
The Gaussian Integral
The function in the Gaussian integral is a very hard function to integrate. However, when analyzed over the whole real line – from minus infinity to infinity – the answer is surprisingly neat. This formula is of extreme use and has a range of applications. For example, it is used to calculate the normalising constant of the normal distribution.
The Analytic Continuation of the Factorial
The factorial function is commonly defined as
However, this only works for positive integers. Therefore, by using this integral:
mathematicians are able to compute factorials for fractions, decimals, negative numbers and even complex numbers. The gamma function is an extension of this, using n – 1 instead of n.
It’s used in various probability-distribution functions, and so highly applicable to probability, statistics and combinatorics.
The Explicit Formula for the Fibonacci Sequence
This formula, derived by Binet in 1843 (although the result was known to Euler, Daniel Bernoulli and de Moivre more than a century earlier) can be used to calculate the nth Fibonacci number in the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, etc., where each number is the sum of the previous two numbers). Although this sequence a classic and known by the vast majority of people, this formula is known to few. Remarkably, despite the formula having square roots and divisions, the answer is always an exact positive integer.
The phi in the formula represents the golden ratio, where
Two quantities are in the golden ratio when their ratio is the same as the ratio of their sum to the larger of the two quantities. When a > b > 0, this can be expressed as
I personally find the golden ratio a fascinating part of mathematics. Would you like me to do a blog post on this? x