• Adnan Gill

The Most Beautiful Equation

Many mathematicians consider Euler’s identity to be the most beautiful equation in analytical mathematics. Some of the features of this equation that have led to this belief are:

  • three basic arithmetic operations, addition, multiplication, and exponentiation occur exactly once

  • it relates five fundamental mathematical constants:

  • the number 1, the multiplicative identity

  • the number 0, the additive identity

  • the number pi, ratio of circumference to diameter of a circle

  • the number e, the base of natural logarithms

  • the number i¸ the unit of imaginary numbers

  • the number of types of numbers represented in this equation is large

  • the number 1 represents natural numbers which are counting numbers

  • the number 0 represents whole numbers

  • the numbers 1, 0, and represent real numbers

  • the numbers 1 and 0 are rational whereas and represent irrational numbers

  • the number i represents imaginary numbers as well as complex numbers

  • e represents algebraic numbers

  • π represents transcendental numbers

  • if the equation is written as e^iπ = –1 , then the number –1 also shows up in this equation and represents negative numbers and integers

The number 1 is the first number that human beings discovered and started to use. Some primitive languages have only two words, one for the number 1 and one for many, for counting. All initial number systems used a line or a notch to represent the number 1 and several lines or notches to represent larger numbers e.g. the Roman numerals start off as I, II, III. One interesting property of the number one is that any number multiplied or divided by it does not change. It can be safely stated that 1 is the oldest number.

Zero is a dangerous number. It is the additive identity which means that when it is added to any number there is no change in that number. When a number is multiplied with 0, it reduces to zero, no matter how large that number may be. Dividing any number by zero, results in an infinite number. Dividing a number with zero is like sharing a cookie with people in such a way that each person’s share is 0. How many people can you share a cookie with if each person gets nothing? The answer is: “an infinite number of people.” Since early Greek era, mathematicians had rejected this number and because Aristotelian philosophy dominated most of Europe, 0 had been kept out of western mathematics till the renaissance. Indian mathematicians started using zero around fifth century AD, later Arab mathematicians learned from Indians and brought it to Europe in around the ninth century. European mathematicians did not start using 0 confidently till around the end of thirteenth century.

Very ancient civilizations were familiar with the fact that there is a constant ratio between the circumference and diameter of a circle. We now represent this ratio by the Greek alphabet π, equivalent to our English alphabet p. As far back as mid to late 1600s mathematicians were using π/δ for the ratio circumference/diameter. π was the first alphabet for the Greek words from which the English words periphery and perimeter have been derived and δ was the first letter of the word from which we have the word diameter. Later just π was used to represent this ratio. Ancient cultures had obtained values for π which were amazingly accurate. By about 2,000 BC, the Babylonians had arrived at a value of for π and an early eighth century Chinese document states 3.141024 < π < 3.142704. The method now used by powerful computers to determine the value of π is ascribed to Archimedes but may have been known to mathematicians of other cultures before him. In this method perimeter of a regular polygon inscribed inside a circle is taken as an approximation of the circumference and then another polygon with twice as many sides is inscribed inside the circle to get a better approximation of the diameter. Archimedes drew a 96-gon inside a circle and this resulted in an error of only 0.008227%. Since π cannot be expressed as a fraction of two whole numbers, it is irrational. Furthermore it is not the root of a polynomial equation and is therefore, transcendental – it transcends algebra.

Euler’s number, e, the base of natural numbers is another fundamental mathematical constant in Euler’s identity. It first appeared in 1618 in an appendix to John Napier’s work on logarithms in which he gave a table of natural logarithms of various numbers. Like most of the notation used in modern mathematics, e was first used by Euler to represent this number. It is defined as the limit of (1 + 1/n)^n, where n is a whole number and approaches infinity and can also be calculated by the series 2 + 1/2 + 1/2*3 + 1/2*3*4 + 1/2*3*4*5 + ..., both methods giving an approximate value of 2.71828. The number e is important because it is used in a variety of applications ranging from engineering to economics and population growth. Integral, as well differential calculus depend heavily on this number.

The next number in this equation is i, the imaginary unit. It is defined as the square root of –1, or in other words i^2 = –1. When students are first introduced to this number they are understandably shocked and many do not come out of this shock for many years. It is a number that defies common observation and if squaring a number denotes an area then how can the area be negative. Although ancient Greeks regarded numbers as lengths and could never have understood mathematics that uses imaginary numbers, Heron of Alexandria, a Greek mathematician and engineer was the first person to have hinted about such numbers. Cardano used imaginary numbers while solving cubic equations, but most mathematicians remained skeptical and in fact the term imaginary was used in a derogative sense by Descartes. Euler and later Gauss studied complex numbers extensively and today they are used in control theory, signal processing, electromagnetism, quantum mechanics and a host of other subjects.

This brief introduction of only the numbers in Euler’s identity should be proof enough for its mathematical beauty and significance, but interested readers can explore its applications in a myriad of fields to convince themselves of the utility and beauty of this equation that many mathematicians believe to be the most beautiful of all mathematical equations.


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