Whenever there is a beauty contest among mathematical formulas, Euler's formula is the one that gets the most votes. The formulation above is not the most common but the most natural and elegant in my opinion. Euler's formula is better known in the following forms:

The first version combines three fundamental numbers from three different areas of mathematics; π comes from geometry, i belongs to algebra and e is a part of analysis. The result of combining them is not some complex or strange number but simply -1.

The second version combines in one simple formula, equality and the three basic operations of arithmetics; addition, multiplication and exponentiation. It contains the additive identity 0 and the multiplicative identity 1 and of course the three basic numbers e, i and π. Quite amazing.

The third version tells us that if you combine four of the most fundamental achievements from centuries far apart, first geometry with π from antiquity, then the discovery and extension of numbers to include negative numbers followed by the extension of real numbers to include complex numbers such as i to solve equations and finally the number e, a starting point for analysis and limiting procedures then you come back to where all mathematics started from, the number one, the first step when you learn to count.

Definitions and values

Pi is defined by a circle and its value can be calculated with arbitrary precision from a sum or an integral.

Pi is not a solution to any polynomial equation with integer coefficients. It is neither rational nor algebraic, it is irrational and transcendental with an infinity of non-repeating digits.

The imaginary unit i is a number that solves the equation: x² + 1 = 0.

The number i leads to the complex numbers and the fundamental theorem of algebra, stating that every polynomial of degree n has n complex roots when counted with multiplicity.

The constant e is irrational and transcendental just like π. It's the natural base for exponentiation since no other base has a derivative equal to itself D(e^x) = e^x. It's the base of the natural logarithm and plays a central role in analysis, probability, number theory and physics. It can be defined in many ways:

The last definition based on a differential equation assumes the Picard-Lindelöf theorem that deals with existence and uniqueness of solutions to y'(t) = f(t,y(t)) with y(t₀) = y₀.

The definition of π is the biggest mistake in the history of mathematics, τ corrects that mistake.

What is wrong with π and the diameter? For starters, a circle is most naturally defined from a radius. A circle consists of all points in a plane with a given distance (radius) from a certain point. An engineer might find the diameter more useful since this is what you measure on a physical object. Inner and outer diameters are measured with a caliper. A mathematician's choice is the radius. The unit circle has a radius of one but a diameter of two, which is not very unitary. A consequence of the definition of π is that a unit circle has a circumference of 2π. With τ the circumference of a unit circle is 1τ = 1 turn. The circumference of a circle with radius r is C = r·τ. Both π and τ have their own days of celebration 3/14 for π and 6/28 for τ. More about tau can be read in the document of this link.→

Angles, circles and spheres

360° 180° 90° 60° 45° 30° 2π π π/2 π/3 π/4 π/6 τ τ/2 τ/4 τ/6 τ/8 τ/12 Angles in analysis are given in radians = arc/radius. It's obviously much more natural with τ than with π. Why let a quarter turn be π/2 when it could be τ/4?

Another nice feature of τ is that its value has a neat geometrical illustration. Let the unit circle be inscribed by a regular hexagon. It is made from six equilateral triangles and has a circumfrence of 6. The factor missig to get τ corresponds to the ratio of the arc-length and side-length of an equilateral triangle.

Formulas in mathematics and physics that reflect some underlying rotational symmetry often has a 2π in them. These formulas would look better with a simple τ instead of 2π, but what about the area of a circle isn't πr² nicer than τr²/2? In mathematics, as little as possible should be momorized, better a formula you can derive in your head.

The area of a circle is the limit of the area of a circumscribed regular n-gon as n goes to infinity. Let the circumference of an n-gon be C_n. Each n-gon consists of n triangles with base C_n/n and height r.

This formula better reflects the triangular nature of the sectors that make up a circle. Can we in a similar manner derive the area and volume of a sphere instead of memorizing A=4πr² and V=4πr³/3. The r² and r³ part are obvious from scaling. Let's calculate the area and volume of a unit sphere. Partition the surface into small pieces and add the volumes of the pointy parts. The volume of such a piece is given by Base·Height/Dimension. By starting with a basic pyramid this follows for any shape going up to an apex.

History

Pi, also known as Archimedes' constant was approximated with various fractions without knowledge of its true nature in ancient civilizations long before antiquity. Archimedes gave an algorithm for calculating π in 250 BC. The infinite series π/4 = 1 − 1/3 + 1/5 − ⋯ was first realized by Indian mathematicians in the 14th century. The symbol π comes from the first letter of periphery in Greek. π/δ (periphery/diameter) was used in the mid 17th century to represent 3.14 and sometimes π/ρ was used for 6.28. The single letter π for 3.14 was first used by Euler in 1727 but sometimes he used π to represent 6.28. J.H. Lambert proved that π is irrational in the 1760's and a proof that π is transcendental (not a solution to a polynomial with rational coefficients) was given by F. Lindemann in 1882.

Zero was present in Egyptian hieroglyphs and Babylonian tablets in the 2nd millenium BC. The use of a concept and a symbol representing nothing was problematic for the philosophical Greeks. The origin of the modern zero can be traced to India and China. The Indian Bakhshali manuscript contains the earliest known use of a zero symbol in the form a placeholder dot for decimal numbers. Brahmagupta from 7th century India gave gave clear rules for the use of zero. The Hindu-Arabic numeral system made zero and its symbol 0 something to count with and spread it over the world.

Negative numbers were used early in China. They occur in Nine chapters on the Mathematical Art, a book compiled 200BC−200AD. Brahmagupta used negative numbers in the 7th century. It took European mathematicians thousand years to accept the idea that negative numbers are just as real as positive numbers. A solution to n + 1 = 0 would to them be considered just as absurd as a solution to x² + 1 = 0 would be for a person without knowledge of complex numbers. Fibonacci (ca. 1200) used negative solutions in finance calculations to signify debt. Calculus starting with Leibniz and Newton made negative numbers natural but it wasn't until the middle of the 19th century that negative numbers where fully accepted. The origin of the plus and minus signs from around 1500 are a bit unclear but + could be a simplification of the latin "et" meaning and. The minus sign is a simplification of m for "meno" meaning minus that turned into a ∼ that later became straight.

In 16th century Italy there was a fierce competition in solving polynomial equations of degree three and beyond. Even though only real solutions counted, a formal treatment with roots of negative numbers lead some contestants to real solutions. Notable in this competition were Cardano (1501-1576) and Tartaglia (1500-1557). Rules for complex aritmetic were given by Bombelli in the book l'Algebra from 1572. The name imaginary for these numbers was coined by Descartes. The term i for √-1 was introduced later to avoid the mistake of using √a·b = √a·√b where it does not apply, √(-1)·(-1) ≠ √-1·√-1.

The roots of e can be traced to the natural logarithm that is based on the curve y=1/x. ln x is defined as the integral of 1/x. The first reference to the constant is from 1628 in a work on logarithms by J. Napier. The definition of e as the limit of (1+1/n)ⁿ as n → ∞ comes from J. Bernoulli and his studies of compund interest in 1683. e is sometimes referred to as Euler's number, not to be confused with Euler's constant. It was Euler who chose the symbol e to represent 2.718... . Why he chose the letter e is unclear but it's a lot shorter than the base of the natural logarithm, e^x is the inverse of ln x. Euler also proved that e is an irrational number. A proof that e is transcendental was given by C. Hermite in 1873.

Analysis

To calculate e^iπ we start with real analysis and f(x) = e^x. A function can be approximated around a point with a polynomial. The precision increases with the degree of the polynomial. This is the content of Taylor's theorem. If a real function f is k times differentiable at a then there is a function h defined around a with:

If f is infinitely differentiable at 0 then it has a Taylor series that is defined both on R and C.

The Cauchy-Hadamard theorem gives the radius of convergence R for a formal power series.

For proofs of these statements look them up in a book on analysis or the Wikipedia. If all of this is applied to f(x) = e^x with D(e^x) = e^x and f^(k)(0) = 1 then we get a Taylor series and a definition for e^z valid in all of C.

The expansion of e^x from real to complex numbers via Taylor series works for matrices as well, is perfectly calculable. Checking coefficients in Taylor series shows that e^z+w = e^z·e^w for complex numbers. This does not apply to exponentiation of matrices since the derivation depends on multiplication being commutative. Real and complex numbers form an algebraic structure called a field. A field is a set F with two operators ⊕ and ⊗ that fulfills:

Associativity	(a⊕b)⊕c = a⊕(b⊕c)	(a⊗b)⊗c = a⊗(b⊗c)
Commutativity	a⊕b = b⊕a	a⊗b = b⊗a
Existence of identities	∃a0∀a: a⊕a0 = a	∃a1∀a: a⊗a1 = a
Existence of inverses	∀a∃b: a⊕b = a0	∀a≠a0∃b: a⊗b = a1
Distributivity	a⊗(b⊕c) = (a⊗b)⊕(a⊗c)

The set R⨯R with operators (a₁,b₁)⊕(a₂,b₂) ≡ (a₁+a₂ , b₁+b₂) and (a₁,b₁)⊗(a₂,b₂) ≡ (a₁a₂−b₁b₂ , a₁b₂+b₁a₂) is a fancy way of defining the complex number field C with elements a+i·b and i² = −1.

Complex numbers can be visualized in the vector space R2 but bear in mind that C has more structure than R2, vectors can't be multiplied but complex numbers can. Going from R to C you gain algebraic closure but you lose order. R has an order relation with: a > b ⇒ a+c > b+c and a > 0, b > 0 ⇒ ab > 0. C can not be given such a total order.

To calculate e^z = e^a+ib = e^a·e^ib we need to know e^ix when x∈R. To do this you need to know the Taylor series for the trigonometric functions. With arguments given in radians D(sinx) = cosx and D(cosx) = −sinx:

This leads to polar representation of complex numbers, the best way to derive trigonometric identities.