Richard Feynman called Euler's formula "our jewel"

^{[2]} and "one of the most remarkable, almost astounding, formulas in all of mathematics."

^{[3]}
Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills by Paul J. Nahin, Professor Emeritus, University of New Hampshire, PhDEE

From

+ Plus magazine ... Living Mathematics Review by Lewis Dartnell

The hero of this book is Euler's formula:

*e*^{iπ} + 1 = 0

This simple equation has been widely considered through the last two centuries to be one of the most beautiful formulae of mathematics, and Nahin tells us why.

The constant *e* is the base of the natural logarithm (and you might have used it in calculations on radioactive decay in physics lessons, for example), whereas *&pi*, as we all know, is the ratio of a circle's circumference to its diameter. Both*e* and *&pi* are irrational numbers, that is, you could never write down all of their decimal places as they can be proven to continue for ever. The symbol *i* denotes the square root of -1, a value that does not even exist on the standard number line. Each of these three quantities is therefore curious in its own right, and they were originally developed within very different areas of maths. So how on Earth does *e*^{i&pi} equal *exactly* -1? It seems like the biggest fluke in the Universe — and this is part the formula's exquisite beauty. It is also fairly easy to derive the formula yourself, and the proof can be found in any textbook on complex numbers. And as Nahin's book shows, it is also one of the most influential formulae in the history of mathematics.

The book starts off gently enough, with an Introduction leading the reader through a few examples of the most fundamental skill in mathematics; constructing a proof. We see, for example, why √2 must be irrational through a simple proof by contradiction. The process of proving things is mostly ignored at GCSE and A-level, but really does give an eye-opening insight into how real mathematics is often like solving a intriguing puzzle, rather than slogging through homework exercises to practice the basic skills.

Through the following six chapters of the book, Nahin shows us many of the applications of imaginary numbers, Euler's formula, and other mathematical formulae and techniques that have been built on this eighteenth century foundation, both in pure and applied maths. Regular *Plus*readers might already know a little bit about some of these, and the sections in the book include: drawing regular prime number polygons, such as the 17-gon, using only a ruler and a compass; how to deconstruct any continuous function, such as a sound wave, into a set of sine waves — a technique that is crucial to modern gadgets like mp3 players. Nahin also shows us the maths of complex numbers lying behind the uncertainty principle of quantum mechanics, listening to the radio, and even how to build a telephone voice scrambler from simple electronics.

The book is well-written and illustrated with numerous diagrams and graphs. For those wanting a little more detail, or to follow up on the bibliography, the book is usefully cross-referenced and has hordes of end notes. But I do have one major warning for

*Plus* readers. Although this book is written to be more easy-reading and popularist than a text book, it is still very heavy-going. More importantly it is pitched at the level of university undergraduates. However, if you're enjoying maths A-level, then this book has a lot to offer, even if you don't understand everything. Every chapter begins with an interesting introduction on the history of a particular problem and the lives of the mathematicians involved, before heading into streams of equations and derivations. The final section of the book gives a detailed biography of the genius behind all of this mathematics through the ages,

Leonhard Euler. If you have a class project to write an essay on an influential mathematician, then Euler would certainly be an inspirational choice, and this final chapter a good place to start your research.

We will be discussing two equations. The ring illustrated above is Euler's Identity, which we will discuss second. Feynman's quote refers to Euler's formula, which we will discuss first. From Wikipedia:

**Euler's formula**, named after

Leonhard Euler, is a

mathematical formula in

complex analysis that establishes the deep relationship between the

trigonometric functions and the

complex exponential function. Euler's formula states that, for any

real number *x*,

where

*e* is the

base of the natural logarithm,

*i* is the

imaginary unit, and cos and sin are the

trigonometric functions cosine and sine respectively, with the argument

*x* given in

radians. This complex exponential function is sometimes called

**cis**(

*x*). The formula is still valid if

*x* is a

complex number, and so some authors refer to the more general complex version as Euler's formula.

^{[1]}
## History

It was

Bernoulli [1702] who noted that

And since

the above equation tells us something about complex logarithms. Bernoulli, however, did not evaluate the integral. His correspondence with Euler (who also knew the above equation) shows that he didn't fully understand logarithms. Euler also suggested that the complex logarithms can have infinitely many values.

Meanwhile,

Roger Cotes, in 1714, discovered

(where "ln" means

natural logarithm, i.e. log with base

*e*).

^{[4]} We now know that the above equation is only true

modulo integer multiples of

2π*i*, but Cotes missed the fact that a complex logarithm can have infinitely many values which owes to the periodicity of the trigonometric functions.

It was Euler (presumably around 1740) who turned his attention to the exponential function instead of logarithms, and obtained the correct formula now coined after his name. It was published in 1748, and his proof was based on the

infinite series of both sides being equal. Neither of these men saw the geometrical interpretation of the formula: the view of complex numbers as points in the

complex plane arose only some 50 years later (see

Caspar Wessel).

## Applications in complex number theory

This formula can be interpreted as saying that the function

*e*^{ix} traces out the

unit circle in the

complex number plane as

*x* ranges through the real numbers. Here,

*x* is the

angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in

radians.

The original proof is based on the

Taylor series expansions of the

exponential function *e*^{z} (where

*z* is a complex number) and of sin

*x* and cos

*x* for real numbers

*x* (see below). In fact, the same proof shows that Euler's formula is even valid for all

*complex* numbers

*z*.

A point in the

complex plane can be represented by a complex number written in

cartesian coordinates. Euler's formula provides a means of conversion between cartesian coordinates and

polar coordinates. The polar form reduces the number of

terms from two to one, which simplifies the mathematics when used in multiplication or powers of complex numbers. Any complex number

*z* =

*x* +

*iy* can be written as

where

- the real part
- the imaginary part
- the magnitude of
*z* - atan2(
*y*, *x*) .

is the

*argument* of

*z*—i.e., the angle between the

*x* axis and the vector

*z* measured counterclockwise and in

radians—which is defined

up to addition of 2π. Many texts write tan

^{-1}(

*y*/

*x*) instead of atan2(

*y*,

*x*) but this needs adjustment when

*x* ≤ 0.

Now, taking this derived formula, we can use Euler's formula to define the

logarithm of a complex number. To do this, we also use the definition of the logarithm (as the inverse operator of exponentiation) that

and that

both valid for any complex numbers

*a* and

*b*.

Therefore, one can write:

for any

*z* ≠ 0. Taking the logarithm of both sides shows that:

and in fact this can be used as the definition for the

complex logarithm. The logarithm of a complex number is thus a

multi-valued function, because

φ is multi-valued.

Finally, the other exponential law

which can be seen to hold for all integers

*k*, together with Euler's formula, implies several

trigonometric identities as well as

de Moivre's formula.

## Relationship to trigonometry

Euler's formula provides a powerful connection between

analysis and

trigonometry, and provides an interpretation of the sine and cosine functions as

weighted sums of the exponential function

**:**

The two equations above can be derived by adding or subtracting Euler's formulas

**:**

and solving for either cosine or sine.

These formulas can even serve as the definition of the trigonometric functions for complex arguments

*x*. For example, letting

*x* =

*iy*, we have

**:**

Complex exponentials can simplify trigonometry, because they are easier to manipulate than their sinusoidal components. One technique is simply to convert sinusoids into equivalent expressions in terms of exponentials.

After the manipulations, the simplified result is still real-valued. For example

**:**

Another technique is to represent the sinusoids in terms of the

real part of a more complex expression, and perform the manipulations on the complex expression. For example

**:**

This formula is used for recursive generation of cos(

*nx*) for integer values of

*n* and arbitrary

*x* (in radians).

## Other applications

In

differential equations, the function

*e*^{ix} is often used to simplify derivations, even if the final answer is a real function involving sine and cosine. The reason for this is that the complex exponential is the

eigenfunction of differentiation.

Euler's identity is an easy consequence of Euler's formula.

In

electrical engineering and other fields, signals that vary periodically over time are often described as a combination of sine and cosine functions (see

Fourier analysis), and these are more conveniently expressed as the real part of exponential functions with

imaginary exponents, using Euler's formula. Also,

phasor analysis of circuits can include Euler's formula to represent the impedance of a capacitor or an inductor.

## Definitions of complex exponentiation

The exponential function

*e*^{x} for real values of

*x* may be defined in a few different equivalent ways (see

Characterizations of the exponential function). Several of these methods may be directly extended to give definitions of

*e*^{z} for complex values of

*z* simply by substituting

*z* in place of

*x* and using the complex algebraic operations. In particular we may use either of the two following definitions which are equivalent. From a more advanced perspective, each of these definitions may be interpreted as giving the unique

analytic continuation of

*e*^{x} to the complex plane.

### Power series definition

For complex

*z*

Using the

ratio test it is possible to show that this

power series has an infinite

radius of convergence, and so defines

*e*^{z} for all complex

*z*.

### Limit definition

For complex

*z*

## Proofs

Various proofs of the formula are possible.

### Using power series

Here is a proof of Euler's formula using power series expansions as well as basic facts about the powers of

*i*:

and so on. Using now the power series definition from above we see that for real values of

*x*

In the last step we have simply recognized the

Taylor series for

*sin(x)* and

*cos(x)*. The rearrangement of terms is justified because each series is

absolutely convergent.

### Using calculus

Treating

*i* as a constant, albeit an imaginary constant, note that

Then define the function

Because the product rule holds for complex valued functions of a real variable for the same reason as in the real case, the

derivative of ƒ(

*x*) according to the

product rule is:

Therefore, ƒ(

*x*) must be a

constant function in

*x*. Because ƒ(0) = 1 by inspection, ƒ(

*x*) = 1, giving

Multiplying both sides by cos

*x* +

*i* sin

*x*, we obtain

### Using differential equations

Here is another proof that follows from the differential identity above. Define a new function ƒ(

*x*) of the real variable

*x* as

Then we may check that

Thus ƒ(

*x*) and

*e*^{ix} satisfy the same first-order

ordinary differential equation (here the complex values are considered as points in the plane ℝ

^{2}). Note also that both functions are equal to 1 at

*x* = 0, then by the uniqueness of solutions to ordinary differential equations they must be equal everywhere (see

Picard–Lindelöf theorem and note the comments concerning global uniqueness in the proof section there).

## See also

## References

**^** Moskowitz, Martin A. (2002). *A Course in Complex Analysis in One Variable*. World Scientific Publishing Co.. pp. 7. ISBN 981-02-4780-X.
**^** Feynman, Richard P. (1977). *The Feynman Lectures on Physics, vol. I*. Addison-Wesley. pp. 22–10. ISBN 0-201-02010-6.
**^** Feynman, Richard P. (1977). *The Feynman Lectures on Physics, vol. I*. Addison-Wesley. pp. 22–1. ISBN 0-201-02010-6.
**^** John Stillwell (2002). *Mathematics and Its History*. Springer.

## External links

The

exponential function *e*^{z} can be defined as the

limit of

(1 + *z*/*N*)^{N}, as

*N* approaches infinity, and thus

*e*^{iπ} is the limit of

(1 + *iπ/N*)^{N}. In this animation

*N* takes various increasing values from 1 to 100. The computation of

(1 + *iπ/N*)^{N} is displayed as the combined effect of

*N* repeated multiplications in the

complex plane, with the final point being the actual value of

(1 + *iπ/N*)^{N}. It can be seen that as

*N* gets larger

(1 + *iπ/N*)^{N} approaches a limit of −1.

In

analytical mathematics,

**Euler's Identity**, named for the Swiss-German

mathematician Leonhard Euler, is the equality

where

- is Euler's number, the base of natural logarithms,
- is the imaginary unit, which satisfies
*i*^{2} = −1, and - is pi, the ratio of the circumference of a circle to its diameter.

Euler's Identity is also sometimes called

**Euler's Equation**.

## Beauty

Euler's identity is considered by many to be remarkable for its

mathematical beauty. These three basic

arithmetic operations occur exactly once each:

addition,

multiplication, and

exponentiation. The identity also links five fundamental

mathematical constants:

- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The number
*π*, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics (π = 3.14159265...)
- The number
*e*, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828...). Both π and e are transcendental numbers.
- The number
*i*, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.

Furthermore, in

algebra and other areas of mathematics,

equations are commonly written with zero on one side of the

equals sign.

A poll of readers conducted by

*The Mathematical Intelligencer* magazine named

Euler's Identity as the "most beautiful theorem in mathematics".

^{[1]} Another poll of readers that was conducted by

*Physics World* magazine, in 2004, chose Euler's Identity tied with

Maxwell equations (of

electromagnetism) as the "greatest equation ever".

^{[2]}
An entire 400-page mathematics book,

*Dr. Euler's Fabulous Formula* (published in 2006), written by Dr. Paul Nahin (a

Professor Emeritus at the

University of New Hampshire), is devoted to Euler's Identity. This monograph states that Euler's Identity sets "the gold standard for mathematical beauty."

^{[3]}
Constance Reid claimed that Euler's Identity was "the most famous formula in all mathematics."

^{[4]}
The

mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician.

^{[5]}
After proving Euler's Identity during a lecture,

Benjamin Peirce, a noted American 19th century

philosopher/mathematician and a professor at

Harvard University, stated that "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth."

^{[6]}
The

Stanford University mathematics professor, Dr.

Keith Devlin, said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."

^{[7]}
## Derivation

Euler's formula for a general angle

The identity is a special case of

Euler's formula from

complex analysis, which states that

for any

real number *x*. (Note that the arguments to the

trigonometric functions

*sine* and

*cosine* are taken to be in

radians, and not in degrees.) In particular,

Since

and

it follows that

which gives the identity

## Generalizations

Euler's Identity is actually a special case of the more general identity that the

*n*th

roots of unity, for

*n* > 1, add up to 0:

Euler's identity is the case where

*n* = 2.

In another field of mathematics, by using

quaternion exponentiation, one can show that a similar identity also applies to quaternions:

## Attribution

While Euler wrote about his formula that relates

*e* with

*cosine* and

*sine* terms, in the field of complex numbers, there is no known record of Euler's actually stating or deriving the simplified identity equation itself.

Furthermore, Euler's formula was probably known before the life of Euler.

^{[8]} (If so, then this usage would be an example of

Stigler's law of eponymy.) Thus, the question of whether or not this identity should be attributed to Euler is unanswerable.

## See also

## Notes

**^** Nahin, 2006, p.2–3 (poll published in the summer 1990 issue of the magazine).
**^** Crease, 2004.
**^** Cited in Crease, 2007.
**^** Reid.
**^** Derbyshire p.210.
**^** Maor p.160 and Kasner & Newman p.103–104.
**^** Nahin, 2006, p.1.
**^** Sandifer.

## References

- Crease, Robert P., "The greatest equations ever", PhysicsWeb, October 2004 (registration required).
- Crease, Robert P. "Equations as icons," PhysicsWeb, March 2007 (registration required).
- Derbyshire, J.
*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics* (New York: Penguin, 2004).
- Kasner, E., and Newman, J.,
*Mathematics and the Imagination* (Bell and Sons, 1949).
- Maor, Eli,
*e: The Story of a number* (Princeton University Press, 1998), ISBN 0-691-05854-7
- Nahin, Paul J.,
*Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills* (Princeton University Press, 2006), ISBN 978-0691118222
- Reid, Constance,
*From Zero to Infinity* (Mathematical Association of America, various editions).
- Sandifer, Ed, "Euler's Greatest Hits", MAA Online, February 2007.