$$e^{i\pi} + 1 = 0$$
Wooah, slow down! How can we believe an
equation like this when it doesn't even make sense to raise
a number to the power of i. Or does it?
We learn at a young age that xn
means multiplying x by itself n
times, and indeed that interpretation does not make sense for
an exponent that is an imaginary number.
But in mathematics, there are other ways to represent this
exponentiation function, specifically as a power series,
which gives the same result for real numbers as what we are
taught in grade school.
And with the power series representation, we can put
i in the exponent.
This brave idea opened mathematics to many new directions of
research and beautiful results.
Despite the new concept for imaginary exponents, nobody would have ever expected that the the five most important numbers in all of mathematics are related in such a simple, beautiful equation. Each of these numbers - 0, 1, e, i, and π — is independently important as follows:
| Number | What it is | Why it matters |
|---|---|---|
| 0 | The additive identity and multiplicative annihilator | Adding it to any number leaves that number unchanged, while multiplying it brings the product to 0. |
| 1 | The multiplicative identity | Multiplying by it leaves any number unchanged, and it constructs every natural number via addition. |
| e | The base of natural growth, ≈ 2.71828... | Discovered by Jacob Bernoulli as the limit of (1 + 1/n)n while studying compound interest, but this constant turns up everywhere. In calculus, e is the number where exponential functions grow at a rate equal to its own value. |
| i | The imaginary unit, √−1 | i brings about complex numbers, which brings about the fundamental
theorem of algebra, making it possible to factor any polynomial into linear components.
|
| π | The ratio of a circle's circumference to its diameter, ≈ 3.14159... | It not only is involved in the circumference, but also the area of the circle. π
is another number that shows up in many surprising ways in mathematics: just keep
reading Pi Facts to see other examples. |
History
Leonhard Euler (1707–1783) was born in the Swiss Confederacy (Switzerland). Throughout his lifetime Euler became the most prolific and one of the greatest mathematicians ever, achieving this despite eyesight problems for more than half of his life. At the age of 32, Euler lost eyesight in on eye and was completely blind by age 64 yet he continued to publish lots of mathematical research, with help from family members and assistants to write down his ideas.
This result was in his 1748 book Introductio in analysin infinitorum,
at the age of 31 (still having full eyesight).
Euler used Taylor expansions, a form of power series, for the exponential
function and noticed certain patterns that came out when he substituted imaginary
exponents.
It came down to what is known as Euler's formula:
$$ e^{i x} = \cos{x} + i \sin{x}$$
Euler's identity pops out of Euler's formula by substituting x=π,
since cos(π) = -1 and sin(π) = 0.
Show the math
We start with the Taylor series for the exponential function:
$$e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$
Substitute x = iθ and multiply out the powers of i,
and then regroup so that those terms with an i in them are together,
we arrive at:
The first term after the equal sign is exactly the Taylor expansion
of cos(θ) and the second is i⋅sin(θ) (see any of the references below), so the above
is:
Now set θ = π:
and moving the -1 to the other side gives the result:
Remark: We have over-simplified the math by skipping over the radius of convergence of the power series, but that is important and the interested reader should verify the convergence.