## Euler rule is almost mystical

In his wonderful book “Road to Reality” Roger Penrose mentions that the Euler rule is “almost mystical” because it relates the 5 fundamental numbers 0, 1, i, π and e to each other.

The Euler rule

$\displaystyle e^{i \pi } + 1 = 0$

is a special case of the Cotes-Euler identity

$\displaystyle e^{i \phi } = cos(\phi) + i sin(\phi)$

if $\phi=\pi$ then the Cotes-Euler identity would be written as

$\displaystyle e^{i \pi } = cos(\pi) + i sin(\pi)$

we know that $cos(\pi)=-1$ and $sin(\pi)=0$; the Euler rule follows.

The Cotes-Euler identity is a powerful tool. It makes the multiplication and division of complex numbers much easier. Instead of using trigonometry which involves awkward computations we use the simple algebra of exponential numbers. For example

C1 = r1[cos(φ1) + i sin (φ1)]

C2 = r2[cos(φ2) + i sin (φ2)]

We multiply these two complex numbers easily

C1 C2 = r1 e1  r2 e2 = r1r2ei(φ12)

The division operation is very easy too

C1 / C2 = r1 e1 / r2 e2 = (r1/r2) ei(φ12)

In the multiplication operation we add the phases, and in the division operation we subtract the phases in the exponent.

Computing the roots of a complex number becomes very easy too

Square root of C= (r)1/2 e(iφ/2)

Cubic root of C= (r)1/3 e(iφ/3)

N’th root of C= (r)1/N e(iφ/N)

In the N’th root operation we divide the phase by N.

Similarly, computing the powers of a complex number is very easy

N’th power of C= (r)N e(iNφ)

In the N’th power operation we multiply the phase by N.

In multiplication, division, root and power operations we simply applied the rules of  exponential numbers

ea eb = e(a+b)

ea/eb = e(a-b)

The (natural) logarithm of a complex number is easily computed as well

ln(C)= ln(r e) = ln(r) + iφ

Here we simply applied the ‘multiplication-to-addition’ rule of logarithms.

ln(ab) = ln(a) + ln(b)