Question

How does it relate \[e\] to \[\pi \] ?

Answer 1

Hint: The relationship between \[e\] and \[i\] can illustrated using the Euler’s Equation, whose mathematical form is \[{e^{i\pi }} + 1 = 0\]

Complete Step by Step Solution:
The relationship between \[e\]and\[i\], can be shown using the Euler’s’ Equation, \[{e^{i\pi }} + 1 = 0\]
Where
\[e\] is Euler's number, the base of natural logarithms,
\[i\] is the imaginary unit, which by definition satisfies , and
\[\pi \] is pi, the ratio of the circumference of a circle to its diameter.
Its geometric interpretation can be understood as-
Any complex number \[z = x + iy\] can be represented by the point \[(x,y)\]on the complex plane. This point can also be represented in polar coordinates as \[(r,\theta )\], where r is the absolute value of z (distance from the origin), and \[\theta \]is the argument of z (angle counterclockwise from the positive x-axis). By the definitions of sine and cosine, this point has cartesian coordinates of\[\left( {r{{\cos }^{}}\theta ,r{{\sin }^{}}\theta } \right)\], implying that\[z = r\left( {{{\cos }^{}}\theta + i{{\sin }^{}}\theta } \right)\]. According to Euler's formula, this is equivalent to saying\[z = {r^{i\theta }}\].
Euler's identity says that\[ - 1 = {e^{i\pi }}\]. Since \[{e^{i\pi }}\] is \[{r^{i\theta }}\] for \[r = 1\] and\[\theta = \pi \], this can be interpreted as a fact about the number −1 on the complex plane: its distance from the origin is 1, and its angle from the positive x-axis is \[\pi \] radians.
Additionally, when any complex number z is multiplied by\[\theta \], it has the effect of rotating z counterclockwise by an angle of on the complex plane. Since multiplication by −1 reflects a point across the origin, Euler's identity can be interpreted as saying that rotating any point \[\pi \] radians around the origin has the same effect as reflecting the point across the origin.

Note: Euler’s formula explains the relationship between complex exponentials and trigonometric functions.
The Euler's equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
This is an important equation with its application in AC (Alternating Current) Analysis.
Euler’s law states that ‘for any real number $x,{e^{ix}} = \cos x + i\sin x.$ where, $x = $ angle in the radians, $i = $ imaginary unit e= base of natural logarithm.