Exponential Form of Complex Numbers
Complex numbers, the lifeblood of so many mathematical computations, can be portrayed not only in their familiar rectangular form but also in an alternative format: the exponential or polar form. We denote this as $$ z = re^{iθ} $$
Here's how to break it down:
- "r" is the magnitude of the complex number, figured out by the formula √(a² + b²)
- "θ" stands for the argument or phase of the complex number, determined by the calculation arctan(b/a)
- "i" is the unassuming imaginary unit
- "e" signifies the base of the natural logarithm (approximating to e=2.7182...)
From a visual standpoint, a complex number expressed in exponential form takes on a unique shape.
Now, you might wonder - why bother representing complex numbers in this manner? The power of this approach lies in its elegance; the exponential form has the uncanny ability to simplify various operations like division, multiplication, root extraction, and exponentiation among complex numbers.
Allow me to walk you through the derivation of this intriguing formula.
You're probably already acquainted with a complex number's Cartesian form, represented by:
$$ z = a+bi $$
Here, "a" refers to the coefficient of the real part, "b" indicates the coefficient of the imaginary part, and "i" is our trusty imaginary unit.
Now, this very same number can be painted in a new light, a trigonometric form.
$$ z = r \cdot ( \cos θ + i \cdot \sin θ ) $$
In this case, "r" is the magnitude and "θ" is the argument of the complex number.
Then comes Euler's formula, a beautiful piece of mathematical genius that ties the complex exponential function eix with trigonometric functions.
$$ e^{iθ} = \cos θ + i \cdot \sin θ $$
By weaving Euler's formula into the trigonometric form, we obtain the exponential form of a complex number:
$$ z = r \cdot ( \cos θ + i \cdot \sin θ ) $$
Transforms into
$$ z = r \cdot ( e^{iθ} ) $$
And simplifies to
$$ z = r e^{iθ} $$
In this representation, the magnitude "r" is akin to the distance from a point to the center of the complex plane, and "θ" is the angle created between the positive real axis and the point within the complex plane.
The utility of the exponential form becomes evident when performing multiplication or division among complex numbers. The operations, in this case, turn out to be more straightforward.
Imagine two complex numbers in exponential form:
$$ z_1 = r_1 e^{iθ_1} $$
$$ z_2 = r_2 e^{iθ_2} $$
Multiplying these two numbers involves simply calculating the product of the magnitudes and adding the arguments.
$$ z_1 \cdot z_2 = (r_1 \cdot r_1) e^{i (θ_1+θ_1)} $$
On the other hand, division requires determining the ratio of the magnitudes and subtracting the arguments.
$$ \frac{z_1}{z_2} = ( \frac{r_1}{r_2} ) e^{i (θ_1-θ_1)} $$
When it comes to raising a complex number to the power of "n", you raise the magnitude to the "n" power and multiply the argument by "n".
$$ z_1^n = ( z_1^n) e^{i (θ_1 \cdot n)} $$
With the exponential form, computations involving complex numbers become surprisingly more intuitive and easier to handle, leaving you free to tackle the mathematical challenges that truly matter.
