Why Squaring a Complex Number Doubles Its Angle

The complex plane and the properties of complex numbers are fascinating subjects in mathematics. One of the most intriguing properties of complex numbers is how the operation of squaring affects their angles. This article delves into the reasons why squaring a complex number doubles its angle, using concepts from polar form and Euler's formula.

Understanding Complex Numbers in Polar Form

Any complex number can be expressed in polar form, which is a compact and powerful way to represent it. A complex number z is written as:

z rcosθ isinθ reiθ

Here, r is the modulus (or magnitude) of the complex number, and θ is the argument (or angle) in radians.

Let's explore what happens when we square a complex number in polar form:

z2 (reiθ)2 r2e2iθ r2cos(2θ) isin(2θ)

This transformation shows two key points:

The modulus r is squared, resulting in r2. The angle θ is doubled, resulting in 2θ.

This doubling of the angle is a consequence of the rules of exponentiation in the context of Euler's formula, which connects complex exponentials to trigonometric functions.

Multiplication and Angle Doubling

Another way to understand why squaring a complex number doubles its angle is through the concept of multiplication. When we square a complex number, we are essentially applying the transformation to itself. This can be visualized as follows:

Multiplying by a complex number z means scaling by its length and rotating by its angle. Squaring a complex number, therefore, scales the length by a factor of r2 and doubles the angle, resulting in 2θ.

For instance, consider a complex number z abi, which can be expressed in polar form as z reiθ.

Graphical Representation and Trigonometric Interpretation

Graphing a complex number z abi in an xy-plane with the real x-axis and the imaginary y-axis, we can think of it as a right triangle with the points of interest at the origin (0,0) and at (a,b).

The third vertex of this right triangle can be either (a,0) or (0,b), but for our purposes, we want to define the triangle such that θ spans the arc between the line segments (0,0) to (a,0) and (0,0) to (a,b).

Using Euler's formula: cosθ isinθ eiθ, we can express z as:

abi reiθ, where r √(a2 b2)

To get θ from a and b, we need to note that our circle spans 2π, while arctanθ has period π. Therefore, we rectify this by knowing that we want:

arctanθ when b > 0, and π - arctanθ when b Choosing arctanθ for a > 0 and π - arctanθ for a . If a 0 b, then r 0 and θ is trivial.

The square of the complex number in polar form is then:

bi reiθ, and squaring it gives:

(bi}2) (reiθ)2 r2e2iθ r2ei2πθ r2ei2θ

It is evident that squaring the angle θ doubles it, which is a direct result of the exponential relationship in Euler's formula.

Conclusion

The property that squaring a complex number doubles its angle can be explained through the polar form of complex numbers and Euler's formula. This fundamental insight has deep implications in various areas of mathematics, including signal processing, quantum mechanics, and more.