Question

If \(z\) is a complex number, prove that there exists an \(r \geq 0\) and a complex number \(w\) with \(|w|=1\) such that \(z=r w\). Are \(w\) and \(r\) always uniquely determined by \(z\) ?

Short answer

Are these values of r and w unique for a given z? Answer: Yes, there exists an r ≥ 0 and a complex number w with |w| = 1 such that z = rw. The values of r and w are uniquely determined by z when z ≠ 0. However, they are not uniquely determined when z = 0.

Step-by-step solution

Write z in its polar form

Recall that any complex number z = a + bi (where a and b are real numbers) can be written in its polar form as: z = r(cos(θ) + i sin(θ)), where r = |z| and θ is the angle formed by the complex number in the complex plane. First, let's find r and θ in terms of a and b: r = |z| = √(a^2 + b^2) θ = arctan(b/a) Now, z can be written as: z = r(cos(θ) + i sin(θ))

Identify r and w

Let's consider w = cos(θ) + i sin(θ). We can clearly see that |w| = √(cos^2(θ) + sin^2(θ)) = 1, which satisfies the condition given for w. Now, we can rewrite z as: z = r * w This equation satisfies the given condition, that z should be the product of a complex number r with a non-negative real part and a complex number w with a magnitude equal to one.

Check for the uniqueness of r and w

To check if r and w are unique for a given z, we need to find if there is any other feasible pair of r and w that would result in the same z. If z = 0, then we can have r = 0, and w could be any complex number with a magnitude of 1 (since 0 * w = 0). In this case, w is not uniquely determined by z. If z ≠ 0, we can show that r and w are unique. Suppose there is another pair of r' and w' such that z = r' * w' with r' ≥ 0 and |w'| = 1. Then, r * w = r' * w'. By dividing both sides by r', we get: w * (r/r') = w' Since |w| = |w'| = 1, and r and r' are both non-negative, it follows that r = r' and w = w' for z ≠ 0. Thus, r and w are uniquely determined by z when z ≠ 0. In conclusion, there exists an r ≥ 0 and a complex number w with |w| = 1 such that z = rw. The values of r and w are uniquely determined by z when z ≠ 0, but they are not uniquely determined when z = 0.

Key concepts

Complex Number Polar Representation

When it comes to complex numbers, understanding their polar representation opens up a new perspective beyond the standard 'a + bi' form used on the Cartesian plane. Here's how it works:

In polar form, a complex number is defined by two parameters: the magnitude (or modulus) and the angle (or argument). The magnitude, denoted as 'r', is the distance of the complex number from the origin in the complex plane. To find the magnitude, you use the formula:
\r

\r
• \r \( r = |z| = \sqrt{a^2 + b^2} \)
\r

\rThe angle, denoted as \( \theta \), represents the counterclockwise rotation from the positive x-axis to the line segment connecting the origin to the complex number. This angle is calculated using the arctangent function:
\r

\r
• \r \( \theta = \arctan(\frac{b}{a}) \)
\r

\rGiven these two values, the polar representation of a complex number \( z \) can be compactly written as:
\r

\r
• \r \( z = r(\cos(\theta) + i \sin(\theta)) \)
\r

\rThis format is particularly helpful when dealing with multiplication or division of complex numbers, as well as finding powers and roots.

Magnitude of a Complex Number

The magnitude, or absolute value, of a complex number is central to its identity in the polar form. It is the real number that represents the distance between the point in the complex plane corresponding to the complex number and the origin.

To find the magnitude of a complex number \( z = a + bi \), simply use the Pythagorean theorem:
\r

\r
• \r\( |z| = r = \sqrt{a^2 + b^2} \)
\r

\rNote that the magnitude is always a non-negative number. It is zero only when both a and b are zero, which corresponds to the origin point. The concept of magnitude is crucial when solving problems in complex analysis and in various applications such as signal processing, where it may represent amplitude or strength.

Uniqueness of Polar Form

One might wonder if each complex number has a unique polar form. The answer depends on the context of the complex number we're examining. When the complex number \( z \) is non-zero, its polar form is indeed unique. The magnitude \( r \) and the angle \( \theta \) can be precisely determined, and the complex number \( w \), with magnitude 1, that corresponds to \( \cos(\theta) + i \sin(\theta) \), can also be uniquely found.

However, if we consider \( z = 0 \), the magnitude of the complex number is 0, and the angle is undefined—any angle would satisfy the polar form since a magnitude of 0 multiplied by any unit complex number still results in 0. This scenario presents an exception to the uniqueness of the polar form where the angle is not uniquely determined.

In summary, every non-zero complex number has a unique polar representation defined by its magnitude and angle. When the complex number is zero, although the magnitude is uniquely zero, the angle is not, leading to multiple valid representations.