Problem 38 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 4 $$

Step-by-Step Solution

Answer

The polar form of 4 is \(4(\cos 0 + i\sin 0)\).

Step 1: Identify the Real and Imaginary Parts

The complex number given is just 4, which can be thought of as \(4 + 0i\). This means that the real part is 4, and the imaginary part is 0.

Step 2: Calculate the Magnitude

The magnitude (or modulus) of a complex number \(a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). For our number, this is \(|z| = \sqrt{4^2 + 0^2} = \sqrt{16} = 4\).

Step 3: Find the Argument \(\theta\)

The argument \(\theta\) of a complex number is calculated using \(\tan^{-1}(\frac{b}{a})\). Since the imaginary part \(b = 0\), our complex number lies on the positive real axis. Therefore, \(\theta = 0\).

Step 4: Write in Polar Form

Using the polar form representation \(z = r(\cos \theta + i\sin \theta)\), we have \(r = 4\) and \(\theta = 0\). So in polar form, \(z = 4(\cos 0 + i\sin 0)\). This simplifies to \(z = 4\left(1 + 0i\right) = 4\).

Key Concepts

Polar Form

The polar form of a complex number is an elegant way of expressing a number in terms of its magnitude and angle relative to the positive real axis. It is often used in scenarios where multiplication or division of complex numbers is involved, due to its ability to simplify these operations. A complex number in polar form is given by: \[ z = r(\cos \theta + i\sin \theta) \] where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle) of the complex number. In simpler terms, think of \( r \) as how "far" the number is from the origin and \( \theta \) as the direction of the line connecting the origin to the point representing the complex number. Polar representation is particularly useful in engineering and physics.

Magnitude of Complex Numbers

The magnitude of a complex number can be visualized as the distance from the origin on the complex plane. For a complex number \( a + bi \), the magnitude is computed as: \[ |z| = \sqrt{a^2 + b^2} \] This formula comes directly from the Pythagorean theorem, as we are essentially calculating the hypotenuse of a right-angled triangle where \( a \) and \( b \) are the other two sides.

• In our example, the complex number 4 can be rewritten as \( 4 + 0i \), so its magnitude is simply \( 4 \).
• If the real or imaginary parts were different from zero, we would use the same method to find the magnitude.

Understanding magnitude helps determine how 'large' a complex number is, a critical parameter in various mathematical applications.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis, moving counterclockwise towards the complex number on the complex plane. It helps define the direction or orientation of the complex number in polar coordinates. Mathematically, the argument \( \theta \) is calculated as: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] However, when the imaginary part is zero (as in our number \(4 + 0i\)), the number lies directly on the positive real axis, and thus the argument is \(\theta = 0\). To summarize:

• If \(\theta = 0\), the complex number is on the positive real axis.
• If \(b\) were negative or different, \(\theta\) would have to be calculated differently, often requiring adjustments to ensure it lies between \(0\) and \(2\pi\).

Recognizing the argument is key in converting complex numbers to their polar form.

Real and Imaginary Parts

Complex numbers are built from two parts: the real part and the imaginary part. For any complex number \( a + bi \), \(a\) is the real part and \(bi\) is the imaginary.

• In this problem, identifying the real and imaginary parts helps simplify the task of expressing the number in polar form.
• From the given number \(4 + 0i\), it's clear that the real part is 4 and the imaginary part is 0.

These components define the position of the complex number on the complex plane, setting the foundation for determining the magnitude and argument. Understanding these elements makes it easier to work with and manipulate complex numbers.