Question

Write the following in polar form, \(z=r e^{i \theta}\). a. \(i-1\). b. \(-2 i\). c. \(\sqrt{3}+3 i\).

Short answer

\(i-1 = \sqrt{2} e^{i \frac{3\pi}{4}}\), \(-2i = 2 e^{i \frac{3\pi}{2}}\), and \(\sqrt{3} + 3i = 2\sqrt{3} e^{i \frac{\pi}{3}}\).

Step-by-step solution

Compute the modulus

Find the modulus (or magnitude) of the complex number using the formula \(r=\sqrt{a^2 + b^2}\). For example, for the complex number \(i-1\), \(a=-1\) and \(b=1\). Hence, \(r=\sqrt{(-1)^2 + (1)^2} = \sqrt{2}\).

Calculate the Argument

Calculate the argument (or angle) of the complex number with the formula \(\theta = \arctan(\frac{b}{a})\) for \(a>0\), and \(\theta = \arctan(\frac{b}{a})+\pi\) for \(a<0\). For the complex number \(i-1\), \(a=-1\) and \(b=1\). Since \(a\) is negative, we get \(\theta = \arctan(\frac{1}{-1})+\pi = \frac{3\pi}{4}\).

Write the polar form

The polar form of a complex number is \(z=r e^{i \theta}\). For the complex number \(i-1\), \(r=\sqrt{2}\) and \(\theta = \frac{3\pi}{4}\), so the polar form is \(z=\sqrt{2} e^{i \frac{3\pi}{4}}\).

Repeat steps for all complex numbers

Repeat Steps 1, 2 and 3 for the remaining complex numbers. For \(-2i\), \(r=2\) and \(\theta = \frac{3\pi}{2}\), so the polar form is \(z=2 e^{i \frac{3\pi}{2}}\). For \(\sqrt{3} + 3i\), \(r=2\sqrt{3}\) and \(\theta = \frac{\pi}{3}\), so the polar form is \(z=2\sqrt{3} e^{i \frac{\pi}{3}}\).

Key concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in two main formats: standard form and polar form. Polar form presents a complex number as a product of its modulus and a complex exponential. It is denoted as \(z = r e^{i \theta}\), where \(r\) is the modulus and \(\theta\) is the argument. This form is particularly useful when multiplying or dividing complex numbers, as it simplifies the calculations significantly.

• The modulus \(r\) tells us the distance from the origin on the complex plane.
• The argument \(\theta\) gives the angle between the positive real axis and the line representing the complex number.

To convert a complex number from standard to polar form, first calculate the modulus and argument, then express the number as \(z = r e^{i \theta}\). This transformation allows one to easily visualize complex numbers on the polar plot.

Modulus of a Complex Number

The modulus of a complex number measures its "size" or "magnitude". For a complex number \(a + bi\), the modulus is calculated as \(r = \sqrt{a^2 + b^2}\). This formula applies the Pythagorean Theorem to find the distance of the complex number from the origin in the complex plane.

• For example, for the number \(i - 1\), since \(a = -1\) and \(b = 1\), the modulus is \(\sqrt{(-1)^2 + (1)^2} = \sqrt{2}\).
• For \(-2i\), where \(a = 0\) and \(b = -2\), the modulus is \(\sqrt{0^2 + (-2)^2} = 2\).
• For \(\sqrt{3} + 3i\), \(a = \sqrt{3}\) and \(b = 3\), resulting in a modulus of \(\sqrt{(\sqrt{3})^2 + (3)^2} = 2\sqrt{3}\).

Calculating the modulus is an essential step in expressing complex numbers in polar form, as it provides the magnitude component.

Argument of a Complex Number

The argument of a complex number is the angle it makes with the positive real axis on the complex plane. It is denoted as \(\theta\), and often calculated using the formula \(\theta = \arctan\left(\frac{b}{a}\right)\). Adjustments may be needed based on the signs of \(a\) and \(b\) to ensure the correct quadrant is used.

• For \(i - 1\), since \(a = -1\) and \(b = 1\), we find \(\theta = \arctan\left(\frac{1}{-1}\right) + \pi = \frac{3\pi}{4}\), placing it in the second quadrant.
• For \(-2i\), \(a = 0\) and \(b = -2\), the argument is \(\theta = \frac{3\pi}{2}\), directly on the negative imaginary axis.
• For \(\sqrt{3} + 3i\), \(a = \sqrt{3}\) and \(b = 3\), giving \(\theta = \frac{\pi}{3}\) in the first quadrant.

The argument gives the orientation of the complex number on the plane, crucial for the polar representation.