Question

Express \(-5+5 i\) in polar form and in exponential form.

Short answer

\(-5 + 5i = 5\sqrt{2}(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) = 5\sqrt{2}e^{i\frac{3\pi}{4}}\).

Step-by-step solution

Understand the Components for Polar Form

To convert a complex number into polar form, identify the magnitude and the angle. The complex number is given as \(-5 + 5i\).

Calculate the Magnitude

The magnitude \(|z|\) of the complex number \(-5 + 5i\) can be calculated using the formula \(|z| = \sqrt{x^2 + y^2}\). Here, \(x = -5\) and \(y = 5\). Hence, \(|z| = \sqrt{(-5)^2 + (5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}\).

Calculate the Argument

The argument (angle) \(\theta\) is determined using \(\tan^{-1}(\frac{y}{x})\). Here, \(\theta = \tan^{-1}(\frac{5}{-5}) = \tan^{-1}(-1)\). Considering the quadrant where \(-5 + 5i\) is located (second quadrant), \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Express in Polar Form

With the magnitude and the argument determined, the polar form of \(-5 + 5i\) is \(|z|(\cos \theta + i\sin \theta)\). Hence, \(-5 + 5i = 5\sqrt{2}\left(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4}\right)\).

Convert to Exponential Form

The exponential form of a complex number in polar form is \(|z|e^{i\theta}\). Therefore, converting \(-5 + 5i\) into exponential form gives \(-5 + 5i = 5\sqrt{2} e^{i\frac{3\pi}{4}}\).

Key concepts

Magnitude of Complex Numbers

The magnitude of a complex number, also known as the modulus, tells us how far the number is from the origin in the complex plane.
It is essentially the 'length' of the vector representing the complex number.
For a complex number in the form of \(a + bi\), the magnitude is found using the formula \( |z| = \sqrt{a^2 + b^2} \).
In our exercise, we have \( -5 + 5i \).
To find the magnitude:
\[-5 + 5i = \sqrt{((-5)^2 + (5)^2)} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]
This means the magnitude is 5√2.

Argument of Complex Numbers

The argument of a complex number, often denoted as \( \theta \), indicates the angle the number forms with the positive real axis in the complex plane.
For a complex number \( a + bi \), the argument can be calculated using the inverse tangent function:
\( \theta = \tan^{-1} \left( \frac{b}{a} \right) \)
For \( -5 + 5i \), we need to determine the angle formed with the negative real axis.
Thus, \( \theta = \tan^{-1} \left( \frac{5}{-5} \right) = \tan^{-1}(-1) \).
Because our complex number lies in the second quadrant, we adjust the angle: \( \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).

Polar Form of Complex Numbers

The polar form of a complex number represents it in terms of its magnitude and argument.
It is written as:
\( |z| \left( \cos \theta + i \sin \theta \right) \).
Previously, we found that the magnitude \ is \( 5\sqrt{2} \) and the argument \ \( \theta \) is \ \( \frac{3\pi}{4} \).
Thus, in polar form, our complex number is:
\[-5 + 5i = 5\sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \].
This form helps us easily visualize the number's position and rotation in the complex plane.

Exponential Form of Complex Numbers

The exponential form of a complex number relies on Euler's formula, \( e^{i\theta} = \cos \theta + i\sin \theta \), which links complex exponentials to trigonometry.
It offers a more compact way to express the polar form.
For our polar form:
\[-5 + 5i = 5\sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]
We convert it by employing Euler's formula:
\[-5 + 5i = 5\sqrt{2} e^{i\frac{3\pi}{4}} \]
This simplifies calculations, especially for multiplication and division of complex numbers.