Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$1.6-2.7 i$$

Short answer

The number is \( 1.6 - 2.7i \) in Cartesian form and approximately \( 3.14 e^{-1.04i} \) in polar form. Plot the point (1.6, -2.7).

Step-by-step solution

Identify Real and Imaginary Parts

Recognize that the given complex number can be written in the form of \( x + i y \), where \( x \) is the real part and \( y \) is the imaginary part. For the number given, \( 1.6 - 2.7i \), the real part \( x = 1.6 \) and the imaginary part \( y = -2.7 \).

Convert to Polar Form

To convert the complex number to polar form, calculate the magnitude \( r \) and the angle \( \theta \): \[ r = \sqrt{x^2 + y^2} = \sqrt{(1.6)^2 + (-2.7)^2} = \sqrt{2.56 + 7.29} = \sqrt{9.85} \approx 3.14\] \[ \theta = \tan^{-1} \left(\frac{y}{x}\right) = \tan^{-1} \left(\frac{-2.7}{1.6}\right) \approx -1.04 \text{ radians} \] Hence, the polar form is \[ r e^{i\theta} = 3.14 e^{-1.04i} \]

Plot in Complex Plane

On the complex plane, the real part \( 1.6 \) is plotted on the x-axis, and the imaginary part \( -2.7 \) is plotted on the y-axis. The point \( (1.6, -2.7) \) gives the desired location of the complex number.

Key concepts

polar form of complex numbers

Complex numbers can be expressed in two common forms: standard form and polar form. The standard form is written as \(x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. However, the polar form represents a complex number in terms of its magnitude and angle. To convert a complex number \(x + iy\) to polar form, we need to find two components: magnitude \(r\) and angle \(\theta\).
Magnitude \(r\) is found using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \].
The angle \(\theta\) (in radians) can be found using the arctangent function: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \].
This angle indicates the direction of the complex number in the complex plane, and it is measured against the positive x-axis. Together, these give us the polar form \[ r e^{i\theta} \].
The benefit of using polar form is that it simplifies multiplication and division of complex numbers.

complex plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to represent complex numbers graphically. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
To plot a complex number on this plane, you take the real part \(x\) and plot it along the x-axis. The imaginary part \(iy\) is plotted along the y-axis. For example, the complex number \(1.6 - 2.7i\) is plotted at the point \( (1.6, -2.7) \).
This graphical representation helps in visualizing operations on complex numbers. For instance, addition of complex numbers is simple vector addition in the complex plane, and the modulus or magnitude represents the distance from the origin to the point. The argument or angle gives the direction from the positive real axis to the point.
Using the complex plane allows for a deeper understanding of functions involving complex numbers, and it is particularly useful in engineering and physics.

converting complex numbers

Converting complex numbers between standard form and polar form involves calculating their magnitude and angle. Let's walk through the conversion process using the number \(1.6 - 2.7i\).
First, identify the real part \(x = 1.6\) and the imaginary part \(y = -2.7\).
Next, calculate the magnitude \[ r = \sqrt{x^2 + y^2} = \sqrt{(1.6)^2 + (-2.7)^2} = \sqrt{2.56 + 7.29} = \sqrt{9.85} \approx 3.14 \]
Then, find the angle \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) = \tan^{-1} \left( \frac{-2.7}{1.6} \right) \approx -1.04 \text{ radians} \]
Thus, converting the number to polar form gives us \[ 3.14 e^{-1.04i} \].
This conversion process is useful, especially when dealing with multiplication and division of complex numbers, making the operations simpler to handle compared to the standard form.