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. $$2.8 e^{-i(1.1)}$$

Short answer

The number in Cartesian form is \(1.27 - 2.50i\). Plot at \((1.27, -2.50)\) in the complex plane.

Step-by-step solution

- Understanding the form

Identify this form is the polar form of the complex number, where \(r=2.8\) and \(\theta=-1.1\).

- Convert to Cartesian Form

The Cartesian form of a complex number is given by: $$r e^{i \theta} = r(\cos \theta + i \sin \theta)$$. Substituting the values, we have: $$2.8 e^{-i(1.1)} = 2.8(\cos(-1.1) + i \sin(-1.1)).$$

- Calculate Cosine and Sine

Compute the values of \(\cos(-1.1)\) and \(\sin(-1.1)\) (Recall that \(\cos(-\theta)=\cos(\theta)\) and \(\sin(-\theta)=-\sin(\theta)\)): $$\cos(-1.1) = 0.4536$$ and $$\sin(-1.1) = -0.8912$$.

- Multiply by the Magnitude

Substitute the values back into the equation: $$2.8(0.4536 - 0.8912i) = 2.8 \cdot 0.4536 + 2.8 \cdot (-0.8912i) = (1.27 - 2.50i) $$. Therefore, the Cartesian form is \(1.27-2.50i\).

- Plotting in the Complex Plane

To plot \(1.27-2.50i\) in the complex plane, locate the point \((1.27, -2.50)\). The real part \(1.27\) is on the x-axis and the imaginary part \(-2.50\) is on the y-axis.

Key concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in different forms, one of which is the polar form. In polar form, a complex number is represented as \( r e^{i \theta} \), where \( r \) is the magnitude (or modulus) of the complex number and \( \theta \) is the phase (or argument). The magnitude \( r \) tells you how far the point is from the origin, and the phase \( \theta \) indicates the angle the line to the point makes with the positive real axis. - Magnitude \( r \) is calculated as \( |z| = \sqrt{a^2 + b^2} \). - Phase \( \theta \) is calculated as \( \theta = \tan^{-1} \left( \frac{b}{a} \right) \). Imagine you have a complex number given in polar form, for example, \( 2.8 e^{-i(1.1)} \). Here, the magnitude \( r = 2.8 \) and the phase \( \theta = -1.1 \). Knowing this form allows us to convert it into the Cartesian form, which is often more intuitive for visual representation.

Cartesian Form of Complex Numbers

The Cartesian form of a complex number expresses it as a sum of its real and imaginary parts: \( z = a + bi \).You can think of \( a \) as the x-coordinate and \( b \) as the y-coordinate on the complex plane. To convert from polar form \( r e^{i \theta} \) to Cartesian form, we use Euler's formula: \[ r e^{i \theta} = r( \cos( \theta ) + i \sin( \theta ) ) \]. For example, to convert \( 2.8 e^{-i(1.1)} \) to Cartesian form: 1. Take \( r = 2.8 \) and \( \theta = -1.1 \). 2. Compute \( \cos(-1.1) = 0.4536 \) and \( \sin(-1.1) = -0.8912 \). 3. Plug these values into the equation: \( 2.8(0.4536 + i(-0.8912)) \). 4. Finally, multiply by the magnitude: \( 2.8 \cdot 0.4536 + 2.8 \cdot (-0.8912i) = 1.27 - 2.50i \). So, the Cartesian form of \( 2.8 e^{-i(1.1)} \) is \( 1.27 - 2.50i \).

Complex Plane Plotting

Plotting a complex number in the complex plane helps to visualize the number more clearly. The complex plane can be thought of as a coordinate system where the x-axis (real axis) represents the real part, and the y-axis (imaginary axis) represents the imaginary part of the complex number. To plot \( 1.27 - 2.50i \): 1. Locate the real part, \( 1.27 \), on the x-axis. 2. Find the imaginary part, \( -2.50 \), on the y-axis. 3. Plot the point \( (1.27, -2.50) \). Imagine an arrow starting from the origin (0,0) to this point. The arrow's length corresponds to the magnitude \( r \) of the complex number, and the angle it makes with the positive x-axis represents the phase \( \theta \). This visual approach makes understanding the relationship between the number's magnitude and angle easier, providing an intuitive grasp of complex numbers.