Problem 23 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ -3+4 i $$

Step-by-Step Solution

Answer

The complex number can be plotted at the point (-3,4) on the complex plane and its polar form equivalent is \[5 \, cis(-126.87^\circ)\] or \[5 \, cis(2.0344 \, rad)\].

Step 1: Plot the Complex Number

Plot the complex number \(-3 + 4i\) in the complex plane. This can be done similarly to plotting points on the 2-dimensional plane. The real part of the complex number (-3) represents the x-coordinate and the imaginary part (4) represents the y-coordinate. We're looking to plot this point on the plane.

Step 2: Translate into Polar Coordinates

Now we have to convert our Cartesian coordinates into polar coordinates. The magnitude (r) of a complex number can be found by the formula \[r = \sqrt{x^2 + y^2}\], and the argument or angle in polar form (θ) can be found by \[θ = \arctan(\frac{y}{x})\]. Substitute x=-3 and y=4 into the formulas.

Step 3: Calculation of Modulus and Argument

In this step, we calculate the magnitude(r) using the formula. \[r = \sqrt{(-3)^2 + 4^2}\] and the argument (θ) \[θ = \arctan(\frac{4}{-3})\]. Ensure to correct for quadrant by adding π to the angle as we are in the second quadrant (x is negative and y is positive).

Step 4: Final Answer In Polar Form

The final answer should be in the polar form: \[z = r(\cos θ + i \sin θ)\]. So, put the calculated r and θ into the formula. Also, note that \( \cos( θ ) + i \sin( θ ) \) can also be written as \( cis ( θ ) \).

Key Concepts

Polar Coordinates

Complex numbers can be expressed in both Cartesian and polar forms. To convert a complex number into polar coordinates, we need to find its modulus and argument. This form is particularly useful for multiplication and division of complex numbers. In polar coordinates, any complex number is represented as \(z = r(cos\,θ + i\,sin\,θ)\), where \(r\) is the modulus, and \(θ\) (theta) is the argument.

Think of polar coordinates like plotting a point on a circle. Instead of using a traditional x and y coordinate (as in Cartesian), we use an angle and a distance from the origin. This is helpful when dealing with rotations and angles.

In the example of \(-3 + 4i\), the modulus \(r\) describes how far the point is from the origin, and the argument \(θ\) gives the angle relative to the positive x-axis. In this format, it's easier to visualize complex numbers geometrically.

Cartesian Coordinates

Cartesian coordinates are the most common system to plot numbers on a plane, using a two-number format \((x, y)\). For a complex number, like \(-3 + 4i\), this means we plot the real part, \(-3\), on the x-axis and the imaginary part, \(4\), on the y-axis.

This pairing uniquely identifies each point in the 2D plane, making it simple to convert complex numbers to visual form. For example:

• The real part of \(-3 + 4i\) is \(-3\), so we move 3 units to the left on the x-axis.
• The imaginary part is \(4\), so we move 4 units up along the y-axis.

Thus, Cartesian coordinates give us an intuitive picture of where a complex number stands within the realm of all numbers.

Modulus

The modulus of a complex number is its distance from the origin in the complex plane. It's like the hypotenuse of a right triangle formed by the axes and the complex number. Using the Pythagorean theorem, we calculate the modulus \(r\) as \(r = \sqrt{x^2 + y^2}\).

For the complex number \(-3 + 4i\):

• \(x = -3\)
• \(y = 4\)

Substitute these into the modulus formula to get \(r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

The modulus tells us how far \(-3 + 4i\) is from the origin, helping convert it from Cartesian to polar form. It's a crucial step in understanding the magnitude of a complex number.

Argument (Angle)

The argument of a complex number is the angle it makes with the positive x-axis. This angle, \(θ\), is found using the tangent function: \(θ = \arctan\left(\frac{y}{x}\right)\). However, care must be taken to adjust \(θ\) based on the quadrant where the number lies.

For \(-3 + 4i\),

• \(y = 4\)
• \(x = -3\)

This means it's in the second quadrant, where x is negative and y is positive. So, \(θ = \arctan\left(\frac{4}{-3}\right)\). To convert to the correct angle, add \(\pi\) since the Cartesian angles go counterclockwise.

The argument provides the direction of the number from the origin, crucial for its polar representation.