Question

Do the following by calculator. Round to three significant digits, where necessary. Write each complex number in polar form. $$5+4 i$$

Short answer

The complex number 5 + 4i in polar form is approximately \( 6.403(\cos(0.6745) + i\sin(0.6745)) \) or \( 6.403(\cos(38.66^\circ) + i\sin(38.66^\circ)) \) when rounded to three significant digits.

Step-by-step solution

Find the magnitude (r)

To convert a complex number in the form of a + bi to polar form, we first need to find the magnitude, which is the distance from the origin to the point (a, b) on the complex plane. The magnitude (r) is calculated using the formula: \( \text{r} = \sqrt{a^2 + b^2} \)Using the given complex number 5 + 4i, we calculate as follows: \( \text{r} = \sqrt{5^2 + 4^2} \) which simplifies to: \( \text{r} = \sqrt{25 + 16} \) which simplifies to: \( \text{r} = \sqrt{41} \) which is approximate to: \( \text{r} \approx 6.403 \)

Find the angle θ

We find the angle θ (in radians or degrees) by using the formula: \( \text{θ} = \arctan\left(\frac{b}{a}\right) \) where a is the real part and b is the imaginary part of the complex number. For the complex number 5 + 4i, it would be: \( \text{θ} = \arctan\left(\frac{4}{5}\right) \)Using a calculator, we find the value of θ to be approximately: \( \text{θ} \approx 0.6745 \text{ radians} \) or converted to degrees it becomes: \( \text{θ} \approx 38.66^\circ \)

Write in Polar Form

A complex number in polar form is written as: \( r(\cos(\text{θ}) + i\sin(\text{θ})) \)Now, using the magnitude and angle obtained in the earlier steps, we can write the complex number 5 + 4i as: \( 6.403(\cos(0.6745) + i\sin(0.6745)) \)Or in degrees, \( 6.403(\cos(38.66^\circ) + i\sin(38.66^\circ)) \)This is the polar form, rounded to three significant digits as requested.

Key concepts

Magnitude of Complex Numbers

The magnitude of a complex number is a measure of how far the number is from the origin on the complex plane. This concept is analogous to the length of a vector or the distance of a point from the origin in a two-dimensional space. The complex number is expressed as a combination of a real part (a) and an imaginary part (b), written as a + bi.

To find the magnitude, denoted as r, we apply the Pythagorean theorem to the triangle with sides a and b. The formula is as follows:\( r = \sqrt{a^2 + b^2} \)
The magnitude is always a non-negative number, and it represents the absolute value of the complex number. In the case of the complex number 5 + 4i, the magnitude calculated using the given formula would be approximately 6.403 when rounded to three significant digits. This magnitude aids in visualizing the complex number's position on the complex plane and is essential when converting to polar form.

Angle of Complex Numbers

The angle of a complex number, commonly referred to as the argument, is the measure of the rotation required from the positive real axis to the line joining the origin to the point representing the complex number on the complex plane. In mathematical terms, this angle is denoted as θ (theta).

The angle is calculated using the arctangent function, yielding:
\( \theta = \arctan\left(\frac{b}{a}\right) \)
where a is the real part and b is the imaginary part of the complex number. For our example of the complex number 5 + 4i, we get an angle θ of approximately 0.6745 radians, or 38.66 degrees when converted. Understanding the angle is crucial as it gives a clear geometric representation of the complex number, showing the direction in which it lies from the real axis. This angle is a key element in expressing the complex number in polar coordinates.

Conversion of Complex Numbers to Polar Form

Converting complex numbers to polar form involves using both the magnitude and the angle calculated from the previous steps. The polar form provides a different perspective on complex numbers, focusing on the magnitude and the direction rather than the rectangular coordinates (real and imaginary parts).

To write a complex number in polar form, the expression is given by:\( r(\cos(\theta) + i\sin(\theta)) \)
Here, r stands for the magnitude, while θ is the angle of the complex number. For the example 5 + 4i, we use r = 6.403 and θ ≈ 0.6745 radians to get the polar form as:\( 6.403(\cos(0.6745) + i\sin(0.6745)) \)

Why Polar Form?

Polar form is particularly useful in various fields such as electrical engineering and physics, where it simplifies the multiplication and division of complex numbers, as well as the representation of waveforms and alternating currents. Once students understand how to transition from the standard form to polar form, it allows for a deeper comprehension of complex number operations and their geometric interpretations.