Question

Do the following by calculator. Round to three significant digits, where necessary. Write each complex number in polar form. $$8+4 i$$

Short answer

The polar form of the complex number \( 8 + 4i \) is approximately \( 8.94(\cos(0.464) + i\sin(0.464)) \), with \( r = 8.94 \) and \( \theta \) rounded to three significant digits.

Step-by-step solution

Find the Magnitude

Calculate the magnitude of the complex number using the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number \( a + bi \). For the given complex number \( 8 + 4i \), the magnitude is \( r = \sqrt{8^2 + 4^2} \).

Calculate the Argument

Determine the argument (angle) \( \theta \) of the complex number, which is the angle the line representing the complex number makes with the positive real axis. Use the formula \( \theta = \arctan(\frac{b}{a}) \) for the complex number \( a + bi \). Here, \( \theta = \arctan(\frac{4}{8}) \).

Express in Polar Form

Write the complex number in polar form using the calculated magnitude \( r \) and argument \( \theta \). The polar form is \( r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the magnitude and \( \theta \) is the angle. Round \( r \) and \( \theta \) to three significant digits.

Key concepts

Magnitude of Complex Number

The magnitude of a complex number is essentially the distance from the origin (0,0) to the point in the complex plane that represents the number. It's like the length of the vector pointing to that point.

To find the magnitude, we use the formula \( r = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number \( a + bi \). For instance, if we have the complex number \( 8 + 4i \), its magnitude is calculated as \( r = \sqrt{8^2 + 4^2} = \sqrt{64 + 16} = \sqrt{80} \), which approximately equals 8.944 to three significant digits.

It's important to note that the magnitude is always a non-negative real number, regardless of the values of \( a \) and \( b \).

Argument of Complex Number

The argument of a complex number is the angle it makes with the positive real axis, measured counterclockwise in the complex plane. This angle helps us understand the direction in which the complex number points.

To calculate the argument, we use the arctangent function: \( \theta = \arctan(\frac{b}{a}) \). In our example of the complex number \( 8 + 4i \), the argument would be \( \theta = \arctan(\frac{4}{8}) = \arctan(0.5) \), which yields an angle in radians. When working with angles in trigonometry, it's customary to measure them in degrees, and converting from radians to degrees can be helpful for better comprehension.

Keep in mind that the range of the arctangent function is \(-\pi/2\) to \(\pi/2\), so when the complex number lies in the second or third quadrant of the complex plane, you need to add \(\pi\) to the principal value to find the correct argument.

Polar Coordinates

Polar coordinates provide a different way of representing points in a plane, using a radius and an angle, compared to the standard Cartesian coordinates that use an x (horizontal) and y (vertical) axis.

The radius in polar coordinates corresponds to the magnitude of a complex number, and the angle corresponds to its argument. For any given complex number, you can imagine plotting it on a graph where the x-axis represents the real part, and the y-axis represents the imaginary part. To find the polar coordinates, draw a line from the origin to the point, and measure how long that line is (the radius/magnitude) and what angle it makes with the positive x-axis (the argument).

Using polar coordinates can greatly simplify some mathematical problems, especially those involving multiplication or division of complex numbers or their powers, as it turns these operations into simple manipulations of magnitudes and sums or differences of angles.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number is a concise way to express the number through its magnitude and argument with the help of trigonometric functions.

It is given by the formula \( r(\cos(\theta) + i\sin(\theta)) \), where \( r \) is the magnitude of the complex number and \( \theta \) is its argument. Let's consider the complex number \( 8 + 4i \) again. Its polar form is \( r(\cos(\theta) + i\sin(\theta)) \) where \( r \) is about 8.944 and \( \theta \) is the angle derived through \( \arctan(0.5) \), which is approximately 26.565 degrees.

Real-world Connections

This form is not just an abstract mathematical concept; it has practical applications in fields like electrical engineering, where alternating current is often represented in this trigonometric form.